# Thermodynamics

### Chapter 10 Thermal Physics

As its name suggests, thermodynamics is about temperature and heat. It is a subset of our knowledge about macroscopic objects, i.e. objects which are made up of many atoms and molecules.

#### 10.1 Temperature and the Zeroth Law of Thermodynamics

##### The definition of temperature

The typical common-sense definition of temperature: Hotter objects have higher temperature and colder objects have lower temperature.

For this definition to be useful, we have to define hotness. Otherwise the only information contained in the previous statement is that two objects of different temperature are different.

The definition of temperature: When two systems are in thermal equilibrium, they share a common property. This property is called temperature.

Thermal equilibrium: If two systems A and B are in thermal contact for a sufficiently long time, they eventually reaches a state which is called the thermal equilibrium.

An important characteristics of thermal equilibrium is heralded as

The zeroth law of thermodynamics: If two systems A and B is each in thermal equilibrium with a third system C, they (A and B) must be in thermal equilibrium with each other.

The zeroth law of thermodynamics asserts that two systems in thermal equilibrium must have something is common. This something is what we have defined as the temperature.

Summary:

• There is such a thing that is called the thermal equilibrium.
• Two objects in thermal equilibrium have something in common.
• This common property is called the temperature.

#### 10.2 Thermometers and Temperature Scales

We measure temperature by making use of its correlation with other properties of matter such as length and pressure.

thermometer can be any instrument which has an observable that depends on temperature and is properly caliberized.

Temperature scale: It is for the most part a matter of convention, even though some conventions are better than others.

Celsius

Definition: The ice point of water at atmospheric pressure is defined as 0 degrees celsius, the steam point of water at atmospheric pressure is defined as 100 degrees celsius.

Kelvin

Definition: Differs from the celsius only in its definition of zero point. The triple point of water is defined as 273.16 degrees kelvin. On the Celsius scale, the triple point of water is at 0.01 degrees Celsius.

Fahrenheit (make the least sense to me)

Definition: The ice point of water at atmospheric pressure is defined as 32 degrees Fahrenheit, the steam point of water at atmospheric pressure is defined as 212 degrees Fahrenheit.

Relationships between different temperature scales:

TC = T - 273.15

TF = (9/5)TC + 32

DTC = DT

DTF = (9/5)DTC

#### 10.3 Thermal Expansion of Solids and Liquids

Thermal Expansion: The dimension of a system of solid or liquid (how about gas?) changes (usually increases) as temperature increases. For small variations of temperature, the change of dimension is proportional to the change of temperature.

For the length of an object, we have

DL = aL0DT

where

• a is called the average coefficient of linear expansion
• DT = T - T0 is the change in temperature
• DL = L - L0 is the change in length

For the area of a surface,

DA = gA0DT

where g is called the average coefficient of area expansion.

For the volume of an object,

DV = bV0DT

where b is called the average coefficient of volume expansion.

There is nothing mysterious about the change of dimension being proportional to the change in temperature. It is basically a mathematical description of the common sense that a dimension changes only gradually as temperature is varied. The real physics comes in when these linear relationships fail.

It should not be surprising that the change of length is proportional to length itself. Just imagine a rod as being made up of two identical pieces. Each piece is going to expand (or shrink) by an equal amount. The total change of length is therefore twice the change of each individual piece.

Putting it in words

The description is more concise if we write the same equation as

Describing the thermal expansion coefficient a:

The fractional change of length per unit (change of) temperature.

a = (DL/L0)/DT

a should obviously have a dimension of 1/Temperature, which can specifically be 1/ºC.

g = 2a

b = 3a

if the object is made up of a material that is isotropic.

Proof

#### 10.4 Macroscopic Description of an Ideal Gas

The state of a gas is described in terms of three macroscopic variables P, V, and T. These three variables are not independent. They are related by a equation that is called the equation of state.

A very important and simple equation of state is the one for ideal gas:

Definition of ideal gas: Briefly, idea gas is a gas of sufficiently low density. More precisely, an ideal gas is a gas of sufficiently low density such that the interactions between the constituent atoms or molecules can be ignored.

Ideal gas is a good approximation to most gases at room temperature and atmospheric pressure.

Equation of state for an ideal gas:

PV = nRT

where

• n = m/M is the number of moles. m is the mass of the gas. M is the molar mass.
• R = 8.31 J/mol·K is called the universal gas constant.

One mole of any substance is an amount that contains Avogadro's number

NA = 6.02 × 1023 of atoms or molecules. From this definition,

n = N/NA where N is the total number of atoms or molecules contained in the gas. We can rewrite the equation of state for an ideal gas as

PV = NkBT

where

• N is the total number of atoms or molecules contained in the gas.
• kB = R/NA =1.38 × 10-23 J/K is called the Boltzmann's constant.

Putting it in words

Rewrite the equation as

P = kB(N/V)T

• Pressure of an idea gas is proportional to the temperature (in kelvin)
• and it is proportional to the number density of constituent atoms or molecules (number of molecules per unit volume)

Neither of which is surprising. They will become even more transparent as we learn how to understand a gas based on our atomic picture of matter.

P-V, P-T, and V-T Diagrams

Because variables P, V, and T are generally related by an equation of state, the state of a gas is uniquely determined by any two out of these three variables

(P, V)

(P, T)

(V, T)

These pairs of numbers correspond to points on P-V, P-T, or V-T diagrams. A point on these diagrams corresponds to a state. A curve corresponds to a thermodynamic process in the system changes from one state to another.

What is so magic about the Avogadro's number or the definition of mole?

NA = 6.0221367 × 1023

Its relationship with the atomic picture of matter:

NA = 1/[mu (in gram)]

where

mu = m(12C)/12 = 1.6605402 × 10-27 kg = 1.6605402 × 10-24 g

From the atomic picture of matter, we can estimate the molar mass, i.e., the mass of a pure substance that contains Avogadro's number of molecules, as follows

M = NA×(mass of a single molecule)
» NA×mu(in gram)×(number of nucleon contained in the molecule)
» number of nucleon contained in the molecule (grams).

Avogadro's number is defined such that the mass of this number of molecules (i.e. the molar mass), when expressed in gram, is approximately equal to the number of nucleon contained in the molecule.

#### 10.6 Kinetic Theory of Gases

Microscopic understanding of gases

At any finite temperature (Kelvin), atoms and molecules are undergoing random motion.

• Temperature is a measure of the average energy associated with this random motion.
• Pressure is a result of atoms and molecules bombarding the surface of the enclosure.

More specifically,

Average energy per active degree of freedom = (1/2)kBT

Degrees of freedom: the number of independent coordinates required to specify the state of a particle. An atom has 3 external degrees of freedom. A diatomic molecules has 6 degrees of freedoms: 3 translational, 2 rotational, and 1 vibrational.

The translational degrees of freedom are always active. The rotational and vibrational degrees of freedom may or may not be active depending on the temperature and what molecule it is. The inactive degrees of freedom are also called frozen.

Average translational energy per particle
= (1/2)m(average of v2) (by definition)
= (3/2)kBT

Note that it depends only on the temperature.

Root-mean-square speed vrms
= square root of (average of v2) (definition)
= (3kBT/m)1/2 = (3RT/M)1/2

where

• m is the mass of the atom or molecule
• M is the molar mass

(Recall R = NAkB and M = NAm)

See Table 10.2 in the book for some typical values of rms speed at room temperature.

At a fixed temperature, lighter molecules are moving faster (on average) than the heavier ones since vrms µ m-1/2, but they all have the same average kinetic energy, (3/2)kBT, regardless of mass. (This is one more piece of information to make your atomic picture of matter more precise and quantitative.)

Total translational energy:

Etranslation = (3/2)NkBT

*

For a monoatomic gas Etranslation is the total thermal energy U (until very high temperature).

For diatomic or more complex molecules, there are additional thermal energies associated with the rotational and vibrational degrees of freedom. For example,

U = (5/2)NkBT

for a diatomic molecular gas if the rotational degrees of freedom are active while the vibrational degree of freedom is still frozen.

*

Microscopic understanding of the pressure of a gas

P = F/A

F = momentum transfer per unit time

Take the wall of the enclosure to be perpendicular to the x axis, and consider a steady state in which a beam of particles with a single velocity is bouncing off the wall.

momentum transfer per collision

Dpx = 2mvx

Number of collisions per unit time

(1/2)(N/V)Avx

where (1/2) comes from the fact that out of all the particles in the volume Avx , only half of them have vx positive and thus be able to collide with the surface. (The other half is coming off the wall.)

F = (N/V)Amvx2

P = (N/V)mvx2

In reality particles can have different velocities so we need to average them over.

From

v2 = vx2 + vy2 + vz2

and

(average of vx2) = (average of vy2) = (average of vz2) (All directions are equivalent.)

we have

(average of vx2) = (1/3)(average of v2)

Putting everything together gives us

P = (N/V)m(1/3)(average of v2)
= (N/V)(2/3)(average kinetic energy per particle)
= (N/V)kBT

We have just derived the ideal gas law!

How do you like the atomic picture of matter now?

Contents | Assignments | Thermodynamics | Vibrations and Waves