EQUILIBRIUM OF TWO SUBSTANCES IN THERMAL CONTACT

Two identical blocks (A & B) at different initial temperatures ( TA > TB).
m = mA = mB c = cA = cB
Let T'A & T' B = Temperatures of the Blocks after some Energy has flowed between the two blocks.

Conservation of Energy:
Change in Entropy :

Second Law:

Statement 1 of the second law - that heat flows from hot to cold - implies that the final temperature T_f of the two subsystems must be the same. Otherwise heat would continue to flow.

Let us show that this implies statement 2 - that ΔS_sys >= 0.

Now ln(x) > 0 if x > 1. With a little algebra we can show that

If T_A > T_B then ΔS_sys > 0 and the entropy increases as heat flows.

If T_A = T_B then ΔS_sys = 0 and there is no change in entropy.

Next let us use statement 2 - that entropy increases - to show that this does imply statement 1 - that heat will flow from the hot body to the cold body.

2.

Using the conservation of energy equation above, we can eliminate T_B from the entropy expression

If T_A and T_B are fixed, then ΔS_sys is only a function of T_A The value of T_A that will make ΔS_sys a maximum can be found by taking the derivative of ΔS_sys with respect to T_A and setting it equal to zero. First recall that

so that

This last expression is equal to zero if,

Thus the entropy will be maximum when the final temperature of the two systems are equal. Recalling that T_A > T_B we can examine the direction of the heat flow in the two subsystems,

Thus the increasing of the entropy of an isolated system shows that heat does flow from A to B.