Simple Nature - Light and Matter

dependence onE 1 , so their derivatives are zero, and we find that

the molecular masses can have no effect on the energy sharing.

Setting the derivative equal to zero, we have

0 =

∂

∂E 1

(

n 1 klnV+n 2 klnV+

3

2

n 1 klnm 1 +

3

2

n 2 klnm 2

+

3

2

n 1 klnE 1 +

3

2

n 2 kln(E−E 1 ) +...

)

=

3

2

k

(

n 1

E 1

−

n 2

E−E 1

)

0 =

n 1

E 1

−

n 2

E−E 1

n 1

E 1

=

n 2

E 2

.

In other words, each gas gets a share of the energy in proportion

to the number of its atoms, and therefore every atom gets, on

average, the same amount of energy, regardless of its mass. The

result for the average energy per atom is exactly the same as for

an unmixed gas,K ̄ = (3/2)k T.

5.4.4 Equipartition
Betsy Salazar of Redwood Cove, California, has 37 pet raccoons,
which is theoretically illegal. She admits that she has trouble telling
them apart, but she tries to give them all plenty of care and affection
(which they reciprocate). There are only so many hours in a day,
so there is a fixed total amount of love. The raccoons share this
love unequally on any given day, buton the average they all get
the same amount. This kind of equal-sharing-on-the-average-out-
of-some-total-amount is more concisely described using the term
equipartition, meaning equal partitioning, or equal sharing. If Betsy
did keep track of how much love she lavished on each animal, using
some numerical scale, we would have 37 numbers to keep track of.
We say that the love is partitioned among 37degrees of freedom.

For a monoatomic ideal gas, the analysis in section 5.2.2, p. 317,

leads to the following simple and useful fact about how kinetic en-

ergy is shared among all the atoms’x,y, andzdegrees of freedom:

Equipartition theorem: restricted form

For a monoatomic ideal gas containingnatoms, each of the 3n

degrees of freedom contains, on the average, a kinetic energy

1

2 kT.

As applications of this fact, we can easily find the amount of heat
needed to raise the temperature of the gas by one unit (its specific
heat), or estimate the typical thermal velocities of the atoms given
the temperature. Equipartition gives us a more rigorous and quan-
titative statement of the idea that temperature is a measure of how
concentrated the heat is, or of how much energy there is per particle.

Section 5.4 Entropy as a microscopic quantity 333