34   Entropy of Internal Energy and the Second Law

If for an amount of material comprising  N = n · NA jparticles at temperature T j, a small amount ∆Q jof energy is added, then the temperature of the material will increase by ∆T . The corresponding entropy change is given by energy is added, then the temperature of the material will increase by ∆T. The corresponding entropy change is given by

where f jstands for the number of degrees of freedom. Again we immediately see from the expression that entropy must transform as k jdoes, regardless by which factor temperatures are transformed.

In an ideal gas, the (point-like) particles have only the three translational degrees of freedom of movement. When heated isochoric, this results in the following formula:

But even with further internal degrees of freedom, the ratio ∆T / T jis the fundamental measure of changes in entropy; all temperature-independent factors cancel out as constants in the first derivative of the logarithmic function. For solids at room temperature we have f j= 6, the atoms are in a lattice structure and have three vibrational degrees of freedom. These vibrational degrees of freedom have to be counted twice, their energy can be split in potential and kinetic energy. So we obtain

All solids at room temperature should have the same molar heat Cv j= 3 · R . This is indeed the case. One speaks of the Dulong-Petit law. And it is clear that molar heat must transform the same as entropy and also the same as k jand R .


If you place a disc of aluminum (10 moles, temperature 350 K) on another disc of the same material (also 10 moles, but temperature 290 K), the warmer plate will always impart energy to the cooler one, until the two discs have the same temperature of about 320 K. The reverse process can never be observed, although conservation of energy would also apply here. The famous time arrow is thus disclosed. It is independent of the concept of entropy, but can be elegantly summarized: total entropy increases for all irreversible processes.

In our example, the entropy of the warmer plate decreases by approximately 10·3·R·1/350 as it cools from 350 K to 349 K. Meanwhile, the entropy
of the cold plate increases to 10·3·R·1/290. Overall, this first step in the convergence of the temperatures gives an entropy increase of about
10·3·R·(1/290 - 1/350) J/K, or about 0.147 J/K.


The second law of thermodynamics can be formulated in various ways. One says that thermal energy only flows spontaneously from higher to lower temperatures. Another formulation is the following: processes occur spontaneously only so that the total entropy remains constant (reversible process) or increases. With the diffusion of a gas, the volume entropy increases and with heat exchange the heat entropy increases. For complicated processes, a partial entropy can also decrease. But the sum of all entropies, that is, the total entropy S jalways increases when the operation is not reversible.