EP204 – part 2: Thermodynamics and Kinetic Theory

Ref: Serway, Physics for Scientists and Engineers (4

th ed.)

- chapters 19, 20, 21 (Temperature, Heat and the First Law of Thermodynamics, the

Kinetic Theory of Gases)

Lectures 1 to 3: Temperature, Zeroth Law of Thermodynamics, Thermometers and

temperature scales, Constant Volume gas thermometer and the Kelvin scale, thermal

expansion of solids and liquids, macroscopic description of an ideal gas.

Thermodynamics - relationships between bulk (“macroscopic”) properties of systems, eg.

volume, temperature, pressure, specific heat capacity.

Kinetic Theory of Gases – explains bulk properties at the atomic/molecular scale

(“microscopic”).

Temperature and the Zeroth Law of Thermodynamics:

Firstly, we should distinguish between the temperature we sense and the temperature that is

objectively measured using some form of thermometer – the former is quite often

misleading. Eg. bare feet on stone or tiled flooring on a cold winter morning; this feels much

colder than walking on a rug or a carpet, but if the stone and rug are in the same room, then

both have the same temperature. The reason they feel different is because stone conducts

heat better than carpet, which leads to heat flowing out of your feet faster on the stone floor.

This rapid heat loss makes you feel cold, but can give the wrong impression as regards the

temperature of the object you are touching.

For the same reason, a windy day at 6°C feels much colder than a calm day at 6°C because

moving air removes our body heat faster (the “wind chill” factor).

Conclude: humans are not reliable thermometers.

To understand temperature, we must define ‘thermal contact’ and ‘thermal equilibrium’.

Thermal contact: consider two objects, at different temperatures, inside an insulated box (i.e.

no heat flow permitted into or out of the box). Energy will be exchanged between the

objects. This flow of energy which occurs because of the temperature difference is HEAT,

i.e. heat is energy in motion. Such objects are in ‘thermal contact’ if heat can be exchanged

between them.

Thermal equilibrium: if objects are in thermal contact, and there is no net heat flow between

them, then the objects are in thermal equilibrium.

Zeroth law: consider objects A and B, not in thermal contact, and a third object C (a

thermometer). To find out if A and B would be in thermal equilibrium if they were in contact

with each other, first bring C in contact with A and wait for the temperature to stabilize –

now record the temperature. Then bring C in contact with B, and again record the

temperature once thermal equilibrium is reached; if the two readings are the same, then A

and B are in thermal equilibrium.

“If objects A and B are separately in thermal equilibrium with object C, the A and B are in

thermal equilibrium with each other if placed in thermal contact.”

This “Zeroth Law” (so-called as its necessity was realized after the “first” law had become

well established) is used to define temperature: two objects in thermal equilibrium are at the

same temperature. Conversely, if objects have different temperatures then they are not in

thermal equilibrium.

Thermometers and Temperature Scales:

Thermometer – makes use of change of some physical property as a function of temperature:

e.g. change in volume of a liquid, change in length of a solid, change in pressure of a gas (at

constant volume), change in resistance of a conductor, etc.

A thermometer must be calibrated, typically using two points:

i) mixture of water and ice in thermal equilibrium at a pressure of one atmosphere –

“ice point of water” – gives 0°C,

ii) mixture of water and steam in thermal equilibrium at a pressure of one

atmosphere (1Atm) – “steam point of water” – gives 100°C.

Example: mercury thermometer – note levels of liquid in the capillary tube at 0°C and

100°C, and then divide the span into 100 equal segments. Each segment equals one degree.

(diagram: two-point calibration of mercury thermometer)

Problems with this approach:

i) we are assuming a linear relationship between L and T – this is not always true

(particularly if we move far below or far above the calibration points),

ii) particular thermometers have a limited range (e.g. mercury freezes at -39°C;

ethanol boils at 79°C, so cannot use an alcohol thermometer to measure the

boiling temperature of water).

Ideally, we wish to have a ‘universal’ thermometer whose readings are independent of the

substance used, and whose response is linear over a wide range of temperatures.

The Constant-volume Gas Thermometer and the Kelvin Temperature Scale:

In a gas thermometer, temperature readings are nearly independent of the gas used.

(diagram: constant-volume gas thermometer)

Suppose T increases => pressure of gas in bulb rises => it pushes mercury below zero mark

on fixed scale. The position of the reservoir is now adjusted until the mercury level is back at

the zero mark => the volume of gas in the bulb is held constant. The height h in the open

column is then a measure of the pressure of the gas. This is the principle of the “constant

volume” gas thermometer. It could be calibrated using the ice point and steam point of

water, as discussed above.

It is found that the thermometer readings are almost independent of the type of gas used,

provided:

i) the gas pressure is low,

ii) T is well above the point where the gas liquefies.

Even with slight differences, however, it is found that in every case when the calibration

curve is extrapolated to the point of zero pressure, the temperature at which this occurs is

always the same, namely –273.15°C.

(diagram: extrapolation of p-T curves for constant-volume gas thermometer)

This significant temperature is used as the basis of the Kelvin Temperature Scale, which sets

–273.15°C as its zero point (0K). For convenience, the size of the degree on the Kelvin scale

is the same as that of the Celsius (“centigrade”) scale. Note that, by convention, the degree

symbol (°) is not used for the Kelvin scale. To convert from one scale to the other, use:

TC = TK – 273.15

e.g. T = 373.15K converts to T=100°C, i.e. the boiling point of water (at a pressure of

1Atm).

Temperatures on the Kelvin scale are also known as ‘absolute’ temperatures.

The existence of this fundamental ‘absolute zero’ temperature permits a new procedure for

calibrating thermometers. Before, we needed two points to find the relationship

L = aT + b

where L is the length of the mercury column (for example), and a and b are the slope and

intercept of the linear relationship between column length and temperature. As there are two

unknowns (a and b) we need two independent calibration points. However, if we use

absolute zero as one of our calibration points, then ‘b’ is zero (zero intercept) and we simply

have that

L = aT

so that we only require one calibration point to determine the slope ‘a’.

(diagram: single-point calibration procedure using the absolute scale)

The single point that was officially adopted in 1954 is the ‘triple point of water’, which is a

unique temperature at which liquid water, water vapour, and ice can coexist in equilibrium.

It occurs when the pressure is about 608Pa (0.6% of atmospheric pressure – 4.58mm Hg,

where 1Atm is 760mm Hg), and on the Celsius scale it corresponds to a temperature of

0.01°C, i.e. 273.16K. Note that it is just slightly above the temperature of the ice point (as

measured at a pressure of 1Atm = 1.01x105Pa). This triple point of water is used as it can be

readily reproduced in laboratories in different locations.

The scale based on 0K and the triple point at 273.16K is called the ‘thermodynamic

temperature scale’. The SI unit of temperature is the ‘kelvin’ and is defined as (1/273.16) of

the temperature of the triple point of water.

The Fahrenheit scale:

An older temperature scale (devised by Fahrenheit in 1714); now obsolete (except in the

USA). Fahrenheit set 100° to be the temperature of the human body, and set 0° to be the

lowest temperature that he could then achieve (corresponding to about -18°C). The ice and

steam points on this scale have values of 32.0°C and 212.0°C. Note that there are therefore

180 ‘degrees’ between these two calibration points, and so the Fahrenheit degree is almost

half the size of the Celsius or Kelvin degree. To convert from Celsius to Fahrenheit, use

TF = (9/5)TC +32

e.g. if TC = 100, then TF = (900/5) + 32 = 212.

The Fahrenheit scale is not used in scientific work and will not be mentioned again in this

course.

Thermal Expansion of Solids and Liquids:

Most solids/liquids expand when heated (important exception – water; see below).

Reason – asymmetric shape of potential energy curve associated with binding between atoms

in solids (similar PE curve is used in liquids, but packing is usually looser).

(diagram: binding energy curve)

If solid has no thermal energy, atoms are stationary and separations = Ro (see diagram). Now

suppose solid is heated so that the potential energy is raised from Eo to E1: atoms now

vibrate between R1 and R2.

But (R2 – Ro) > (Ro – R1) because of asymmetrical shape of curve => atoms spend more time

in region (R2 – Ro) than in region (Ro – R1) => mean separation increases => solid expands.

(Note: if curve were symmetrical, then the mean separation would stay at Ro even if the

amplitude of vibrations increases due to heating – i.e. no expansion).

Provided the expansion is small compared to the initial dimensions, the change in any

dimension is approximately proportional to the first power of the change in T, i.e.

δL ∝ δT

If the object has initial length Lo, then it is also found that

δL ∝ Lo

(this makes sense – the more atomic separations along the length, the more will be the

overall expansion)

thus δL ∝ Lo.δT

or δL = α.Lo.δT note – do not confuse ‘∝’ (proportional to) with ‘α’ (alpha)

or (L – Lo) = α.Lo(T – To)

where L is the final length, T the final temperature, To is the initial temperature, and α is the

proportionality constant (‘coefficient of linear expansion’).

Units of α: [m] = α [m].[°C] => α must have units of °C-1

or K-1

NB – expansion acts on all dimensions of an object - think of it as being similar to a

photographic enlargement. Eg. when a circular washer is heated, both the outer radius of the

washer and the radius of the hole expand – the hole is enlarged.

(diagram: enlargement of washer on heating)

For most substances, α > 0 => expansion as T increases

Solids – typical α ~ 10-5

K-1

Relationship between linear and volume expansion coefficients:

As all of the dimensions of a solid or liquid expand when T is increased (in most cases) it is

evident that the overall volume must expand:

δV = β.Vo.δT

where β is the volume expansion coefficient. There is a simple relationship between α and β

for a solid:

consider object shaped like a solid box, with side lengths l,w,h such that the initial volume is

Vo = l.w.h

If T now increases by δT, the volume changes by δV:

Vo+δV = (l+δl).(w+δw).(h+δh) = (l+αlδT).(w+αwδT).(h+αhδT)

(because δw = α.w.δT etc., from above)

=> Vo+δV = (l.w.h)(1+αδT)3

= Vo(1 + 3αδT + 3(αδT)2 + (αδT)

3 )

Divide across by Vo:

1 + δV/Vo = 1 + 3αδT + 3(αδT)2 + (αδT)

3

We know that α is small, so for relatively small δT we can say that

αδT >> (αδT)2 >> (αδT)

3

so that δV/Vo ~ 3α.δT

But δV = β.Vo.δT

and so β = 3α.

Similarly it can be shown that the change in area of a plate is

δA = 2α.Ao.δT

where the factor ‘2’ is the number of dimensions involved (2 for area; 3 for volume).

Thermal switch – Bimetallic strip:

An application of thermal expansion – exploits differences in values of α to make a thermal

switch in electrical circuits:

e.g. α(steel) = 1.1x10-5

K-1

and α(brass) = 1.9x10-5

K-1

=> brass expands more than steel for

the same value of δT. Suppose a strip of brass and a strip of steel are bonded together – the

strip will curve as it is heated because of the different expansion rates. This curvature is used

in thermostats to control switching on/off of heating supply (e.g. in hot-water boilers).

(diagram: bimetallic strip and mode of use as a circuit breaker)

Example: expansion of railroad track – suppose steel railroad track has length = 30m at T =

0°C. What is L when T = 40°C?

Coefficient of linear expansion for steel: α = 1.1x10-5

K-1

δL = α.L.δT

= (1.1x10-5

)(30)(40-0) = 1.3x10-2

m => L at 40°C is 30.013m

The track has therefore expanded by more than 1cm, and this must be allowed for during

construction to prevent buckling.

Note: it is not necessary to convert from °C to K in a problem such as this, as we are

working with a temperature difference, which will be the same on either scale. There are

other situations, however, where it is essential to convert to the absolute scale (whenever you

are calculating ratios of temperatures, or when you are working with a single temperature in

isolation, such as in the Stefan-Boltzmann radiation law). Simple rule – if in doubt, always

convert all temperatures to the absolute scale.

Expansion of water as a function of temperature:

Most solids and liquids expand as T is increased; the volume expansion coefficient β for

liquids is typically ten times that of solids, but the range of values is large in both categories

(e.g. β for aluminium is 7x10-5

K-1

; β for ethanol is 1.1x10-3

K-1

). The typical β value for

water is lower – about 2x10-4

K-1

at 20°C – but at low temperatures the volume expansion

coefficient of water becomes negative, i.e. cold water contracts when heated between 0°C

and 4°C (at a pressure of 1Atm). Thus water has a maximum density at 4°C; above that

temperature water behaves in a manner similar to other liquids.

(diagram: density profile of water in vicinity of density maximum at 4°C)

The mechanism responsible for this density anomaly in liquid water is presumably the same

as that responsible for the anomalous phase change in water at 0°C: solid water is less dense

than liquid water – ice floats on water – which is most unusual (very few other substances

exhibit this anomalous behaviour; examples are bismuth, gallium, silicon). In the case of

water, the anomaly is thought to arise due to a balance between optimal packing of the

molecules and optimisation of directional hydrogen bonding which leads to a less dense

structure. The direct consequence of both the 4°C liquid anomaly and the freezing anomaly

is that colder water stays near the surface, and ice forms and stays near the surface. Without

these anomalies, lakes would freeze from the bottom up.

Macroscopic description of an ideal gas:

Consider gas, mass m, contained in volume V at pressure P and temperature T – how are

these quantities related? The equation relating such quantities is called an ‘equation of state’

and is usually rather complicated. However, for gases at very low pressure (or low density),

we find that the equation of state is simple – ‘ideal gas’. Most gases at room T and

atmospheric P behave approx. as ideal gases.

Usual to express amount of gas in moles, n:

- 1 mole = 6.02x1023

molecules = NA – Avagadro’s number

- n = m/M, where m=total mass (in grams) and M=’molar mass’ in grams per mole.

Example: oxygen has atomic weight = 16, so molecular O2 has atomic weight = 32, and the

molar mass of molecular oxygen is 32 grams per mole.

Suppose an ideal gas is confined in a cylinder with a piston, such that the volume can be

varied (the amount of gas remains constant):

(diagram: gas in cylinder with piston)

We find that P ∝ 1/V for constant T (Boyle’s law)

and that V ∝ T for constant P (Charles’ law)

We can rewrite these as PV = constant, and V/T = constant (not the same constants)

or P1V1 = P2V2 and V1/T1 = V2/T2 in going from state 1 to state 2.

These can be combined into one formula:

PV/T = constant, or PV = const.T

To see why, consider P1V1/T1 = P2V2/T2

if T1 = T2 we get P1V1 = P2V2 (Boyle’s law retrieved from combined formula);

if P1 = P2 we get V1/T1 = V2/T2 (Charles’ law retrieved from combined formula).

This combined formula is the equation of state for an ideal gas, and is usually written as

PV = nRT Ideal Gas Law

where n=number of moles, T is the absolute temperature (i.e. using the Kelvin scale), and R

is a constant which is the same for all (ideal) gases – the ‘molar gas constant’ or ‘universal

gas constant’

R = 8.31 J.mole-1

.K-1

Molar Gas Constant

Example: calculate the volume occupied by one mole of an ideal gas at standard temperature

and pressure (STP).

STP is specified as 0°C (273.15K) and 1Atm (1.013x105Pa) (1Pa = 1N.m

-2)

NB. when using the ideal gas law you MUST convert T to the absolute scale (otherwise, at

STP you would get that PV=0 which would make no sense)

PV = nRT => V = RT/P, for n=1 mole

thus V = (8.31)(273.15)/(1.013x105) = 0.0224m

3

(check for yourself that the units of nRT/P gives m3)

Recall that 1m3 = 1000 litres, so the volume occupied by 1 mole of

ideal gas at STP is 22.4 litre.

Example: ideal gas occupies a volume of 200cm3 at 20°C and at a pressure of 100Pa – how

many moles of gas are present?

V = 200cm3 = 2x10

-4m

3 (recall that 1m

3 = 10

6cm

3)

T = 20°C = 293.15K

PV = nRT => n = PV/RT = (100)(2x10-4

)/(8.31)(293.15) = 8.2x10-6

mole

Number of molecules present?

1 mole contains NA molecules, so 8.2x10-6

mole contains (8.2x10-6

)(6.02x1023

)

= 4.9x1018

molecules.

The ideal gas law can also be expressed in terms of the total number of molecules, N:

N = nNA, where n = number of moles, and NA is the Avagadro number.

Thus PV = nRT = (N/NA).RT

or PV = NkT, where k = R/NA = 8.31/6.02x1023

= 1.38x10-23

J.K-1

k is ‘Boltzmann’s constant’, and is frequently encountered when discussing systems at the

atomic/molecular level (R, on the other hand, is used when discussing macroscopic systems).

Note that R, NA, and k are related – if you know two of them, you can calculate the third.

P,V,T are called the ‘thermodynamic’ variables of the system. If the equation of state of the

system is known, then one of the variables can always be expressed as a function of the other

two. Different systems may use different variables in the equation of state, e.g. for a wire

under tension at constant pressure, the appropriate variables would be the tension force, the

length of the wire, and the temperature of the wire.