Chapter 13 Temperature and Kinetic Theory. 13-1 Atomic Theory of Matter Atomic and molecular masses are measured in unified atomic mass units (u). This

Chapter 13

Temperature and Kinetic Theory

13-1 Atomic Theory of Matter

Atomic and molecular masses are measured in unified atomic mass units (u). This unit is defined so that the carbon-12 atom has a mass of exactly 12.0000 u. Expressed in kilograms:

Brownian motion is the jittery motion of tiny flecks in water; these are the result of collisions with individual water molecules.

13-1 Atomic Theory of Matter

On a microscopic scale, the arrangements of molecules in solids (a), liquids (b), and gases (c) are quite different.

13-2 Temperature and Thermometers

Temperature is a measure of how hot or cold something is.

Most materials expand when heated.

Applications of Thermal Expansion – Bimetallic Strip

• Thermostats– Use a bimetallic strip– Two metals expand differently

13-2 Temperature and Thermometers

Common thermometers used today include the liquid-in-glass type and the bimetallic strip.

Comparing Temperature Scales

Converting Among Temperature Scales

CF

CF

KC

T5

9T

32T5

9T

15.273TT

13-3 Thermal Equilibrium and the Zeroth Law of Thermodynamics

Two objects placed in thermal contact will eventually come to the same temperature. When they do, we say they are in thermal equilibrium.

The zeroth law of thermodynamics says that if two objects are each in equilibrium with a third object, they are also in thermal equilibrium with each other.

13-4 Thermal Expansion

Linear expansion occurs when an object is heated.

Here, α is the coefficient of linear expansion.

ΔL = αLoΔT


Volume expansion is similar, except that it is relevant for liquids and gases as well as solids:

(13-2)

Here, β is the coefficient of volume expansion.

For uniform solids,


12.4 Linear Thermal Expansion

Example 3 The Buckling of a Sidewalk

A concrete sidewalk is constructed between two buildings on a day when the temperature is 25oC. As the temperature rises to 38oC, the slabs expand, but no space is provided for thermal expansion. Determine the distance y in part (b) of the drawing.


m 00047.0C 13m 0.3C101216

TLL o

m 053.0m 00000.3m 00047.3 22 y


THE BIMETALLIC STRIP


THE EXPANSION OF HOLES

Conceptual Example 5 The Expansion of Holes

The figure shows eight square tiles that are arranged to form a square pattern with a hole in the center. If the tiled are heated, what happens to the size of the hole?


A hole in a piece of solid material expands when heated and contracts when cooled, just as if it were filled with the material that surrounds it.

12.5 Volume Thermal Expansion

VOLUME THERMAL EXPANSION

The volume of an object changes when its temperature changes:

TVV o

coefficient of volume expansion

Common Unit for the Coefficient of Volume Expansion: 1C

C

1


Example 8 An Automobile Radiator

A small plastic container, called the coolant reservoir, catchesthe radiator fluid that overflows when an automobile enginebecomes hot. The radiator is made of copper and the coolant has an expansion coefficient of 4.0x10-4 (Co)-1. If the radiator is filled to its 15-quart capacitywhen the engine is cold (6oC),how much overflow will spill into the reservoir when the coolant reaches its operating temperature (92oC)?


quarts 53.0C 86quarts 15C1010.414

coolant V

quarts 066.0C 86quarts 15C105116

radiator V

quarts 0.46quarts 066.0quarts 53.0spill V


Expansion of water.

Solid and Liquid Water

Liquid Solid

Gas Laws

Copyright© by Houghton Mifflin Company. All rights reserved.

24

P1V1 = P2V2

T1 T2

V1 = V2

T1 T2

P1V1 = P2V2

P1 = P2

T1 T2n = mass molar mass

PV = nRTPV = nkT

Ptotal = Pa + Pb + Pc

Boyle’s Law

Gay-Lussac’s Law

Charles’s LawIdeal Gas Law

Dalton’s Law of Partial Pressure

Combined Gas Law

13-7 The Ideal Gas Law

We can now write the ideal gas law:

(13-3)

where n is the number of moles and R is the universal gas constant.

13-8 Problem Solving with the Ideal Gas Law

Useful facts and definitions:

• Standard temperature and pressure (STP)

• Volume of 1 mol of an ideal gas is 22.4 L

• If the amount of gas does not change:

• Always measure T in kelvins

• P must be the absolute pressure

13-9 Ideal Gas Law in Terms of Molecules: Avogadro’s Number

Since the gas constant is universal, the number of molecules in one mole is the same for all gases. That number is called Avogadro’s number:

The number of molecules in a gas is the number of moles times Avogadro’s number:

13-9 Ideal Gas Law in Terms of Molecules: Avogadro’s Number

Therefore we can write:

where k is called Boltzmann’s constant.

(13-4)

14.1 Molecular Mass, the Mole, and Avogadro’s Number

To facilitate comparison of the mass of one atom with another, a mass scale know as the atomic mass scale has been established.

The unit is called the atomic mass unit (symbol u). The reference element is chosen to be the most abundant isotope of carbon, which is called carbon-12.

kg106605.1u 1 27

The atomic mass is given in atomicmass units. For example, a Li atom has a mass of 6.941u.


One mole of a substance contains as manyparticles as there are atoms in 12 grams ofthe isotope cabron-12.

The number of atoms per mole is known asAvogadro’s number, NA.

123 mol10022.6 AN

AN

Nn

number ofmoles

number ofatoms


moleper Massparticle

particle m

Nm

Nmn

A

The mass per mole (in g/mol) of a substancehas the same numerical value as the atomic or molecular mass of the substance (in atomicmass units).

For example Hydrogen has an atomic massof 1.00794 g/mol, while the mass of a single hydrogen atom is 1.00794 u.

molesmoleg

gn

molesmoleg

gn

065.0/99.22

5.1Sodium. of 1.5gin moles ofnumber theFind

33.3/01.9

30Beryllium. of 30gin moles ofnumber theFind


Example 1 The Hope Diamond and the Rosser Reeves Ruby

The Hope diamond (44.5 carats) is almost pure carbon. The RosserReeves ruby (138 carats) is primarily aluminum oxide (Al2O3). Onecarat is equivalent to a mass of 0.200 g. Determine (a) the number ofcarbon atoms in the Hope diamond and (b) the number of Al2O3 molecules in the ruby.


mol 741.0

molg011.12

carat 1g 200.0carats 5.44

moleper Mass

mn(a)

(b)

mol 271.0molg96.101

carat 1g 200.0carats 138

moleper Mass99.15398.262

mn

atoms1046.4mol10022.6mol 741.0 23123 AnNN

atoms1063.1mol10022.6mol 271.0 23123 AnNN

14.2 The Ideal Gas Law

An ideal gas is an idealized model for real gases that have sufficiently low densities.

The condition of low density means that the molecules are so far apart that they do not interact except during collisions, which are effectively elastic.

•Ideal gas behavior breaks down at high pressure and low temperature …why?

TP

At constant volume the pressureis proportional to the temperature.


At constant temperature, the pressure is inversely proportional to the volume.

VP 1

The pressure is also proportionalto the amount of gas.

nP


THE IDEAL GAS LAW

The absolute pressure of an ideal gas is directly proportional to the Kelvintemperature and the number of moles of the gas and is inversely proportional to the volume of the gas.

V

nRTP

nRTPV

KmolJ31.8 R


NkTTN

RNnRTPV

A

AN

Nn

KJ1038.1

mol106.022

KmolJ31.8 23123

AN

Rk

k is called Boltzmann’s Constant

particles ofnumber theis and moles ofnumber theis where

or :Therefore

Nn

NkTPVnRTPV


Example 2 Oxygen in the Lungs

In the lungs, the respiratory membrane separates tiny sacs of air(pressure 1.00x105Pa) from the blood in the capillaries. These sacsare called alveoli. The average radius of the alveoli is 0.125 mm, andthe air inside contains 14% oxygen. Assuming that the air behaves asan ideal gas at 310K, find the number of oxygen molecules in one ofthese sacs.

NkTPV


14

23

33345

109.1

K 310KJ1038.1

m10125.0Pa1000.1

kT

PVN

AtomsOxygen 107.214.0109.1 1314


Consider a sample of an ideal gas that is taken from an initial to a finalstate, with the amount of the gas remaining constant.

nRTPV

i

ii

f

ff

T

VP

T

VP

constant nRT

PV

Combined Gas Law


i

ii

f

ff

T

VP

T

VP

Constant T, constant n:

iiff VPVP Boyle’s law

Constant P, constant n:

i

i

f

f

T

V

T

V

Charles’ law

i

i

f

f

T

P

T

P

Constant V, constant n:

Gas Law Example

If a quantity of gas is at a pressure of 12 atm, a volume of 23 liters, and a temperature of 200 K, and the pressure is raised to 14 atm and the temperature is increased to 300 K, what is the new volumeof the gas?

lKatm

Klatm

TP

TVPV

T

VP

T

VP

if

fiif

i

ii

f

ff

6.29)200)(14(

)300)(23)(12(

Gas Law Example

A student heats a balloon in the oven. If the balloon initially has a volume of 0.4 liters and a temperature of 20 0C, what will the volume of the balloon be after he heats it to a temperature of 250 0C?

lK

Kl

T

TVV

T

V

T

V

T

VP

T

VP

i

fif

i

i

f

f

i

ii

f

ff

71.0)293(

)523)(4.0(

constantP

14.3 Kinetic Theory of Gases

The particles are in constant, randommotion, colliding with each otherand with the walls of the container.

Each collision changes the particle’s velocity.

As a result, the atoms and molecules have different velocities.


THE DISTRIBUTION OF MOLECULAR SPEEDS


V

vmNP

2

3

221

322

31

rmsrms mvNmvNPV

NkT KE

kTmvrms 232

21KE


Example 6 The Speed of Molecules in Air

Air is primarily a mixture of nitrogen N2 molecules (molecular mass 28.0u) and oxygen O2 molecules (molecular mass 32.0u). Assumethat each behaves as an ideal gas and determine the rms speedsof the nitrogen and oxygen molecules when the temperature of the airis 293K.

kTmvrms 232

21

m

kTvrms

3


sm511

kg1065.4

K293KJ1038.13326

23

m

kTvrms

For nitrogen…

kg1065.4g1065.4mol106.022

molg0.28 2623123

m

For O2, vrms=478 m/s


kTmvrms 232

21KE

THE INTERNAL ENERGY OF A MONATOMIC IDEAL GAS

nRTkTNU 23

23

13-10 Kinetic Theory and the Molecular Interpretation of Temperature

Assumptions of kinetic theory:

• large number of molecules, moving in random directions with a variety of speeds

• molecules are far apart, on average

• molecules obey laws of classical mechanics and interact only when colliding

• collisions are perfectly elastic

13-6 The Gas Laws and Absolute Temperature

The relationship between the volume, pressure, temperature, and mass of a gas is called an equation of state.

We will deal here with gases that are not too dense.

Boyle’s Law: the volume of a given amount of gas is inversely proportional to the pressure as long as the temperature is constant.


The volume is linearly proportional to the temperature, as long as the temperature is somewhat above the condensation point and the pressure is constant:

Extrapolating, the volume becomes zero at −273.15°C; this temperature is called absolute zero.


The concept of absolute zero allows us to define a third temperature scale – the absolute, or Kelvin, scale.

This scale starts with 0 K at absolute zero, but otherwise is the same as the Celsius scale.

Therefore, the freezing point of water is 273.15 K, and the boiling point is 373.15 K.

Finally, when the volume is constant, the pressure is directly proportional to the temperature:


We can combine the three relations just derived into a single relation:

What about the amount of gas present? If the temperature and pressure are constant, the volume is proportional to the amount of gas:


A mole (mol) is defined as the number of grams of a substance that is numerically equal to the molecular mass of the substance:

1 mol H2 has a mass of 2 g

1 mol Ne has a mass of 20 g

1 mol CO2 has a mass of 44 g

The number of moles in a certain mass of material:


The force exerted on the wall by the collision of one molecule is

Then the force due to all molecules colliding with that wall is


The averages of the squares of the speeds in all three directions are equal:

So the pressure is:

(13-6)


Rewriting, (13-7)

so

(13-8)

The average translational kinetic energy of the molecules in an ideal gas is directly proportional to the temperature of the gas.


We can invert this to find the average speed of molecules in a gas as a function of temperature:

(13-9)

13-11 Distribution of Molecular Speeds

These two graphs show the distribution of speeds of molecules in a gas, as derived by Maxwell. The most probable speed, vP, is not quite the same as the rms speed.

As expected, the curves shift to the right with temperature.

Summary of Chapter 13

• All matter is made of atoms.

• Atomic and molecular masses are measured in atomic mass units, u.

• Temperature is a measure of how hot or cold something is, and is measured by thermometers.

• There are three temperature scales in use: Celsius, Fahrenheit, and Kelvin.

• When heated, a solid will get longer by a fraction given by the coefficient of linear expansion.

Summary of Chapter 13• The fractional change in volume of gases, liquids, and solids is given by the coefficient of volume expansion.

• Ideal gas law:

• One mole of a substance is the number of grams equal to the atomic or molecular mass.

• Each mole contains Avogadro’s number of atoms or molecules.

• The average kinetic energy of molecules in a gas is proportional to the temperature:

Summary of Chapter 13• Below the critical temperature, a gas can liquefy if the pressure is high enough.

• At the triple point, all three phases are in equilibrium.

• Evaporation occurs when the fastest moving molecules escape from the surface of a liquid.

• Saturated vapor pressure occurs when the two phases are in equilibrium.

• Relative humidity is the ratio of the actual vapor pressure to the saturated vapor pressure.

• Diffusion is the process whereby the concentration of a substance becomes uniform.

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Chapter 13 Temperature and Kinetic Theory. 13-1 Atomic Theory of Matter Atomic and molecular masses are measured in unified atomic mass units (u). This