Molecular Motions and Heat Capacities (English)

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3-2: Molecular Motions and Heat Capacities

The quantity

of heat required

to raise the temperature

of a substance

is called

heat capacity;

especially for 1 g

of the substance,

it is called

specific heat capacity

specific heat.

(The adjective

specific means

"per unit mass".)

For example,

since the heat

necessary to warm up

1 g of water by

is 1 cal,

the specific heat

of water is

If such a physical

quantity

in the macroscopic

world as specific heat

can be derived

from the molecular

or atomic degrees of freedom

in the microscopic world,

how wonderful

it would be!

To explain this,

let us study

the law of

equipartition of energy.

[The Law of Equipartition of Energy]

When we learned

about the motion

of molecules on the page, 1-4: The Motion of Molecules,

we estimated

the speed

of the molecules

in a gas.

There,

putting a gas

in a cubic container

as shown

in Fig. (A)

and

assuming the x -component

of the average velocity

of the molecule

to be

v_x ,

we calculated the pressure

applied onto the wall A

of the container.

Consequently, we obtained

a relation

where P,

V and M

are the pressure,

the volume and the

total mass of the gas,

respectively.

Now, consider

1 mole of gas.

Let m be

the mass of a single molecule

and T the absolute

temperature of the gas.

The total mass

of the gas

is M = m N_A,

where N_A

is the Avogadro constant.

Combining the above relation and

famous Boyle-Charles's law

we have

Here

k is

a fundamental constant

called

the Boltzmann constant.

The quantity

on the left-hand side

of the above equation

is the kinetic energy

of a molecule.

This implies

that the mean value

of the kinetic energy

of one molecule is kT /2.

In this discussion,

we consider

that the molecular motion

is only in the

x direction.

In a realistic case,

a molecule has also

the y- and

z-components

of the velocity.

Let the

x- ,

and z-components

of the average molecular velocity

v_x ,

v_y ,

and

v_z

respectively.

Then the

average kinetic energy

of the molecule, ,

is written

This result tells us

that the mean value

of the kinetic energy of

each degree of freedom

in the

y and z

directions

is kT /2.

In other words,

the equal amount of energy

kT /2

is distributed to

each of all the degrees

of freedom of the molecules.

This is called

the law of equipartition of energy.

According to the

theories of

statistical mechanics,

this equipartition law

can be generalized

to be said that

the average kinetic energy of each dynamical degree of freedom is equal to kT /2.

The proof of this theorem

is given

on the following page

3-2-A: Generalization of the Law of Equipartition of Energy .

But you may

skip this page,

if you feel it difficult or tedious

because of mathematical

formulae and complicated

expressions.

[Heat Capacities of Gases]

Consider an ideal

(or perfect) gas.

It means a gas

in which the atomic

or molecular interactions

can be neglected.

In 1 mole

of the gas,

there are

as many molecules

as Avogadro's constant

N_A.

In the case

of a monoatomic molecular gas

a noble gas

(inert gas),

the number of the degrees

of freedom

of one molecule is 3,

which means the directions

of the

x-,

y-,

and

z- axes.

Hence the total number

of degrees of freedom

in 1 mole of the gas

is 3N_A.

An ordinary gas

like hydrogen or oxygen

is in general

diatomic molecular gas.

In this case,

as seen

in the following figure,

there are 5 degrees of freedom;

3 of them are those

of the center-of-mass

motion

in the directions of the

x-,

y-,

and

z- axes,

and the rest 2

are those of

the rotational motion

around the center-of-mass,

i.e. the degrees of freedom of

and .

Therefore the total number

of degrees of freedom

in 1 mole of

a diatomic molecular gas

is 5N_A.

Degrees of freedom of a diatomic molecule

Since there are

2 degrees of freedom

of the rotational motion,

and ,

of the rotational motion,

in addition to

3 degrees of freedom of the

translational

center-of-mass motion,

the total number

of degrees of freedom is 5.

If we consider

that the same amount

of kinetic energy

kT /2

is partitioned

onto these

3N_A

5N_A

degrees of freedom

according to the law

of equipartition of energy,

the total energy E

of 1 mole of the gas

would be

Accordingly, the molar

heat capacities

(heat capacities per mole)

of gases are

These theoretical results

are compared

with the experimental data

in the following Table 1.

We can see

from Table 1 that

the law of equipartition

of energy

is well satisfied.

The molecular structure

of carbon monoxide resembles

to that of oxygen,

so that those molar

heat capacities are

rather similar

to each other.

However the molecular structures

of carbon dioxide,

water and methane

are more complicated

than such a diatomic molecule

as oxygen,

and their heat capacities

appear to differ a little.

[Heat Capacities of Solids]

In a solid,

the inter-molecular interaction

is so strong

that the molecules

cannot freely move.

Therefore we cannot

count the number of degrees of freedom

as done in a gas or liquid.

A Collection of Springs

A solid is considered

to be a collection

of oscillators.

This figure is drawn

on a two-dimensional plane,

but

it should be noted

that a realistic solid

three-dimensional.

It is considered

that each molecule

in a solid

connects with adjacent ones

via springs as shown

in the above figure.

These molecules

are infinitesimally oscillating

around their equilibrium

positions.

Namely,

we can consider

that there are

as many harmonic oscillators

as the number of molecules

in the solid.

One molecule has

three directions of motion,

i.e. the directions of the

x-,

and

z- axes,

so that we can consider

that there are

3N_A

harmonic oscillators

in 1 mole of solid.

As stated

on the page,

3-2-A: Generalization of the Law of Equipartition of Energy,

the constant amount

of energy kT

is partitioned

to every harmonic oscillator.

Hence the total energy

of a solid is

and

the molar heat capacity

of a solid is

This result is

nothing other than

Dulong and Petit's law

discovered experimentally

in 1819.

Some experimental data

of molar heat capacities

of solids

are shown

in Table 2.

[Problems on Heat Capacities]

As discussed above,

the molar heat capacities

appear to be explained

well by using

the law of equipartition

of energy.

Certainly,

there is no problem

when the temperature

of the system

is high,

whereas problems might

appear when the temperature

becomes low.

For example,

as seen in

the following figure,

the molar heat capacity

of zinc

becomes smaller and smaller

and approaches 0

when the temperature

comes close to 0 K.

Every solid shows

a similar property.

This is a serious problem

that can never

be explained

with the classical theory.

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