Specific Heat of the Vacuum (English)

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3-3: The Heat Capacity of a Vacuum

We have discussed

the heat capacities

of matter or gas

on the preceding page.

Let us study

the heat capacity of a vacuum here.

You might feel

it strange that

we will discuss

the heat capacity

of a vacuum which

contains nothing.

Does there exist

a concept of temperature

or energy

in a vacuum

where nothing exists?

Yes there is.

There is an energy

named radiation.

For example,

heat (or mainly infrared

light) is

emitted from

a highly heated electric

heater (stove).

This warms up

human bodies

and others around.

This is called

heat radiation.

(See the above figure).

Heat radiation

is not a heat conduction

through the air.

Even if there

is nothing in between,

heat can be conducted

even through a vacuum.

Namely, radiation

can exist even

in a vacuum;

i.e.,

there exists an energy

in a form of

light or

electromagnetic wave

in a vacuum.

Accordingly,

since even a vacuum

can contain energy,

we can measure

the heat capacity

of a vacuum,

if we can define

the temperature of the vacuum.

How can we consider

the temperature of a vacuum?

[Cavity Radiation]

Suppose that

we put an object

in a heat bath

(or heat reservoir)

of the absolute temperature T.

If the temperature

of the object

is lower than T,

the object absorbs heat

from the heat bath,

but if it

is higher than T,

then the object

emits heat into the bath.

After a while,

the object would be

of the same temperature T

and comes

in a thermal equilibrium.

Looking at this thermal

equilibrium in detail,

the object is always

emitting and absorbing

some amount of heat.

The amounts of the emitting

or absorbing heat

are equal on average

and they are keeping

a balance.

Consider that we

put a vacuum

in the heat bath

instead of an object.

For example,

make a hollow space

(cavity)

inside a block of iron

and keep the temperature

of this iron block to be T.

In a thermal equilibrium,

the outer part

of the iron block

would be a kind of heat bath

and the temperature

of the cavity

(vacuum) can be thought

to be T.

In this situation,

the cavity would be filled

with radiation,

i.e. lights

or electromagnetic waves.

This radiation

is called the

cavity radiation.

(See the following figure.)

In order to know

what frequency

of light exists

in the cavity radiation,

and how strong the intensity

of the radiation is,

we may observe the inside

of the cavity

through a small hole

on the wall.

It should be small enough

so as not to disturb the inside

too much.

Practically,

the inside of a blast furnace

in a steel industry

may be considered

to present a kind

of state of cavity radiation.

Therefore we can observe

an example of cavity radiation

through a small window

on the wall

of the blast furnace.

[Difficulty in the Heat Capacity of a Vacuum]

As studied

in the part of Heat Capacities of Solids

on the preceding page,

a solid is

a collection

of harmonic oscillators.

We obtained

the heat capacity

of this solid,

considering that

the constant amount

of energy kT is

partitioned to these

harmonic oscillators.

The result is well fit

to experiment

as far as the temperature

of the solid

is not so low.

According to Maxwellian

electromagnetism,

electromagnetic wave (light) in a vacuum

is oscillation

of electromagnetic field.

It is understood to be

equivalent to the vibration

of continuous elastic body.

If it is

a one-dimensional vibration,

we can imagine vibration

of bowstring

like a guitar or a harp,

and if it

is two-dimensional,

we can associate it with

a drum.

In the case of

electromagnetic wave (light),

it is three-dimensional,

so that we can associate

it with the oscillation

of the air,

i.e. the sound,

assuming that the air

is a continuous substance.

These oscillations

are formally equivalent

to collections of

vibrations of springs

(harmonic oscillators).

Therefore the way

to obtain the heat capacities

of these systems

is exactly the same as

that to calculate

the heat capacity

of a solid matter.

Difference is

in the number

of the harmonic oscillators,

i.e.

the number of

the degrees of freedom.

In a case of 1 mole of solid,

the number of

the degrees of freedom

is considered

to be 3 times as large as

the number of the molecules,

i.e. 3 times of

Avogadro's constant,

3N_A.

However, the electromagnetic field

in a vacuum

is a continuous

elastic body.

This is an essentially

different point.

The normal oscillations of a string with length L

The waves with wavelengths,

2L,

2L/2,

2L/3,

2L/4,

and so on,

are possible.

Let us consider

an oscillation of a string

with the length L .

Its normal modes

of oscillation

are shown in

the above figure.

Let the wavelength

of the normal oscillation be

and

the frequency .

Hence the allowed wavelength

Let the sound speed

in the string be c.

Because

the corresponding frequency

Taking the frequency

as one-dimensional coordinate axis,

we plot dots

at the positions

of the values

of the normal mode

frequencies on this axis,

then they distribute

uniformly at equal interval

in units of

c/(2L).

Hence, the number of

the normal oscillations

with frequencies between

and is

Because the electromagnetic

field in a vacuum

(or cavity) is

three-dimensional,

we have to extend

the above discussion

to a three-dimensional space.

This is somewhat tedious

and we omit here,

but the result is

where V

is the volume of the cavity

and

c is the light speed.

Since the oscillation

of the electromagnetic field

in a vacuum

is just the normal oscillation

of a perfectly continuous body,

there must be

no upper limit

in its frequency;

namely any high frequency

is possible.

This means that

the number of

the corresponding harmonic

oscillator

is infinite in the cavity.

In other words,

the number of degrees of freedom

in a vacuum is

infinite.

If the same amount

of energy kT

is partitioned to these

infinite number

of degrees of freedom,

the total energy

of a vacuum becomes infinite,

and the heat capacity

of a vacuum would be infinite.

This implies

that a vacuum can absorb

an infinite amount

of energy.

But such a strange situation

never happens in practice.

Therefore there must be

something wrong

in the above discussion.

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