Some Notes on the Proof of Entropy Increase in the Thermalization of Two Blocks

Abstract

The thermalization of two blocks with different initial temperatures in an insulated recipient is an irreversible process, so the entropy of the system will increase during this process. Lima (Eur. J. Phys. 36 (2015) 068001) has given an elegant and concise proof for that which had been proved by Mungan (Eur. J. Phys. 36 (2015) 048004) with a complex method. However, there are still two problems in Lima’s proof: 1. It is assumed that the heat capacities of two blocks are constants, which is not true in most practical cases. 2. An inequality that describes the concavity of the logarithm function was used but it is still relatively uncommon for beginners. In this article, two stricter and simpler proof were given for the problem 1 by making use of 1/T–Q diagram and T-S diagram, respectively. In the Proof by 1/T–Q diagram, the area under the curve of 1/T over the domain [0, Q0] is the value of the entropy change of the cooler block, which is positive; while the area under the curve of 1/T’ over the domain [Q0, 0] is the value of the entropy change of the hotter block, which is negative. It is rather intuitive to compare these two values by using the monotonicity and domains of T and Т’. A similar method is adopted in the proof by T-S diagram. For the problem 2, another proof for the key inequality in Mungan’s paper was given by using elementary geometric method which is really more suitable for physics beginners.