Printing - Advancing Physics A2 Teacher Edition13.pdf · 1 Advancing Physics Absolute ... It turns out that all gases produce very similar results in that the ... These questions

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Absolute temperatureQuestion 10W: Warm-up Exercise

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This question is about temperature scales and the idea of an absolute zero of temperature.

Below is a picture of a mercury thermometer. It has been placed in a mixture of ice and water, then inboiling water and finally left in the air in the laboratory. The three mercury levels have been markedon the tube:

in ice + water in boiling water in air ruler marked in cm

1. The Celsius scale has 100 degrees between the ice point (0 °C) and the boiling point of water(100 °C). What is the temperature of the air in the room in degrees Celsius?

2. In order to answer question 1 you had to make an important assumption about the way mercuryexpands when it is heated. What was it?

If a constant volume of gas is heated, its pressure increases with temperature. Data for a constantvolume of a particular gas is shown in the table:

Temperature / °C Pressure / kPa

0 93

20 100

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Temperature / °C Pressure / kPa

40 107

60 114

80 120

100 127

120 134

140 141

3. Plot a graph of pressure (y-axis) against temperature (x-axis).

4. What would be the pressure of this gas at a temperature of 27 °C?

5. The graph you have drawn is linear. Use the same data to draw a second graph but on this oneextend the x-axis backwards to find the point at which the trend line intercepts it.

6. What is the temperature (in Celsius) and pressure (in kPa) at this intercept?

It turns out that all gases produce very similar results in that the pressure is linear with temperatureand their trend lines all point back to the same intercept on the temperature-axis. This suggests thatthis particular temperature is very significant, and it is used as the zero of the Kelvin temperaturescale.

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7. What is the temperature in Celsius at absolute zero on the Kelvin scale (0 K)?

8. The Kelvin scale uses a unit of the same size as 1 °C. At what temperature does ice melt on theKelvin scale?

9. At what temperature does water boil (at atmospheric pressure) on the Kelvin scale?

10. Convert a room temperature of 27 °C to a temperature on the Kelvin scale.

Boyle’s lawQuestion 20W: Warm-up Exercise


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Boyle’s Law for a constant mass of gas at constant temperature states that:

pV = constant

where p is the pressure of the gas and V its volume.

1. Sketch a graph of p against V.

2. Sketch a graph of p against 1 / V.

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3. Sketch a graph of pV against V.

4. What is the name given to the relationship between p and V?

If you put your finger over the end of a bicycle pump and push the piston in the pressure of the gasinside the pump increases. A student sets up an experiment to measure the pressure as he halvesthe volume. The initial pressure of gas is 102 kPa.

5. Predict the final pressure assuming that Boyle’s law is obeyed.

The student did the experiment twice. The first time he pushed the piston in very slowly and obtaineda result of 196 kPa. The second time he pushed it in very quickly and obtained a result of 240 kPa.

6. Suggest why the first result deviates from Boyle’s law.

7. Suggest why the second result deviates from Boyle’s law.

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Molecular motionQuestion 30W: Warm-up Exercise


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In kinetic theory the large-scale behaviour of a gas is explained in terms of the motions of themolecules it contains.

1. When the temperature of a gas increases, what happens to the molecules in the gas?

2. If the mean kinetic energy of a gas molecule is E and there are N molecules in the gas, what isthe total internal energy U of the gas (assume there is no potential energy, and that the gas ismonatomic).

3. The mean kinetic energy of a gas molecule is given approximately by kT. Use this and youranswer to question 2 to write an equation for the internal energy of a gas of N molecules andestimate the internal energy of one mole of such a gas at 293 K. (k = 1.38 10–23 J K–1 and NA =

6.02 1023 mol–1).

4. One mole of a gas at room temperature and pressure occupies about 0.024 m3 mol–1. Estimatethe total internal energy of 1 m3 of air at room temperature and pressure.

5. Use your answer to question 4 to estimate the total kinetic energy of air molecules in a schoollaboratory of dimensions 5.0 m 8.0 m 3.0 m.

6. A single air molecule has a mass of about 5 10–26 kg. What is the speed of an air molecule withenergy kT? Take T = 293 K.

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Imagine you are blindfold and you have to walk across a crowded dance floor.

7. Why can’t you walk straight across the dance floor?

8. Estimate how far you would walk before bumping into someone (assume they don’t try to avoidyou!)

9. Let’s say you are walking at 2 m s–1. Use your answer to question 8 to estimate how much time

there will be between collisions. How many collisions per second is this?

10. Molecules move randomly (which is a bit like being blindfold). Air molecules at room temperatureand pressure travel about 10–7 m between collisions. They move at about 400 m s–1 on average.Use these figures to estimate the time between molecular collisions.

11. Use your answer to question 10 to estimate the number of collisions per second for a single airmolecule.

12. Mean molecular speeds are high but gases diffuse very slowly. Explain this in molecular terms.

The ideal gas equationQuestion 40W: Warm-up Exercise


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These questions will help you to use the ideal gas law equations.

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The ideal gas equation is pV = nRT.

1. Write down what each of the terms in the equation stands for.

2. What units must T be measured in? Why?

3. Why is the equation called the ‘ideal’ gas equation rather than just the ‘gas equation’?

The ideal gas equation contains the three gas laws.

4. Show that the ideal gas equation becomes Boyle’s law pV = constant if n and T are constant.

5. Show that the ideal gas equation becomes Charles’s law V / T = constant if n and p are constant.

6. Show that the ideal gas equation becomes the pressure law p / T = constant when n and V areconstant.

A useful form of the ideal gas equation (for a fixed mass of gas) is:

constantT

pV

This means that if a gas changes its state from p1 V1 T1 to p2 V2 T2 then:

1

11

2

22

T

Vp

T

Vp

Use this form of the equation to solve the following problems.

7. An air bubble is released by a diver at a depth where the external pressure and temperature are4.0 105 Pa and 12 °C and rises to the surface where the pressure and temperature are 1.0 105 Pa and 16 °C. If the original volume of the bubble was 1.0 cm3 calculate its volume at thesurface.

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8. At the start of a journey the air pressure inside a car tyre is 2.2 105 Pa and its temperature is 20°C. At the end of the journey the temperature has risen to 38 °C. What is the new pressure?

Assume the volume of the tyre is unchanged.

9. A cylinder of volume 1.0 litres contains an ideal gas at 18 °C and at a pressure of 1.1 105 Pa. Itis rapidly compressed to a volume of 0.25 litres and the pressure rises to 6.6 105 Pa. What is

the final temperature of the gas?

Floating or sinking?Question 10S: Short Answer


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Archimedes’ principleArchimedes was one of the greatest mathematicians of all time. He lived and worked in Syracuse,capital of the Greek colony of Sicily, in the third century BC. His writings in physics, engineering,optics and astronomy influenced later thinkers including Leonardo da Vinci, Galileo and Kepler. Oneof Archimedes’ most famous discoveries is associated with the fact that objects apparently loseweight when supported in a fluid. He showed that this apparent loss of weight equals the weight ofthe fluid which has been displaced. In modern language, this is the ‘buoyancy force’ or ‘upthrust’acting.

Density of air = 1.2 kg m–3

Density of water = 1000 kg m–3

Density of hydrogen = 0.09 kg m–3

Density of helium = 0.18 kg m–3

g = 9.8 N kg–1

QuestionsA mini-submarine is lowered on a cable into water and displaces a volume of water which weighs 1kN.

1. If the submarine weighed 7 kN before being lowered, what is its apparent weight when it is fullyimmersed in water?

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In the 1930s, airships filled with the flammable gas hydrogen were becoming a popular means ofcrossing the Atlantic. However, the airship Hindenberg exploded as it approached its landing tower inNew Jersey and many died. This put a stop to the development of airships for many decades.

Recently helium has been used in place of hydrogen. The volume of gas in an airship is 105 m3.

What load can it lift in addition to the weight of the gas, when filled with:

2. Hydrogen?

3. Helium?

What is the buoyancy force acting on a human body of volume 7.4 10–2 m3 when fully immersed:

4. In air?

5. In water?

Very accurate measurements of mass using a balance must include a buoyancy correction.

6. Would the buoyancy correction be more important for a table-tennis ball or a large ball-bearing ofequal radius? Explain.

An advertising balloon, filled with helium, is tied to the ground so that it hovers above a shopping

centre. The volume of the balloon is 25 m3 and the mass of the balloon itself without the gas is 15 kg.

7. Calculate the tension in the tethering rope on a still day.

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Indian rosewood has a density of 1020 kg m–3 while pine has density of 550 kg m–3.

8. Which type of wood will float on water? Use Archimedes’ principle to explain your answer.

9. What is the buoyant force on a hot air balloon of volume 3000 m3? Is it greater than the weight of

the balloon?

10. A bridge designed to carry a canal across a valley is called an aquaduct. Is the downward forceon an aquaduct greater when a boat passes across it? Explain.

11. 20.0 m of a supertanker are below the surface when it is in seawater of density 1030 kg m–3.Assuming the sides of the tanker are vertical, what depth will be below the surface when it enters

fresh water?

12. Imagine you are served a soft drink, with ice cubes floating in it, full to over-brimming. It is a hotday. Will your glass overflow if the ice melts before you take a drink? Explain.

Using the ideal gas relationshipsQuestion 40S: Short Answer

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These are exercises to allow you to gain confidence in using the gas equation:


p V = n R T. The molar gas constant (R = NA k) is 8.31 J mol–1 K–1.

Try theseA 5 mol sample of nitrogen exerts a pressure of 150 000 Pa at a temperature of 373 K.

1. What is the volume of this sample?

2. The temperature is changed to 273 K and the pressure drops to 100 000 Pa. What is the volumenow?

A sample of gas with a pressure of 100 000 Pa has a volume of 5 litres at a temperature of 7 C. Thepressure now drops to 80 000 Pa and the temperature increases by 40 C.

3. Calculate the new volume.

The atmospheric pressure is about 100 000 Pa and the temperature about 300 K.

4. Estimate the number of moles of air in the room you are in now.

The molar mass of carbon dioxide is 0.045 kg mol–1.

5. Calculate the density of the gas when the temperature is 273 K and the pressure is 120 000 Pa.

The summit of Mount Everest can be at a temperature of – 50 C and the pressure at its summit is

roughly one-third that at sea level. The density of air in your laboratory is about 1.25 kg m–3.

6. Calculate the density of air at the top of the mountain.


Momentum and collisions with a wallQuestion 50S: Short Answer


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The learning questions that follow are designed to revise the relationships between momentum,change of momentum and force. They consider both inelastic and elastic collisions. Beginning with aball colliding with a wall, you will calculate:

the momentum transferred to an end wall in a single collision

the force resulting from a steady series of bangs from one mass

the force resulting from a stream of elastic balls.

Impact of a ball on a wallA ball of mass 2 kg, moving at 12 m s–1, hits a massive wall head-on and stops.

1. How much momentum did the ball have before impact?

2. How much momentum does the ball have when it has stopped, after impact?

3. How much momentum did the ball lose during impact?

4. Assume that Newton’s third law is correct and applies to this case. How much momentum did thewall gain?

Force due to a stream of particles hitting a wallNow suppose the wall is hit by a stream of 2 kg balls, each moving at 12 m s–1. All balls hit the wallhead-on and stop dead. Suppose 1000 such balls hit the wall in the course of 10 s.

5. How much momentum do the 1000 balls lose?

6. How much momentum does the wall gain in that 10 s period?

7. Now calculate the force on the wall. Remember that force time = change of momentum.

What force is it that you have calculated here? The balls arrive one after the other, each making abump on the wall. Yet here you have calculated a force for the whole 10 s period, while 1000 ballsbump into the wall. You have not calculated the very big force that a ball makes for the very short time


it while it is bumping into the wall. You have calculated a ‘smeared out’, average, force.

Force due to a stream of elastic ballsSuppose the wall is hit by a stream of 2 kg balls each moving at 12 m s–1. But in this case each ball

arrives with a speed of 12 m s–1 and bounces straight back with an equal speed of 12 m s–1 in theopposite direction. As previously, 1000 balls arrive at the wall in 10 s.

8. Calculate the change of momentum when one ball arrives at the wall and bounces away.Remember that, if a velocity towards the wall is positive, a velocity away from the wall must benegative.

9. Calculate the change in momentum for all the 1000 balls.

10. Calculate the average force on the wall during that 10 s period.

Kinetic theory by numerical exampleQuestion 60S: Short Answer


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Using the kinetic theory model, you imagine gases being made of tiny molecules making perfectlyelastic collisions with the walls of a container. The learning questions that follow are designed to helpyou develop for yourself a quantitative relationship for an ideal gas, from a numerical example.

Picture a stream of identical elastic balls colliding with a wall. Assume you know their mass andvelocity, and the number arriving each second. With this information, you can calculate themomentum transferred to a wall, and thus the force acting. Here you will go further, calculating thepressure caused by that force. This will be done first for elastic balls, and then for molecules of air.

Pressure on a wallThe force on a wall resulting from a stream of elastic balls hitting and bouncing back from it is 4.8 kN.

1. Calculate the pressure on the wall if the balls hit various places all over a wall 2 m high and 3 mwide. Remember that pressure = force / area.

Pressure on a wall of a boxNow suppose that you have a closed box containing just one elastic ball moving to and fro between

the ends. It is a 2 kg ball moving at 12 m s–1 parallel to the length of the box 4 m long. You are going


to calculate the average force on one end.

In this case, instead of using the number of balls hitting a wall once you must use the number of hitsmade by one ball on its repeated returns to the front end. Calculate how many times the ball hits thefront end in 10 s, by the stages below:

2. How far does the ball travel altogether in 10 s at 12 m s–1?

3. How far does it travel between successive hits of the front wall?

4. Then how many ‘round trips’ does it make front to back to front in 10 s?

5. Using the same method as before, calculate the average force on the front end (3 m × 2 m), andthen the pressure.

Molecules in a boxYou are now ready to calculate the force due to bombardment of molecules and the pressure whichthat causes. Suppose the box is 4 m long by 3 m by 2 m, containing ordinary air at room temperature.

Such a volume contains about 6 1026 molecules of air. At room temperature, air molecules move

with an average speed of 500 m s–1. The average mass of an air molecule is 5 10–26 kg. The realpicture is quite complicated, with molecules moving in slanted directions, rebounding from collisionsin other directions. So imagine that the molecules are arranged in three groups, one moving forwardand back, one moving side to side and the third moving top to bottom. No direction is preferred, so

each group has about the same number of molecules in it. Therefore there are only 2 1026

molecules moving between the front and back of the box.

6. What is the momentum of each molecule, in kg m s–1, as it moves towards the front wall?

7. What is its momentum after rebounding from end the front end?

8. What is the change in momentum when the molecule hits the end wall?


To find how many hits the molecule makes in 10 s, calculate:

9. The total distance that the molecule travels in 10 s.

10. The length of one round trip from front to back to front again.

11. The number of trips in 10 s.

12. Calculate the total change of momentum at the front end for this single molecule in 10 s.

13. What is the total change in momentum for all the molecules travelling front to back, in 10 s?

14. What is the force on the front end, due to those collisions?

15. What is the pressure on the front end?

Kinetic theory algebraicallyQuestion 70S: Short Answer


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Developing a theoryUsing the kinetic theory model, you imagine gases being made of tiny molecules making perfectlyelastic collisions with the walls of a container. The learning questions that follow are designed to helpyou develop for yourself a algebraic relationship for an ideal gas. Beginning with a single mass


moving in a box, you will calculate:

the momentum transferred to an end wall in a single collision

the force resulting from a steady series of bangs from one mass

the force resulting from collisions by all the molecules in a box

the pressure caused by that force.

Molecules in a box:Using algebraSuppose there are N molecules in a box. The box has length a and sides b and c. Each molecule hasmass m and moves with average speed v.

1. What is the momentum of one molecule moving along the length of the box to hit the front end ofthe box?

2. What is the momentum of that molecule when it has just bounced back from the front end? Asbefore, assume the collision is elastic.

3. What is the change of momentum when it hits the front end and bounces back again?

4. How far does the molecule travel altogether in time t with speed v?

5. How far does the molecule travel in one round trip along the length of the box, between one hit onthe front end and the next hit on the front end?

6. How many round trips does it make in time t?

7. What is the total change in momentum at the front end in time t, for one molecule moving to andfro along the length?

Instead of moving in many directions, as they really do, assume now that there are three groups ofmolecules, one moving back and forth along the length, one moving side to side and one moving upand down. This is an artificial assumption, but it will lead to the same result as full statistics of motion


in all directions.

8. So if there are N molecules in a box, write down the number of molecules moving from front toback.

9. Calculate the total change of momentum at the front end in time t.

10. Calculate the force on the front end.

11. Calculate the pressure on the front end, which has area b c.

12. What is the volume of a box with sides a, b and c?

13. Finally, express in symbols the value of pressure volume: p V = ?

Going furtherA further improvement in the model would take account of the fact that molecules have a range ofspeeds, giving

231 vNmpV

where 2v is the mean of the squares of the velocities. Because the density

V

Nm

the equation can also be written as

.231 vp

More about the kinetic theory of gases


Question 80S: Short Answer


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Another view of the theoryThe Advancing Physics A2 student’s book gives one simplified version of the kinetic theory of gases.The questions below lead you through a similar argument, but one open to fewer objections andmaking more realistic assumptions.

Gas molecules hit a wall of molecules

gas moleculemass m

n gasmoleculesper unitvolume

velocitycomponent vxtowards wall

velocitycomponent –vxaway from wall

x

wall ofmoleculesall vibrating

area A

Imagine the gas molecules close to a wall. The wall is itself made of molecules, themselves vibrating.A gas molecule can hit the wall, delivering momentum to the wall. Sooner or later, it will leave thewall, not necessarily with the same velocity as it reached the wall. But, if the gas pressure is constant,and it does not get hotter or colder by being in contact with the wall, the average energy of the gasmolecules must not change as a result of the collisions with the wall. The molecules of the wall mustgive as good as they get.

Numbers of molecules1. Suppose there are n molecules per unit volume in the gas. What is the number within a small

distance x of an area A of the wall?

2. Make an argument that half these molecules will be travelling with some component of velocitytowards the wall, and half travelling with some component of velocity away from the wall.


3. Let the x-component of velocity towards the wall be vx. In time t, from up to what distance x willmolecules travelling towards the wall reach and hit the wall?

4. Show that the number of molecules hitting area A of the wall in time t is ½ nAvx t.

Momentum carried by molecules going towards the wall5. Write down the x-component of momentum which a molecule of mass m, and x-component of

velocity vx, gives to the wall if it hits the wall and sticks there.

6. Combine the answers to questions 4 and 5 to write down an expression for the momentumdelivered to the wall by all the molecules with x-component of velocity vx towards the wall, over an

area A in time t.

7. Use the answer to question 6 to write down an expression for the rate of arrival of momentum inthe x-direction towards area A of the wall.

Momentum carried by molecules going away from the wallWhen a molecule leaves the wall, going in the negative x-direction, there is a recoil on the wall in thepositive x-direction (like firing a gun).

8. What recoil momentum does a molecule give to the wall when it leaves the wall with velocity – vx

away from the wall?

9. Explain why the number of molecules leaving the wall per second is the same on average as thenumber hitting the wall per second. Use the answer to question 4 to write down an expression forthe number leaving the wall per second.

10. Combine the answers to questions 8 and 9 to write an expression for the rate of delivery of recoilmomentum to the wall by the molecules leaving the wall.


Force and pressure on the wall11. Use the answers to questions 7 and 10 to write an expression for the total rate of delivery of

momentum in the positive x-direction to area A of the wall.

12. Why is the answer to question 11 also equal to the force in the x-direction on area A of the wall?

13. Using pressure = force / area, write an expression for the pressure of the gas on the wall.

14. If the velocity component vx increases two things happen: the momentum delivered per moleculeincreases, and the number of molecules hitting the wall increases. Use these facts to explain whythe velocity component vx is squared in the expression for the pressure.

Molecules moving with random speedsIn fact, the x-components of the velocities of molecules vary over a range. Thus the expression for the

pressure p = nvx2 must be replaced by

2xvnp

where 2xv is the average of the squares of the components vx. The velocity v of any one molecule

has x-, y- and z-components vx, vy and vz. There is nothing special about the x-direction, and thepressure must act equally in all directions, so the averages of the squares of the x-, y- andz-components of velocity must be equal. Thus:

.222zyx vvv

But by Pythagoras’s theorem the velocity of any particle is given by

.2222zyx vvvv

Since the averages of the three terms are equal, each must be one-third of the average of the squareof the velocity:

.231222 vvvv zyx

15. Use the above to show that the pressure is given by:

.231 vnmp


16. n is the number of molecules per unit volume. Writing n = N / V, where V is the volume occupiedby N molecules, show that:

.231 vNmpV

17. Show also that:2

31 vp

where is the density of the gas.

The speeds of gas molecules:Some questionsQuestion 90S: Short Answer


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This sequence of questions will help you to think about ways in which physicists measure the averagespeed of a set of moving objects. You are also invited to consider the problem of why the hydrogenthat is released into the atmosphere every day does not stay there.

Measures of the ‘average’ speed of gas moleculesThe table below gives the number of gas molecules moving at a particular speed. In practice, ofcourse, the speeds are not confined to eleven values but have many more values from very slow tovery fast. Also, 150 molecules is an unrealistically small total number.

Number of molecules with

this speedSpeed / m s–1

8 100

17 150

22 200

26 250

24 300


Number of molecules with

this speedSpeed / m s–1

22 350

15 400

10 450

4 500

1 550

1 600

1. Draw a histogram to represent these figures.

2. What is the most probable molecule speed?

3. What is the average speed of the molecules?

4. What is the root mean square speed (rms) of the molecules?

Imagine the release of some air freshener in one corner of a room. The gas molecules are travelling

at typical speeds of about 3 ×102 m s–1.

5. You may have seen a demonstration or a video clip of bromine gas diffusing into air and bromineexpanding into a vacuum. The difference in the rate at which the processes happen may haveimpressed you. Suggest why an aerosol of air freshener sprayed into the corner of a room cansoon be smelt in the opposite corner.


6. Estimate the total distance travelled by a molecule that takes 2 minutes to travel from one cornerof a typical room to another.

Why is this distance so large?

Why the atmosphere contains oxygen but little hydrogenIn order to escape from the Earth, any object has to be travelling faster than the escape speed. Thisspeed is determined only by the mass of the Earth ME, the radius of the Earth r E, and the

gravitational constant (G = 6.67 10–11 N m2 kg–2); the equation for the escape speed is:

.2

E

E

r

GMv

7. Find out the mass and radius of the Earth and, for later, the mass and radius of the Moon.

8. Use the formula above to calculate the escape speed from the Earth.

9. You already know that the pressure of a gas is related to its density and the mean square speed.Write down the equation that relates the pressure of a gas to its density and the mean squarespeed.

10. Use the relationship from question 9 and the ideal gas equation to show that

mM

RTv

32

where 2v is the mean square speed, Mm is the mass of one mole, R is the molar gas constantand T is the temperature in kelvin.


11. Calculate the root mean square (rms) speed for the gases hydrogen, helium, nitrogen and oxygenat 300 K. (The molar masses of these gases are 2 g, 4 g, 28 g and 32 g respectively.)

12. How do these rms speeds compare with the escape speed for the Earth?

13. Why does the atmosphere consist mainly of oxygen and nitrogen with only traces of hydrogenand helium?

14. Account for the lack of an atmosphere on the Moon.

Speed of sound and speed of moleculesQuestion 100S: Short Answer


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Calculating and comparing speedsLook at the values of the speed of sound in a gas and the speed of its molecules. You will find thatthey are comparable in size, with the speed of the molecules always a bit greater, and you can thinkabout why this should be true.

Comparing speeds of sound


Here is a table of measured values of the speed of sound in three gases:

Gas Molar mass / g Speed of sound at 273 K / m s–1 Speed of sound at 300 K / m

Helium 4 972.5 1019

Nitrogen 28 337.0 355.5

Carbon dioxide 44 257.4 269.8

1. Which gas has the highest speed of sound, at either temperature?

2. Which gas has the least massive molecules?

3. Which gas has the lowest speed of sound, at either temperature?

4. Which gas has the most massive molecules?

5. At which temperature is the speed of sound the higher?

6. At which temperature are the molecules moving faster?

Comparing speeds of moleculesThe next questions may suggest to you a reason for the pattern you have seen.

7. If there are N molecules in an ideal gas at temperature T, pressure p, volume V then2

31 vNmNkTpV

where the molecules have mass m and mean square speed 2v , and k is the Boltzmann constant.Show that the mean square speed is given by

.32

m

kTv

8. Calculate the mass of a helium atom, given that 4 g (= 0.004 kg) of helium contains N = 6.02 1023 atoms.


9. Calculate the square root of the mean square speed for helium atoms, at 300 K, given that the

Boltzmann constant k = 1.38 10–23 J K–1.

10. The masses of nitrogen molecules and helium atoms are in the ratio 28 / 4. What should be theratio of their mean square speeds at any given temperature?

11. Using the answer to question 9, predict the square root of the mean square speed (the rmsspeed) for nitrogen molecules at 300 K.

12. Repeat question 9 for carbon dioxide molecules.

Comparing speeds of sound and speeds of moleculesA sound wave in a gas consists of a moving wave of compressions and expansions of the gas. Acompressed region must compress the gas next to it for the wave to move forward. The molecules inthe compressed region must move into, or knock others into, the region next to them. The wave can’thave arrived before the molecules do. So the speed of the wave cannot be larger than the speed ofthe molecules; the two speeds may be comparable.

13. Copy the table of speeds of sound and add to it the values of speeds of molecules calculated forhelium, nitrogen and carbon dioxide. How do the two sets of speeds compare?

Effect of temperature14. If the temperature of a gas falls from 300 K to 273 K, by what factor do you expect the root mean

square speed of its molecules to change?

15. Do the speeds of sound shown in the table follow a similar pattern?


Specific thermal capacity:Some questionsQuestion 110S: Short Answer


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What to doThree of these questions ask you to consider areas in which specific thermal capacity is important:one domestic, one transport-based and one industrial. The remaining questions are calculations thatinvolve the use of specific thermal capacity.

The specific thermal capacity of water is 4200 J kg–1 K–1; the specific thermal capacity of air is about

1000 J kg–1 K–1.

Why does thermal capacity matter?1. Some cooks make toffee. Essentially, this is a process of boiling down a sugar solution to

concentrate it and then allowing the liquid to cool until it sets. Small children are usually warnednot to touch the cooling toffee for a very long time – much longer than the cooling for the samevolume of pure water in the same vessel. Why is the cooling period so long?

2. Why is water commonly used in the cooling system of a motor car? Why is the systempressurised?

3. Find out which materials are used as coolants in nuclear reactors. What do these materials havein common?

Calculations4. The Sun delivers about 1 kW of power to a square metre of the Earth when overhead at the


equator. A parabolic mirror of radius 1 m is used to focus this energy onto a container of water.Estimate the time taken by the mirror to raise 1 litre of water to 100 C. Comment on whetheryour answer is likely to be an over- or an underestimate.

5. Estimate how much energy is required to heat the air in your physics laboratory from a chilly 10°C to a more comfortable 20 °C. Comment on the answer.

Thermal transfers in the homeQuestion 130S: Short Answer


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Using physics to inform thinking can reveal why we arrange thermal changes around the home in theways that we do. One key idea is that when a solid is placed in a liquid (or one liquid is mixed withanother), the energy lost by the hotter material is equal to the energy gained by the cooler material. Inthe questions that follow, ignore any thermal losses to the surroundings.

The specific thermal capacity of water is 4200 J kg–1 K–1 and its density is 1 kg per litre.

In the kitchen1. A 2.5 kW electric kettle is used to boil 1 litre of water which starts at 15 °C.

Estimate how long it takes the kettle to reach boiling point.

Pasta should be cooked near 100 C. For this reason a good cook will bring a generous amount ofwater to the boil before adding the pasta.

2. Use your ideas about energy to explain why it is important to use a generous amount of water.


3. Describe an experiment to find an approximate value for the specific thermal capacity ofuncooked pasta. Assume that you have a thermometer, kitchen scales, a measuring cup, as wellas a saucepan and sufficient supplies of water and raw pasta.

Running the bath4. Some people like to run a bath half full of hot water while doing other things, and then return to

the bathroom to add cold water until the temperature is just right. Hot water is supplied at 50 C,and the cold water is at 10 C.

Calculate the mass of cold water that must be added to 100 kg of hot water, to have a bath at 45C.

Under the bonnetWater is used in most car cooling systems to absorb thermal energy produced by the car engine andcarry it to the radiator.

5. Explain in terms of specific thermal capacity, why water is more effective for this purpose thanmost other liquids.

6. Calculate how much energy is absorbed from a hot engine with 10 kg of water around it, if thewater temperature rises from 10 C to 80 C.

Brownian motionQuestion 140S: Short Answer


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These questions give practice in using the relationship average kinetic energy = (3 / 2) kT, and at thesame time help to dispel the myth that Brown actually observed pollen grains. As we shall see heused somewhat smaller particles. To quote from the title of his published work, he used ‘particlescontained in the pollen of plants’.

Criteria for observing Brownian motionFor Brown to observe the erratic jiggling motion of small particles two conditions had to be met:

the particles must be large enough to see with his microscope

to be observable the speed of the particles’ jiggling motion must be at least 1 mm s–1.

Brown’s microscopeThe microscope used by Brown could resolve particles as small as 1.3 microns (= 1.3 10–6 m).Assume the observations are to be taken on a warm summer’s day with a temperature of 27 C.

1. Pollen grains have a typical dimension of 10 microns and so satisfy the first criterion. Show that

by having a mass of 10–12 kg they fail to meet the second criterion.

2. Show that the mass of a particle that just satisfies the second criterion is 1.3 10–14 kg.

3. Assuming the density of pollen material is similar to that of water (103 kg m–3), show that a pollengranule of 1.3 10–14 has a typical size of 2.4 microns thus satisfying the first criterion. For easeof calculation assume that the granules are little cubes.

Gases and massQuestion 20E: Estimate


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Answer the following questionsData:

Density of gases at normal temperature and pressure: air = 1.29 kg m–3, helium = 0.179 kg m–3.

Gravitational field strength = 9.8 N kg–1.


1. A laboratory table is 750 mm tall, 1200 mm wide and 2500 mm long. What mass of air is underthe table?

2. Estimate the mass of air in the room or space where you are answering this question. Show allestimates and calculations.

3. A helium balloon has a volume of 5000 m3. It carries a travel compartment giving it a total mass of6000 kg. Answer the following sequence of questions to find out the net force acting on it:

What mass of helium does it contain?

What mass of air is displaced by the balloon?

Explain why the balloon rises when free to do so.

What is the net force acting on the balloon?

4. It has been claimed that the air inside the framework of the Eiffel Tower in Paris weighs morethan the iron in the framework. Use the following data to find out whether this is true.

Data: height of Eiffel Tower 300 m; length of side of its square base 126 m; mass of iron in frame7000 tonnes.

The shape of the tower is based on a complex curve designed to reduce the mass of iron neededto build it and reduce wind forces acting on it. Treat the tower as a simple pyramid, although thisexaggerates the volume enclosed.

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The wonderful oddity of waterQuestion 120C: Comprehension

Teaching Notes | Key Terms | Answers | Key Skills

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Read the article and then answer the questions based upon it.

WaterWater is a simple compound of hydrogen and oxygen: H2O. It is one of those chemical compoundsthat are so well known that they have a common name – and it has probably the first chemicalformula that anyone ever learns.

Hydrogen is a very active element and makes up about 70% of the Universe. Oxygen is alsochemically active, but much rarer. The properties of water are very different from other substances wemight expect to find on a planet such as the Earth. They are also different from what chemists wouldexpect if water followed the pattern of similar compounds. The differences are so great as to makewater a unique substance. If water wasn’t so special, the Earth would be a very different place andmuch more hostile to life as we know it. Bear in mind too that almost three-quarters of the Earth’ssurface happens to be covered with water. This reading first considers the strange – anomalous –properties of water, and then looks at the implications of this for life on Earth.

Compared with similar compounds:

water has higher melting and boiling points

water needs more energy to heat it through a given temperature range


water needs more energy to make it evaporate

water can dissolve more substances more easily.

280

260

240

220

200

180

160

1 2 3 4 5 6

row in Periodic Table

prediction for water

measured value

melting point of hydrogen compounds H2X

280

260

240

220

200

180

160

1 2 3 4 5


prediction for water

Boiling point of hydrogen compounds H

300

320

340

360

380measured value

The charts show how the properties of some of the other compounds of hydrogen compared withwater. In each case a chemist might expect the properties of water (hydrogen oxide) to follow thepattern set by the other similar compounds: H2S (hydrogen sulphide), H2Se (hydrogen selenide) andH2Te (hydrogen telluride). These elements are in the same column (VI) of the Periodic Table but arein different rows:

Compound Element bound tohydrogen

Row (as shown in thegraphs)

Water H2O Oxygen 2

Hydrogen sulphide H2S Sulphur 3

Hydrogen selenide H2Se Selenium 4

Hydrogen telluride H2Te Tellurium 5

For example, if the pattern were to be followed the melting point of ice ought to be about 170 K, andwater should boil at about 200 K. There would be no glaciers – and no rivers, seas or rain. The Earthwould just be too hot for water to be other than a gas.

What the figures tell us is that for some reason water needs more energy to change its state than it


should. The value of k T needed to make things happen is higher than predicted. This must mean thatwater – and ice – is held together by stronger forces than expected.

45

40

35

30

25

20

15

10

5

0

1 2 3 4 5 6


predictionfor water

measured value for water

The forces that hold things togetherParticles are held together in compounds or molecules by interatomic forces. These forces are dueto electric charges and will involve the positive nucleus and the negative electrons that surround it.There are two main kinds of bond: the ionic bond (which is a repeating pattern of the kind found inordinary salt, sodium chloride) and the covalent bond (such as in gas molecules like chlorine andcarbon dioxide).

In an ionic bond an electron moves from one atom to become part of the shell of the other atom. Theparticles become ions with an opposite electric charge and the electric attraction holds them togetherin a repeating crystal pattern. This bond is common when one kind of atom involved is a metal.

In a covalent bond all the atoms involved tend to be non-metals. In a simple molecule such aschlorine two chlorine atoms share a pair of electrons, so that each nucleus ‘grabs’ the same pair ofelectrons and so the nuclei are held together. Such bonds are fairly weak compared with the ionicbond. The strength of the bonds can be measured by finding out how much energy is needed to tearthe particles away from each other. The typical energy needed to tear apart a mole of ionicallybonding particles in a solid is 2000 kJ, while it takes just 400 kJ to separate a mole of a solid held bycovalent bonding. Water and the other compounds of the form H2X illustrated are held together bycovalent bonds between the atoms.

But when a solid melts or a liquid boils the forces involved are not interatomic forces butintermolecular forces. These forces are quite a lot weaker than the interatomic bonds describedabove. It is much harder to turn water into a mixture of separate oxygen and hydrogen atoms than it isto separate the molecules – which is what happens when water boils or evaporates. Theseintermolecular forces are called van der Waals forces after the Dutch scientist who suggested thatthey must exist in order to explain why gases didn’t behave in an ‘ideal’ manner – in effect, this is whygases turn into liquids when they are cold enough. The van der Waals forces are short-range forcesand come into play when molecules are close together. Remember that when a gas cools the volumegets less and so molecules get closer together. The energy needed to break apart a mole of


molecules and turn a liquid into a gas is the molar latent heat of evaporation. It is typically about 1 or2 kJ for covalent molecules – which ordinary gases are.

So why is water so much harder to melt and boil than the others?

Shape also countsA water molecule is shaped like a triangle.

oxygenhydrogen

hydrogen

104.5°

water molecule

The angle between the arms linking the hydrogen to the oxygen is 104.5, which is just short of ageometrically significant angle: 109.5. What is special about this angle is that it is the angle betweenthe edges of a solid shape called a tetrahedron. Another odd property of the water molecule ties inwith this. Most molecules are naturally shaped so that the positive and negative charges arepositioned so that they cancel each other out almost perfectly, so that almost no electric force is felt inthe surroundings. This isn’t the case with water. The electrons of the two hydrogen atoms are pulledclose to the oxygen atom. The hydrogen atoms are distorted by this effect so that their nuclei arepartly bared on the outside. This means that they can attract oxygen atoms in other molecules, aswell as their own. The angle between the atoms in the molecule helps to make them align themselvesinto sets of tetrahedra, so that even liquid water contains changing groups of what are almost likecrystals. These help to strengthen the substance with a kind of temporary crystal lattice. Theseeffects add to the van der Waals force between water molecules: the evaporation energy per molenow becomes more than 20 times larger at about 40 kJ per mole. The chart shows the familiarpattern once again.

The effect is even stronger when water solidifies into ice. The extra energy needed to melt ice andboil water comes from having to break all these extra hydrogen bonds between separate watermolecules.

Why does ice float?When water molecules gang up into tetrahedra as they form ice they take up more space than theydid in the more freely organised liquid water, where many molecules did not join in the structure. Theeffect starts as water cools below about 4 C, so that the coldest water and then the ice take up morespace and so becomes less dense. This effect is of great ecological significance. The ice floats on thesurface of a pond say, and acts as a thermal insulator, slowing down the thermal flow of energy fromthe cold water underneath to the probably very much colder air above. This slows down the freezingof water – and of course there is still the effect that more energy must be taken from water than‘ought’ to be taken because of its anomalous behaviour. The pattern of the freezing energy is similarto the charts above. Even in some of the coldest parts of the Earth it is rare for deep lakes and ponds


to freeze solid all the way down. Fish and other aquatic organisms can survive harsh winters. Also,imagine what would happen if ice were denser than water. It would form on the bottom of lakes andtend to stay there even through the summer – it is hard for energy to move downwards in a fluid –convection carries hot fluids upwards. So each winter ice would gradually accumulate on the bottomand in cold climates eventually take up the whole lake.

Water as a solventThe odd electric field around a water molecule allows water to force its way into and between clumpsof particles held together by ionic bonds, so salts are easily broken up into ions. This is one of themost common and useful forms of a solution in water. A very large number of useful biologicalchemicals can be broken up and transported through plant and animal systems for this reason. Also,

the slightest impurity in water allows the water molecule itself to break down into ions, namely the H+

and (OH)– ions that again have such a large chemical and biological usefulness.

Water damps the temperature swingsHere is a chart comparing the high specific thermal capacity of water with other common materials.

4500

4000

3500

3000

2500

2000

1500

1000

500

0

Water covers about three-quarters of the Earth’s surface, with an area of 3.6 1014 m2 and anaverage depth of 3900 metres. The mean temperature at the Earth’s surface is 288 K, but this isgradually increasing due to the greenhouse effect. Of course the temperature varies widely over theEarth’s surface and as the seasons change. The water in the oceans plays two major roles instabilising the Earth’s temperature.

The oceans as a thermostatFirstly, the oceans act as a reservoir of energy, making them a kind of thermostat. The Earth isheated by two main sources: solar radiation and radioactivity in rocks. It is cooled by emission oflow-frequency electromagnetic radiation (infrared) from the surface. If solar radiation and radioactiveenergy stay the same, the Earth settles into a general equilibrium in which the supply of energy is


balanced by the emission. In past geological times the emission from the Sun has varied, probably ina cyclic manner with a period of about 35 million years. Also, on a more irregular basis, the quantity ofcarbon dioxide in the atmosphere has changed. Carbon dioxide is a main contributor to thegreenhouse effect, in which the low-frequency infrared radiation from the Earth is trapped in theatmosphere and re-radiated back to the surface. These effects, together with variations in the Earth'sorbit and other effects that are not well understood, have given rise to periods when the Earth’ssurface has been very cold – Ice Ages – and periods when it has been much warmer than average.

These temperature swings would be much larger were it not for the fact that when the trend is tocolder conditions it takes a long time for the ocean water to cool down. If the surface gets colder thanabout 4 C it will have such a low density that it floats on the surface, so stopping convection ofwarmer deeper water which is now denser than the surface water.

Conduction will take place but fairly slowly: one result is that the surface ice on sea is not as cold asice covering the land. Also the quantity of energy stored in the oceans is so vast that unless the wholesea freezes to the bottom it will tend to stay warmer than the land – and so will any surface ice. Buteven in the coldest and longest of Ice Ages the main open oceans have never been covered with ice.This means that, with the help of winds, the sea will tend to keep the land from getting as cold as itmight as quickly as it might.

When the Earth starts to warm up, a great deal of the energy involved will be stored in the oceans.This is due to the simple fact that if land and sea were to be equally warmed (say by 1 K) the waterwould require a great deal more energy to do this than the equivalent mass of rock.

The oceans as a convection heaterThe second main effect ocean water has on the Earth’s climate is that it evens out the surfacetemperature. The tropics get heated by the Sun far more than temperate or polar regions. The tropicalseas also get hotter, especially on the surface. The cold near-polar water will be denser than the hottropical water and will flow underneath it, causing a huge set of convection currents that move warmsurface water towards the poles. Where these currents exist northern land masses are kept warmerthan they would otherwise be. One such current is the Gulf Stream. This carries warm tropical wateracross the North Atlantic and keeps Britain much warmer in winter than other places at equivalentlatitudes, such as Labrador. The North Pacific Current helps to warm Japan (via the KurishioCurrent). One major ecological effect is due to the fact that the cold water contains a lot of nutrients,built up over the years as erosion carries chemicals to the seas which then fall to the deepest coldestregions of the oceans. As they well up in the tropics they provide food for plankton (small plants andanimals, free-swimming or floating organisms which in turn provide food for fish). It is these currentswhich are involved in the El Niño effect. This is a warm current that displaces the cold upwellingcurrent off the Peruvian coast every few years, appearing in midsummer (just before Christmas – ElNiño is the Christ Child). This means that Peruvian fishermen have a very bad season as theplankton and the fish disappear. But the effect is now known to do more harm than simply reducefishermen’s incomes. By a quirk of the sea–atmosphere interaction a strong El Niño effect results indroughts in Australia and Brazil, with more severe winters in North America.

It is very difficult to predict what will happen to the balance of the Earth’s climate as the greenhouseeffect warms up both land and seas. As the El Niño effect shows, what seems a small change in onepart of the world may have unpredictable and unwelcome effects thousands of miles away. Britainmay get cloudier, wetter and cooler, rather than hotter, for example. The Earth is not a simple system.

Questions1. Water has a much higher melting point than it ‘should’. Suggest two consequences of this fact

(i.e. from biology, geology or the environment as it affects humans).


2. Water has a much higher boiling point than it ‘should’. Suggest two consequences of this fact (i.e.from biology, geology or the environment as it affects humans).

3. The hydrogen atoms in a molecule of water tend to be attracted to the oxygen atom inneighbouring molecules as well to their own. A tetrahedron is a double pyramid. It is like twopyramids fitted together at their square bases.

Use the diagram of a water molecule to sketch how water molecules might join together to makea tetrahedral shape.

Describe the ecological significance of this property of water.

4. Why does water have a larger than expected energy requirement to make it evaporate?

Use the data given here and in the reading to answer the following questions.

specific thermal capacity of seawater 4.2 kJ kg–1 K–1, density of seawater 1030 kg m–3.

5. Calculate the total volume of seawater on the Earth’s surface.

6. Calculate the total mass of the oceans.

7. Calculate the energy is required to heat the Earth’s oceans by 1 K.


The Earth stays at a constant temperature because it reflects and radiates energy into space as fast

as it receives it from the Sun. The rate of arrival of energy from the Sun is 1.4 kW m–2. Of that, abouthalf is reflected back into space by clouds, and half reaches the Earth's surface. The 'greenhouseeffect' temporarily reduces the rate of radiation of energy, so that the Earth warms up until it againradiates energy as fast as it receives it. Suppose that due to the greenhouse effect the Earth'ssurface starts to receive energy from the Sun 1 per cent faster than it radiates it.

8. Estimate the rate of additional delivery of energy to the oceans, remembering that they are insunlight only in the daytime.

9. Show that on these crude assumptions the time it would take for the oceans to warm by 1 K isover 100 years.

10. The warming effects of such an input of energy might become apparent long before this,however. Suggest a reason why.

At its narrowest point the Gulf Stream moves through the Florida Straits at a speed of 6.5 km h–1. TheStraits are 200 km wide and the Gulf Stream here has a depth of about 75 m.

11. Show that the Gulf Stream has a flow of 3 107 m3 s–1 as it passes through the Florida Straits.

12. What mass of water flows per second?

13. The temperature of the water in the Gulf Stream as it passes through the Florida Straits is 25 C.It reaches Norway some months later – keeping the port of Narvik free from ice – at atemperature of about 5 C. Suggest how this energy transfer has an effect on the climate andenvironment of the North Atlantic.


Thermal changesQuestion 140D: Data Handling

Teaching Notes | Key Terms | Answers | Key Skills

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Use these data to answer the questions below, showing how thermal changes apply to a wide rangeof phenomena:

Material Specific thermal capacity / J kg–1 K–1

Aluminium 900

Copper 385

Expanded polystyrene 1300

Iron and steel 450

Ice 2100

Air 1000

Water 4200

A set of varied questionsFor each of the following, find the internal energy difference for a 10 K change in temperature:

1. 5.0 kg of water.

2. The bit of a soldering iron, made from 3.5 g of copper.

3. An expanded polystyrene cup of mass 5.0 g.

4. A steel brake disc of mass 1.5 kg.

5. If you eat a fruit pastry fresh from a hot oven, the pastry may be harmless while the fruit fillingscalds your tongue.

Use your ideas about specific thermal capacity to explain why


6. You can put your hand in an oven at 200 C and even touch a baking cake, without serious harm.But you must avoid touching anything in the oven made from metal.

Why is it not so harmful unless you touch metal?

7. In Fiji, some people will walk barefoot over a bed of hot pumice coals as part of a religious ritual.It is meant to demonstrate supernatural powers over pain and heat. Their feet are generally nothurt.

Pumice has a low specific thermal capacity, low density and is a poor conductor of heat. Explainhow each of these properties helps to make a bed of hot pumice coals relatively safe to walk(quickly) over.

8. In the middle of the nineteenth century, James Joule performed a great series of experiments,which led to the law of conservation of energy. One of them was on his honeymoon, when hemeasured the temperature difference between water at the top and bottom of a waterfall.

If the waterfall was 100 m high, what maximum temperature difference could Joule expect?

Two holes are made in a 1.0 kg block of aluminium. A 48 W electric immersion heater is placed inone hole, and a thermometer in the other. Both objects make good thermal contact with the block.The heater is switched on for exactly 3 minutes and the temperature rises from 20 C to 29 C.

9. Calculate the specific thermal capacity of aluminium.

10. Is this likely to overestimate or underestimate the true value?

In some supermarkets the freezer compartments are upright, with front-opening doors, while in othersupermarkets there are chest-type freezers, with access from the top and no lids. Some peopleconsider the upright design wasteful, because the cold air escapes when the door is opened. Thetemperature inside such freezers might be – 20 C, and the specific thermal capacity of air (at

constant pressure) is about 1000 J kg–1 K–1.


11. Consider a freezer of volume 1.5 m3 and discuss whether you agree or disagree.

A power station needs to get rid of energy at a rate of 800 MW and does so by warming up a riverwhich flows past it.

12. If the river flow rate is 1100 m3 s–1, how much warmer is the river downstream of the powerstation?

Pumping up my tyresQuestion 30X: Explanation–Exposition


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These questions ask you to think about packing more molecules into a fixed space, so increasing thepressure.

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Congested moleculesImagine yourself pumping up a tyre. Each pump pushes a cylinder full of air into the tyre.

1. Using your ideas about particles in a gas to explain why you might expect the pressure to changeas you pump more air molecules in.

2. Use what you have learned about gases to predict how the pressure will vary with the number ofpump cylinders of air emptied into the tyre.


3. Use this model to sketch out a graph of pressure / number of pumps.

4. Do you expect the pressure in the tyre to actually follow this pattern? Say where your model mightbreak down.

You might like to try out your ideas on a real bike tyre / pump.

A fresh start5. Estimate how many molecules of gas are contained in the pump cylinder shown (length 500 mm,

internal diameter 30 mm).

6. Estimate the volume of a tyre (wheel circumference 650 mm, tube diameter 40 mm, density of air

is about 1.2 kg m–3).

Just inflated the tyre will be at atmospheric pressure. Fully inflated it might be at 4 atmospheres. (Oneatmosphere is 15 psi – pounds per square inch – which is an old-fashioned unit you still see in some

tyre pressure gauges. It can also be expressed as 105 Pa, the SI unit for pressure.)

7. How many filled pump cylinders are needed to cause this pressure change?

Physicists take to the air


Reading 10T: Text to Read

Teaching Notes | Key Terms

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Air travel began in the winter of 1783 in and around Paris with three flights in manned gas balloons. In1787, or so it is reported, Jacques Alexandre Charles discovered the law of volume expansion ofgases, also in Paris. It would hardly be surprising to find, on closer examination, that these facts – ascientific discovery following close on the heels of an important invention – are connected, but exactlyhow? It would also be reasonable to ask whether there was anything about Paris or the 1780s whichmight help us understand why these events occurred there and then.

An immediate connection between the technology and the science is Charles himself. TheMontgolfier brothers designed the hot-air balloon used for the first ever successful manned flight on21 November 1783. Charles both designed and piloted the hydrogen balloon used for the two flightson 1 December, the second of these being a swift and daring solo ascent to 6000 feet (around 1800m). Both types of balloon could be regarded as applications of the age-old Archimedes’ principle, andCharles’ law does help to explain the functioning of the Montgolfier balloon. However, since Charles’law was not discovered until four years after the first flights, balloons can hardly qualify as an‘application’ of the law. One scientific precondition for the Charles balloon was the discovery ofhydrogen (‘inflammable air’ as it was then called) in 1766. At much the same time, the rubber solutionused by Charles’ craftsmen collaborators, the Robert brothers, to make the envelope gas-tight, hadbeen introduced from colonial South America.

One of the factors which helped to get ballooning off the ground in the early 1780s was the AmericanWar of Independence. The French had allied themselves with the American rebels against the Britishand so re-ignited the world-wide conflict between the two main colonising powers which had beensmouldering for most of the eighteenth century. Battles on land and sea broke out in North America,the Caribbean, Africa and around the coasts of India and Ceylon (Sri Lanka). Land hostilities inEurope were largely confined to a long siege, by a large French and Spanish force, of the seeminglyimpregnable British fortress of Gibraltar. The quest for some means to penetrate the defences ofGibraltar gave a strong stimulus to the attempts to develop balloon travel in the winter of 1782, but bythe time the balloons were ready to fly, peace was being made. It was not long, however, before theymade their military debut as reconnaissance platforms in the Revolutionary Wars, from 1794onwards.

The American War of Independence also brought Benjamin Franklin to Paris as Ambassador of thebrand new American Republic. His public lectures and demonstrations on electricity made asensational impression and created an intellectual fashion for the study of physics. Charles, already33 years old in 1779, was one of the devotees of the new subject – inspired by Franklin he resolvedto turn himself into a scientist, and did so with such success that already by 1781 he was a renownedpublic lecturer. His spectacular demonstrations in which very he played dangerous-looking gameswith natural lightning became the talk of the town. When he turned to ballooning, he secured thebacking of the rich but popular Duke of Chartres, the so-called ‘king of Paris’, cousin and dangerousrival of King Louis XVI.

From the days of Louis XIV, the French royal family had been enthusiastic supporters of scientificresearch. Louis XV personally financed the far-flung expeditions which confirmed Newton’s theorythat the Earth was flattened at the poles. Pilatre de Rozier, the co-pilot of the Montgolfier balloon, wasemployed as a sort of scientist-in-residence by the younger brother of Louis XVI (who eventuallyreigned as Louis XVIII). Balloonists lifted off from the terraces of royal residences: the rooster, thesheep and the duck from Versailles itself, Charles and Robert from the Tuileries in central Paris.


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The Duke of Chartres led the group of horsemen who tracked Charles’ two-hour flight in the gatheringwinter dusk. They congratulated him on touch-down and then stood amazed as Robert disembarkedand Charles soared terrifyingly aloft on the very first solo manned flight. Charles became the firsthuman to see the sun set twice on the same day, but his fingers soon became too cold to write downreadings from his watch, barometer or thermometer. These exploits gained Charles the patronage ofthe King himself; he was put on the royal pay-roll and was enabled to move his physics laboratoryinside the royal palace of the Louvre. At the revolution, this proximity to the King put him in danger –in 1792 the Paris mob stormed the Tuileries and the adjacent Louvre, but Charles survived themassacre because, it is said, of his star-rating as a ballooning hero.

Outside France, Charles is now remembered for the law of volume expansion with temperature. Butwe only know of Charles’ work on gases from the writings of the chemist Joseph-Louis Gay-Lussacwho first published this law along with accurate experimental proof in 1802. According to Gay-Lussac,Charles had shown him his original apparatus, but this set-up – a mercury barometer with wide tubeand small reservoir – could not have ensured constant pressure during the expansion. It is areasonable conjecture (opinion formed with insufficient evidence) that Charles came to take a closeinterest in the physics of gases through his involvement with balloon technology, but what might havedrawn the young chemist Gay-Lussac to this area of work?

The importance of measuring the expansion coefficient of gases can also be explained in terms of itsscientific significance. In the important new art of quantitative chemistry, scientists needed to be ableto compare the volume yields from gaseous reactions to a standard temperature and pressure. It wasalso necessary to the development of accurate gas thermometers. Gay-Lussac himself, however,was far from uninterested in ballooning: soon after publishing the law of volumes, he took to the airhimself to inaugurate the era of scientific ballooning. Together with the young physicist Jean BaptisteBiot, he undertook a series of systematic observations of atmospheric properties and of the Earth’smagnetic field. While the military applications of balloons came and went long ago, meteorologistsand geophysicists to this day make routine use of balloon measurements.

What happened next…Reading 20T: Text to Read


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Chapter 13 of the Advancing Physics A2 student’s book introduces the story of Jacques AlexanderCesar Charles and his flight in a hydrogen-filled balloon. Here the story continues.

The free balloon, with its expensive filling, lifted off and disappeared over the north-east horizon…

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…and comes down in a crowd of terrified peasants in the village of Gonesse. They attack the devilishcontraption with pitchforks and other weapons. The priest is called to perform an exorcism. Thedefeated monster is torn to shreds by being dragged through the village as a horse’s tail. Theauthorities in the Paris region act to calm the panic by publishing a notice for public display, and to beread out by parish priests after sermon. This is probably the first ever attempt to promote ‘publicunderstanding of science’. The administrators themselves, however, are not fully in the picture, sincethey imply that the Montgolfier balloon was hydrogen-filled. The notice has the desired effect, andsubsequent balloon flights are enthusiastically welcomed even in country areas.

From Paris, August 27, 1783

NOTICE TO THE PEOPLE

Concerning the elevation into the air of Balloons or

Globes; the object in question rose in Paris

on the said August 27, 1783, at five o’clock in the

afternoon, from the Champ de Mars.

A Discovery has been made that the Government has deemed proper to bring to the Public Notice, toforestall the Terrors that it might occasion among the People. By calculating the Difference in Weightbetween that Air which is called inflammable and the Air of our Atmosphere, it has been found that aBalloon filled with this inflammable Air should rise of its own accord toward the Sky, and stop onlywhen the two Airs should be in Equilibrium; which can only be at a very great Height. The firstExperiment was made at Annonay in the Vivarais by the brothers MONTGOLFIER, inventors: a Globeof Cloth and Paper five hundred Feet in circumference and filled with inflammable Air rose of its ownAccord to a Height that it has not been possible to reckon. The same Experiment has just beenrepeated in Paris (on August 27 at five o’clock sharp in the Evening) in the presence of an infinitenumber of persons: a Globe of Taffeta coated with elastic Gum, thirty-six feet around, rose from theChamp de Mars into the Clouds, where it was lost to Sight: it was driven by the Wind in anortheasterly Direction, and it is impossible to predict what a Distance it will be borne. Repetitions ofthis Experiment are planned, with much larger Globes. Anyone who shall perceive such Globes in thesky, presenting the Appearance of a darkened Moon, must therefore be advised that, far from being aterrifying Phenomenon, this is nothing but a Machine, made in every case of Taffeta or light Clothcovered with Paper, which can cause no Harm and for which Applications useful to the Needs ofSociety can be anticipated one day.


Direct measurement of speeds of atoms and moleculesReading 30T: Text to Read


Quick Help

Here you can read about an experiment which measures the speeds of molecules in a gas, bydirectly timing their flight over a short distance. The experiment was first done by Zartman and Ko in1930.

The idea of the experimentHere is a sketch of Zartman and Ko’s experiment.


very high vacuum

oven

screens and slits define molecular beam rotating drum

film recordsarrival ofmolecules

bunch of moleculesenters drum

to vacuum pump

Time of flight measurement of molecular speeds

bunch of moleculescrossing drum

fast movingmoleculesarrive here,earlier

slow movingmoleculesarrive here,later

averagemolecules

drum notrotating

slow movingmolecules

fast movingmolecules

Film builds up record of numbersof molecules at different speeds.

Zartman and Ko’s idea was to chop a bunch of molecules out of a travelling beam of molecules, andtime the bunch as it travelled across a short distance. They chopped the bunch by sending the beamto a spinning drum with a slit in it. As the slit crossed the beam, a short bunch of molecules enteredthe drum. Fast-moving molecules crossed the drum quickly; slow-moving molecules took longer. Sofast-moving molecules hit the other side of the drum first. The spinning drum spread out the placeswhere molecules of different speeds arrived. Arriving molecules were detected by being collected ona sensitive film fixed to the far side of the drum.

Making the beamA gas at a known temperature is produced by evaporating molecules inside a small oven. Zartmanand Ko used bismuth vapour, with the oven at 1100 K. A hole in the side of the oven lets moleculesout. Slits in a screen in the way of the moving molecules shape them into a beam. The beam then


travels across to the spinning drum.

Timing with a spinning drumRemember that molecules travel at hundreds of metres a second. So they cross the drum (say 100mm) in less than 1 millisecond. To spread them out the drum must spin at hundreds of revolutions persecond. The results build up gradually. Each bunch of molecules crossing the drum is recorded onthe film fixed to the back of the drum. They are smeared out, with fast-moving molecules reaching thefilm nearly opposite to the slit, and slow-moving ones meeting it further around. The next bunch issmeared out on top of the first, and so on.

Getting the measurementsThe numbers of molecules at different speeds has to be found by taking the film out of the drum,spreading it out and measuring the density of molecules collected on its surface. The more there are,the darker the film. The speeds have to be worked out from the known speed of the drum and thedifferent distances measured along the film.

Why use a vacuum?The inside of the apparatus has to be kept at a very high vacuum. If not, molecules from the ovenwon’t form a beam, but will collide with molecules in the apparatus and go in random directions. It ishard to get a good enough vacuum because the oven is continually sending molecules into thevacuum chamber. So powerful vacuum pumps are needed.

A tricky pointActually, the average speed of molecules in the beam is not the same as the average speed ofmolecules in the oven. Fast-moving molecules in the oven have a better chance of getting out throughthe hole in the oven wall than slow-moving ones do. Kinetic theory calculations show that the averagekinetic energy of particles in the oven is three-quarters of the average kinetic energy of those that getout into the beam. So the distribution of speeds measured in the beam must be corrected to give thedistribution in the oven. Many books conveniently ignore this slightly awkward point.

Revision ChecklistI can show my understanding of effects, ideas and relationships bydescribing and explaining cases involving:the behaviour of ideal gases

A–Z references: ideal gas, ideal gas laws

Summary diagrams: Boyle's law, density and number of molecules, Changing pressure andvolume by changing temperature, One law, summarising empirical laws

the kinetic theory of ideal gases


A–Z references: kinetic theory of gases

Summary diagrams: Constructing a model of a gas

absolute (Kelvin) temperature as proportional to the average energy per particle, with averageenergy kT as a useful approximation

A–Z references: internal energy

Summary diagrams: The kinetic energy of a single particle

energy transfer producing a change of temperature (in gases, liquids and solids)

A–Z references: specific thermal capacity, conservation of energy

Summary diagrams: Transferring energy to molecules

Random walk of molecules in a gas; distance gone in N steps related to N

A–Z references: Brownian motion, random processes

Summary diagrams: Bromine diffusing

I can use the following words and phrases accurately whendescribing effects and observations:absolute temperature

A–Z references: absolute temperature

ideal gas

A–Z references: ideal gas

root mean square speed

A–Z references: root mean square speed

Summary diagrams: The speed of a nitrogen molecule

internal energy

A–Z references: internal energy, thermal properties of materials, specific thermal capacity

I can sketch, plot and interpret:graphs showing relationships between p, V and T for an ideal gas

A–Z references: ideal gas laws

Summary diagrams: Changing pressure and volume by changing temperature, One law,summarising empirical laws

I can make calculations and estimates involving:


the universal gas law equation pV = NkT where N =nN A and Nk = nR ; number of moles n and

Avogadro constant N A

A–Z references: ideal gas laws

Summary diagrams: Boyle's law, density and number of molecules, Changing pressure andvolume by changing temperature, One law, summarising empirical laws

the equation for the kinetic model of a gas: 2

31 cNmpV

A–Z references: kinetic theory of gases; root mean square speed

Summary diagrams: Constructing a model of a gas, Boltzmann constant and gas molecules,The kinetic energy of a single particle, The speed of a nitrogen molecule

temperature and energy change using E = mc

A–Z references: specific thermal capacity

Summary diagrams: Transferring energy to molecules

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