Some advances textbooks define entropy by the formula
where the sum runs over all microstates accessible to the system and is the probability of the system being in microstate .
(a) For an isolated system, role="math" localid="1647056883940" for all accessible states . Show that in this case the preceding formula reduces to our familiar definition of entropy.
(b) For a system in thermal equilibrium with a reservoir at temperature , role="math" localid="1647057328146" . Show that in this case as well, the preceding formula agrees with what we already know about entropy.
The entropy of the system is defined as
and the probability of finding the system in a microstate is given by
Substitute the value of from equation (2) into equation (1) and simplify to obtain the entropy of the system.
The probability is given by
Take logarithm from both sides of equation (3).
Substitute the values of the parameters from equation (3) and equation (4) into equation (1) and simplify to obtain the required entropy of the system.
The analysis of this section applies also to liner polyatomic molecules, for which no rotation about the axis of symmetry is possible. An example is , with . Estimate the rotational partition function for a molecule at room temperature. (Note that the arrangement of the atoms is , and the two oxygen atoms are identical.)
In the low-temperature limit , each term in the rotational partition function is much smaller than the one before. Since the first term is independent of , cut off the sum after the second term and compute the average energy and the heat capacity in this approximation. Keep only the largest T-dependent term at each stage of the calculation. Is your result consistent with the third law of thermodynamics? Sketch the behavior of the heat capacity at all temperature, interpolating between the high-temperature and low- temperature expressions.
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