Question

The question was asked long time ago on this site, but not answered properly.

In classical thermodynamics, entropy is defined up to a constant. In statistical thermodynamics, there is no such freedom. What exactly is the reason for either point of view?

In classical thermodynamics, only entropy changes play a role. But why does this argument disappear in statistical mechanics?

Also several formulations of the third law allow for entropy to reach a constant value (non-zero) at lowest temperature.

Addenda
1. The paper by Steane arxiv.org/abs/1510.02311 (he added it below) shows that the discussion is pointless, because absolute entropy values can be determined also in classical thermodynamics. He shows that classical entropy is NOT defined up to a constant. A great read.

2. The entropy implied here is observed entropy.

Answer 1

In classical thermodynamics, entropy is considered up to a constant because only entropy changes are measurable, whereas, in statistical thermodynamics, entropy is precisely calculated by counting microstates and is not subject to any arbitrary constants. The third law of thermodynamics sets a reference point for absolute entropy, challenging the view that classical entropy is defined up to a constant.

Step-by-step explanation:

In classical thermodynamics, entropy is a concept used to describe the number of specific ways in which a system may be arranged, commonly referred to as states. Classical thermodynamics does not concern itself with the value of these states, but rather with the difference in entropy between two states. Accordingly, in classical thermodynamics, entropy can be defined up to an arbitrary constant since only changes in entropy (ΔS) are measurable.

In statistical thermodynamics, entropy is more rigorously defined. Here, it is precisely quantifiable by counting the microstates of the system using Boltzmann's entropy formula, S = k ln W, where k is the Boltzmann constant and W is the number of microstates. This approach grounds entropy in an absolute sense, not up to an arbitrary constant. This results from the firm statistical underpinning which links thermodynamic quantities to the behavior of individual particles and their microstates.

The third law of thermodynamics states that the entropy of a perfect crystal at absolute zero is exactly zero, which serves as a reference point for calculating the absolute entropy of substances at any temperature. This contrasts with the perspective that the entropy of a system in classical thermodynamics can be defined up to a constant, as statistical thermodynamics provides absolute values. Moreover, the paper by Steane challenges the notion that classical entropy is defined up to a constant by showing how absolute entropy values can be determined also in classical thermodynamics.