hotter object

A

Unit 5: Physics from creation to collapse

flow of thermal energy .. colder object

B

All matter consists of particles, often in the form of molecules. The th ree common states of matter are solid, liquid, and gas. Whichever state the matter is in, Its molecules are in random motion. In solids the motion is vibration; in liquids and gases the motion is from one place to another.

The molecules have kinetic energy because of their motion. They also have potential energy. There are forces of attraction between them, so when they move apart the potential energy between them increases. The sum of the kinetic and potential energies within a sample of matter is known as its internal energy (symbol U ).

Temperature Each of the following four points is helpful in understanding temperature.

1

2

3

4

If two objects A and B are placed together, and energy (often called thermal energy or heat) moves from A to B, then A started out at a higher temperature (hotter) than B. Adding thermal energy to an objed raises its temperature (so long as it doesn't change state). The molecules in a sample of matter move randomly, with a variety of speeds (see diagram). The temperature of the sample is a measure of the average kinetic energy of its molecules. If you could take all the thermal energy out of an objed, its temperature couldn't fall any further. This point leads to the idea that there is an absolute zero of temperature.

Low temperature

The Celsius scale of temperature is in widespread use, both by scientists and non-scientists. This is defined so that water freezes at OC, and boils at 100e.

High temperature

The absolute zero of temperature is at -273 C (strictly -273.15C, but three significant figures is enough usually) . The absolute or Kelvin scale of temperature has absolute zero as 'zero degrees kelvin', or a K. Other Kelvin temperatures are obtained by adding 273 to the Celsius temperature:

Distribution of molecule speeds for two different temperatures

71K = t/C + 273

This means that an interval of one degree is the same on both Kelvin and Celsius scales.

273K

OK

- 273 ( - absolute zero

Celsius and Kelvin temperatures compared

Red Book 4.1 Blue Book, BLD 4.1, STA 3.3

100 0 ( - water boils o O( - ice melts

Section 4 Ti1ermal energy

Specific heat capacity The specific heat capacity of a material is the energy needed to change the temperature of 1 kg of the material by 1 'C (or 1 K).

where t-.E is the energy suppli ed or removed, m is the mass of the sample, c is the specific heat capacity, and 6.8 is the temperature change . The unit of c is J kg- 1 'C -l or J kg-' K-1

To find a value for c for a material, we need to devise an experiment in which we can measure t-.E, m, and 6.8.

Using an electrical heater makes it easy to measure 6.E, or the power input (the rate at which energy is supplied).

To obtain 6.8, we might use a temperature sensor and data logger to observe how the temperature rises with t ime (see graph).

In a school experiment we might get a value for c accurate to within 10%. To get a more accurate value we have to take care to insulate the material.

Otherwise it will lose heat to the surroundings during the experiment. The accuracy would also be affected because some energy goes into the heater and

connections.

In the experiment shown, the heater receives electrical power at SOW. The mass of the metal block is O.96kg. In 240s (6.t) the temperature rises 14'C (6.8).

a Calculate a value for the specific heat capacity for the metal.

b Discuss whether your value is li kely to be too high or too low.

a State the equation you will use; then re-arrange to make the quantity being asked into the subjed of the equation.

t-.E :. c= --6.E= mcM m6.8

Substitute In the numbers, using the relationship t-.E = power x M .

SOW x 240s c=

O.96kg x 14K

Now use your calculator to work out the answ er, and include the unit.

c = 893 J kg-1 K-1

b Energy may have been lost from the metal block during the process. Also, some energy will have been needed to heat up the heater material and connections . Both these losses would cause 6.8 to be too low, which would mean our value for c is too high.

Q1 a Copper melts at a temperature of 1085 'c. What is this temperature in kelvin?

b The supercooled electromagnets at CERN operate at a temperature of 2 K. What is this temperature in degrees Celsius?

Q2 If no energy moves between objects A and B, what can you say about A and B?

temperature probe

heater

metal

block ---t+-"'==?' insulating material

I 'M I

__________ 1

M

Experiment to measure the specific heat capacity of a meta!

Make the quantity you want (in this case specific heat capacity c) the subject ofthe equation before you put in the numbers. Also, it's a good habit to putthe units in the calculation with the numbers, to remind you in case there's any unit converting to do.

a The specific heat capacity of water (4200J kg-1 K-1) is much

higher than for almost any other liquid. Why does this make water a good liquid to use in central-heating radiators?

b Estimate how long a 3 kW electric kettle should take to boil the water for a cup of tea .

Red Book 4. 1 Blue Book, BLD 4.1, STA 3.3

p

constant pressure

The behaviour of gases can be observed by experiments in a school lab. This observed behaviour agrees with what is predicted by a model called the kinetic theory.

Experimental gas laws The fo llowing quantities can be measured directly in experiments:

p - the pressure within the gas (unit Pa, or N m-' ) V- the volume of space occupied by the gas (unit m3) T - the kelvin temperature of the gas (unit K).

Each experiment is designed to keep one of these three quantities fixed (constant), and to measure how the other two quantities are related . The results of these experiments are:

T constant volume

V constant temperature

1 With P constant, V 0: T 2 With V constant, p 0: T 3 With T constant, p x V

= constant, or V 0: lip, or po: llV

Temperature, volume and pressure relationships for a fixed sample (fixed mass) of gas

The second equation follows from the first. Can you show that? Hint: a fixed mass of gas means N is constant.

You must convert Celsius temperatures to kelvin when using these equations. Thousa nds of candidates every year lose marks by forgetting to do this. Sometimes exa mi ners give a clue because the numbers in Celsius are rather odd, 27 C. But don't rely on th is prompt!

Gas equations These results can be summarised in two equations. First,

where Nis the number of particles in the mass of gas, and k is the Boltzmann constant (k = 1.38 x 10-23 J K- '). Th is equation is sometimes called the ideal gas equation. An idea l gas wou ld be one that perfectly obeys this equation in all circumstances. In reality most gases obey it under most conditions, and in exam questions you can assume that it can be used.

The second equation combines all three experimental laws into one equation:

P2V, p,V,

T2 T, Th is refers to a fixed mass of gas when two or even three of p, Vand Tchange from values 1 (subscript 1) to values 2 (subscript 2).

A closed container of gas is initially at atmospheric pressure (1.00 x 105 Pa) and a temperature of 27 'c. What is the pressure in the container when it is heated to 267 ' C?

The volume is fixed so V, = v,. So the equation becomes h =!l :. p = p, T, T, T , ' T,

Substitute and calculate:

1.00 x 105 Pa x (267 + 273) K p, = (27 + 273)K

1.00 x 10sPa x 540K

300K 1.80 x 105 Pa

Red Book 4.1 Blue Book STA 3.3

Section 4 Thermal energy

Kinetic theory of gases You can use Newton's laws about forces and momentum to model the behaviour of a gas, but you need to start with some assumptions about the gas molecules:

the molecules of the gas are in rapid random motion the molecules make perfectly elastic collisions with each other and with the

container walls the molecules exert no forces on each other or the walls except during collisions the molecules occupy negligible volume compared with the volume of the container the time spent in collis ions is negligible compared with the time between collisions.

These assumptions describe how an ideal gas would behave. From this starting point, the following equation for an ideal gas can be deduced :

~m(c2) = ~kT

where m is the rTlass of one molecule, (is the speed of a molecule, and (2) is the mean (average) value of ( 2 for all the molecules . Thus both sides of this equation represent th e average kinetic energy of the molecu les, and the theory confirms our idea that temperature is a measure of this average kinetic energy. The 'root mean square' (rms) speed is the square root of (2).

a Five gas molecules have speeds of 300ms-" 450ms-' , 520ms-' , 680ms-' and . 730ms-'. Calculate the rms speed for these molecules.

b The molecules of a real gas have both kinetic and potential energy. Explain why an ideal gas has only kinetic energy.

Edexce/ June 2008 Unit Test 2

a Calculate the square of each speed, in (m 5- ')2:

90000, 202500, 270400, 462400, 532900

Find the average of the squared values: 311 640 (m 5-')2 and take the square root of this average value: 558 m s-' .

b One of the five assumptions about an ideal gas is that the molecules exert no forces on each other. Therefore there is no potential energy between them, and all the internal energy is kinetic energy.

Q1 Starting from the ideal gas equation, show that the unit of k is J K-'. Q2 An ideal gas is heated from 300 K to 600 K. By what factor does the

average speed of its molecules change? Q3 The volume of air in a car tyre is 0.02 m3 when the pressure in the tyre is

3.5 atmospheres (3.5 x 105 Pal. What volume would this air occupy when released to atmospheric pressure?

You do not need to learn these assumptions forthe exam. However you will not be able to discuss the kinetic theory unless you understa nd at least the first two points.

In some ways the behaviour of snooker ba lls on a table ~resembles that of gas molecules. How well does each of the five assumptions about gas molecules apply to snooker balls? (very well/pretty well/not well at alii

Red Book 4.1 Blue Book STA 3.3

Unit 5: Physics from creation to collapse

Section 4: Thermal energy checklist By the end of th is section you should be able to:

Revision spread Checkpoints Practice exam questions

Internal energy Explain the concept of internal energy as the random distribution of potential and kinetic energy among molecules.

110

Explain the concept of absolute ,era and how the average kinetic energy of 111

Gas laws and kinetic theory

molecules is related to the absolute temperature.

Recognise and use the expression !J.E ;:;; me tJ.8.

Recognise and use the expression 1 3 ,m(,') = ,kT

109

112

Use the expression pV = NkTas the equation of state for an ideal gas. 113

In July 2003 there was an attempt to fly a manned. spherical balloon to a height of about 40 ki lometres. At this height the atmospheric pressure is on ly one thousandth of its va lue at sea level and the ba lloon would have expanded to a diameter of 210m. The temperature at this height is -60 C The attempt failed because the thin skin of the balloon split while it was being filled with hel ium at sea level. , Make an estimate of the temperature at sea level. and hence obtain the volume of helium the balloon would have contained at sea level if it had been filled successfully. [6J

EdexcelJune 2007 Unit PM5

~ Examiner tip A question like this has a lot of steps. There will be 1 mark or mOfe for each step, so even if you can't immediately see where it's going, you need to start by writing something down - in this case using the hint and the data they've given you, That gives you a better chance of spotting which equation you need to solve it.

Student answer

p,v, = P, V, so V =p, V,T, T, T, I T,p,

Student answer

Pressure is P, at sea level. and P, = 0.001 X P, ~ is a sphere 210m in diameter; so V; = ~.nrJ = ~.7t x (2~O r = 4.8 x 106 m3 Estimate sea level temperature as 20 C. so T, = 120 + 273) K = 293 K T, = 1- 60 +273)K = 213K

~ Examiner tip

Examiner comments

This is the right way 10 start. Show each step in your thinking and calculating. Even jf your final answer is a long way off, the marker can see your steps, and give you some marks for sensible ones.

Examiner comments

This is good. Several of the quantities in the equation are quite complex, so it's a good idea to write each of them out separately before substituting them into the equation.

This question contains two common pitfalls: remembering Celsius to kelvin; and spotting diameter not radiu s. Be on your guard for both of these'

Student answer Examiner comments

V. p,V,T, 0.001xp,x4.8xlO"m3x293K 33

,=---= =6.6x10 m T,p, 213Kxp,

This answer would get full marks.

Section 4 Thermal energy

Practice exam questions

1 A valid set of units for specific heat capacity is A kgJ-1 K-1 B kgJ K-1 C kg-1 J K-1 D kgJ-1 K [1]

2 A large gas holder, of fixed volume, is made up of sheets of metal riveted together. The pressure of gas In the holder is raised from 1 x 105 Pa to 5 x 105 Pa, while the temperature of the gas remains the same. Wh ich of these statements is true? A Each sheet is now struck by fewer molecules. B The mass of gas in the holder is now f ive times as great. C The average speed of the molecules has increased by a factor of five. D The force exerted by the gas on each sheet has not changed . [1]

~ 3 The graphs show the distributions of kinetic energy of the molecules In the "S

~ atmosphere, at sea level and at 40 km above sea level, where it is much colder. "0 Label the sea level graph and give a reason for your answer. [2] ~

Edexcel June 2007 Unit PSA5ii (modified) ru

4 a A student is making calculations about a fuse wire. She assumes that the fuse would initia lly be at a temperature of 20C. Calculate the energy required to raise the temperature of the wire to its melting point of l000~. m (Mass of wire in fuse = 8.70 x 10-5 kg, specific heat capacity = 385J kg-' C-

1.)

.D E ~

z

o~------------~~~--~ Kinetic energy

b Ca lculate the time for the wire in the fuse to reach its melting point. The electrical power going into the fuse is 2.2W [1]

c Discuss whether this is likely to be t he actual time for the fuse to reach its melting point. [2]

Edexcel June 2007 Unit PSA 1 (modified)

5 a What is meant by the absolute zero of temperature? . [1] b The Football Association rules require a football to have a maximum volume of

5.8 x 10-3 m3 and a maximum pressure of 1.1 x 105 Pa above atmospheric pressure (1.0 x lOs Pal . Assuming that the thickness of the material used for the ball is negligible and that the air inside the bal l is at a temperature of 10C, calculate the maximum number of molecules of air inside the football. [4]

ii A football is also required to have a minimum pressure 0.6 x 10sPa above atmospheric pressure. Assuming the volume of the football remains constant, and that It was initially filled to maximum pressure at 10C, calculate the lowest temperature to which the air inside this ball could fall while still meeting the pressure requirements. [3]

Edexcel June 2008 Unit Test 2

6 In a radio programme about space tourism, the presenter says that the Earth's atmosphere stops 100 km above the surface. A student decides to put this claim to the test, initially applying the followi ng equation to gas molecules at this height:

where k is the Boltzmann constant. a State the meanings of the other symbols used in the equation : m, (c 2), T [3] b What physical quantity does each side of the equation represent? [1] c Ca lcu late a valu e for the velocity of an oxygen molecu le at this height, where

the temperature is - 50C. Mass of oxygen molecu le = 5.4 x 10-26 kg. [2] Edexcel June 2008 Unit PSA5ii (modified)