Answer to Question #312484 in Molecular Physics | Thermodynamics for Vikram

Question

Two systems have thermodynamic probabilities of 3.0×10^27 and 1.8×10^28 respectively. Calculate the entropies of the individual systems and as well as their composite system and verify the boltzmann relation.

Answer 1

Answer

We have to find a relation between the entropy and the total number of available microstates describing an isolated thermodynamic system. Because each microstate is equally likely, a macrostate becomes more probable upon being affiliated with a greater number of microstates. For this reason, the total number of microstates, W, is called the thermodynamic probability.

The relation between entropy and thermodynamic probability was discovered by the Austrian physicist Ludwig Boltzmann through a simple thought experiment involving the concept of an irreversible process. Consider the expansion of a gas within a partitioned chamber that is isolated from its environment. Suppose that chamber A originally contains a gas while chamber B is under vacuum. When the valve is opened and the gas expands into the vacuum, the entropy, S, must increase owing to the irreversibility of the process. On the other hand, from a microscopic perspective, the thermodynamic probability must also increase as the final state of the system must be more probable than its initial state. Hence, we can hypothesize that S = f(W).

The functional form involved in the proposed relation can be discerned by considering two independent subsystems (A and B). Because entropy is additive and probability is multiplicative for independent entities, we may assert that

$S_{AB} = S_{A} + S_{B}$

$W_{AB} = W_{A}\times W_{B}$

Only the function can convert a multiplicative operation to an additive operation. Hence, we postulate that the entropy is related to the total number of microstates through the Boltzmann relation,

$S = k\times lnW$ ,

where the constant of proportionality, k, is called Boltzmann’s constant.

This equation can also be regarded as a kind of formulation of the second law of thermodynamics because the second law states that with all processes occurring in nature the entropy, the thermodynamic probability W of the state of all bodies or systems involved in the process, will increase. In other words: every spontaneous process occurring in nature leads to an increase in thermodynamic probability and thus to an increase in the degree of disorder.