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.

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.

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

A *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.

**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.

**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:

*T*_{C}* = T - 273.15*

*T*_{F}* = (9/5)T*_{C}*
+ 32*

*D**T*_{C}*
= **D**T*

*D**T*_{F}*
= (9/5)**D**T*_{C}

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 = aL_{0}DT

where

- a is called the
*average coefficient of linear expansion* - DT = T - T
_{0}is the change in temperature - DL = L - L
_{0}is the change in length

For the area of a surface,

DA = gA_{0}DT

where g is called the average coefficient of area expansion.

For the volume of an object,

DV = bV_{0}DT

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

DL/L_{0} = aDT

Describing the thermal expansion coefficient a:

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

a = (DL/L_{0})/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

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

*N*_{A} = 6.02 × 10^{23} of atoms
or molecules. From this definition,

*n = N/N*_{A} 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 = Nk*_{B}*T*

where

*N*is the total number of atoms or molecules contained in the gas.*k*_{B}=*R/N*_{A}=1.38 × 10^{-23}J/K is called the*Boltzmann's constant*.

*Putting it in words*

Rewrite the equation as

*P = k*_{B}*(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.

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?

*N*_{A} = 6.0221367 × 10^{23}

Its relationship with the atomic picture of matter:

*N*_{A}* = 1/[m*_{u}*
(in gram)]*

where

*m*_{u}* = *m(^{12}C)/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* =* N*_{A}×(mass of a single
molecule)

» *N*_{A}×*m*_{u}(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.

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)k*_{B}*T*

*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 v*^{2}*) *(by
definition)

*= (3/2)k*_{B}*T*

Note that it depends only on the temperature.

*Root-mean-square speed v*_{rms}

*= square root of (average of v*^{2}*) *(definition)

*= (3k*_{B}*T/m)*^{1/2}*
= (3RT/M)*^{1/2}

where

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

(Recall *R = N*_{A}*k*_{B}
and *M = N*_{A}*m*)

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 *v*_{rms }µ *m*^{-1/2}, but they
all have the same average kinetic energy, *(3/2)k*_{B}*T,
*regardless of mass. (This is one more piece of information to
make your atomic picture of matter
more precise and quantitative.)

Total translational energy:

*E*_{translation}* = (3/2)Nk*_{B}*T*

*

For a monoatomic gas *E*_{translation} 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)Nk*_{B}*T*

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

*D**p*_{x}*
= 2mv*_{x}

Number of collisions per unit time

*(1/2)(N/V)Av*_{x}

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

*F = (N/V)Amv*_{x}^{2}* *

*P = (N/V)mv*_{x}^{2}* *

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

From

*v*^{2}* = v*_{x}^{2}*
+ v*_{y}^{2}* + v*_{z}^{2}

and

*(average of v*_{x}^{2}*)
= (average of v*_{y}^{2}*) =
(average of v*_{z}^{2}*)*
(All directions are equivalent.)

we have

*(average of v*_{x}^{2}*)
= (1/3)(average of v*^{2}*)*

Putting everything together gives us

*P = (N/V)m(1/3)(average of v*^{2}*)*

*= (N/V)(2/3)(average kinetic energy per particle)*

*= (N/V)k*_{B}*T*

We have just derived the ideal gas law!

How do you like the atomic picture of matter now?

Contents | Assignments
| Thermodynamics |
Vibrations and
Waves

Copyright © 1997 by Bo Gao. All rights
reserved.