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Eintand the Molecular Specific

Heats

kTKavg2

3

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kTnNKnNEAavgA

23)()(

int

nRTTN

RnNE

A

A2

3

2

3)(int

AN

Rk

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3

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TnCEV int

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A change in the internal energy Ein tof a confined ideal gas

depends on the change in the gastemperature only; it

does not depend on what type of process produces the

change in thetemperature.

TnCEV int

Three paths representing

three different processes

that take an ideal gas from

an initial state i at

temperature T to some final

state f at T+T temperature.

The change in the internal

energy of the gas is the

same for these three

processes and for any others

that result in the same

change of temperature.

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TnCE V int

TnRTnCTnC pV

RCC Vp

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The figure here shows five paths traversed by a gas on a

p-Vdiagram. Rank the paths according to the change in

internal energy of the gas, greatest first.

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All three types of molecules can have translational

motions and rotational motions.

In addition, we would assume that the diatomic and

polyatomic molecules can have oscillatory motions, with

the atoms oscillating slightly toward and away from one

another, as if attached to opposite ends of a spring.

To keep account of the various ways in which energy

can be stored in a gas, James Clerk Maxwell introduced

the theorem of the equipartition of energy:

Every kind of molecule has a certain number fofdegrees

of freedom, which are independent ways in which themolecule can store energy. Each such degree of freedom

has associated with iton averagean energy of kT

per molecule (or RT per mole).

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As the temperature increases, the value ofCV/R gradually increases

to 2.5, implying that two additional degrees of freedom have

become involved.

Quantum theory shows that these two degrees of freedom are

associated with the rotational motion of the hydrogen molecules

and that this motion requires a certain minimum amount of energy.

At very low temperatures (below 80 K), the molecules do not have

enough energy to rotate. As the temperature increases from 80 K,

first a few molecules and then more and more obtain enough

energy to rotate, and CV/R increases, until all of them are rotating

and CV/R = 2.5.

As the temperature increases beyond 1000 K, more molecules have

enough energy to oscillate and CV/R increases, until all of them are

oscillating and CV/R = 3.5.