# Why is Entropy always increasing?

Consider two systems that are exactly the same except that the first one has a higher temperature $T_1$ and the second one has a lower temperature $T_2$. Now suppose $Q$is the heat flowing from the first system to the second system. The change in entropy in the first system is $-Q/T_1$ and the change in entropy in the second system is $Q/T_2$. Since $T_1$ is greater that $T_2$ the total change in entropy is positive.

Now the question is, how do you exactly relate $Q/T$ to the number of microstates so that the magnitude of change in the first system is always less than the magnitude of change in the second system so that the net change in microstates is always positive?

• The equations you wrote for the entropy changes are valid only if the amount of heat transferred is infinitesimal. Aside from that, would you be willing to accept and explanation of why the entropy is generated (but not consumed) based on the transient heat conduction equation? – Chet Miller Oct 21 '17 at 12:05
• There's a nice discussion in section 2.4 in Baierlein's Thermal Physics book that relates $Q/T$ to the statistical mechanics version of entropy for an ideal gas. – march Oct 22 '17 at 1:34

I would like to start stating Clausius theorem

Theorem(Clausius) For a system exchanging heat with external reservoirs and undergoing a cyclic process, one that ultimately returns a system to its original state,

$$\oint {\frac {\delta Q(s)}{T(s)}}ds\leq 0,$$

where $\delta Q$ is the infinitesimal amount of heat absorbed by the system from the reservoir and $T$ is the temperature of the external reservoir at a particular instant in time.Wikipedia statement

In the special case of a reversible process the equality obviously holds since the order in which the path is followed doesn't matter but the weak inequality and its opposite still need to hold (then the equality). This case is particularly important because is used to introduce the entropy state function.

What's have to be keeped in mind is that Clausius theorem is a mathematical explanation of the second law of thermodynamics!. The theorem is the connection betwee the entropy and the heat exchanged between two reservoirs (one more step is needed actually,, but intuitively can be seen very easily)

Clausius developed this in his efforts to explain entropy and define it quantitatively and as already said, it can be used to determine if a cycle is reversible or not.

Intutive idea

Clausius sought to show a proportional relationship between entropy and the energy flow by heating $\delta Q$ into a system. In a system, this heat energy can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that "The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing." In other words, the equation

$$\oint {\frac {\delta Q(s)}{T(s)}}ds=0$$

with $\delta Q$ being energy flow into the system due to heating and $T$ being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible. Clausius then took this a step further and determined that the following equation must be found true for any cyclical process that is possible, reversible or not. This equation is the "Clausius inequality".

$$\oint {\frac {\delta Q}{T}}\leq 0$$

Now that this is known, there must be a relation developed between the Clausius inequality and entropy. The amount of entropy $S$ added to the system during the cycle is defined as

$$\Delta S{=}\oint {\frac {\delta Q}{T}}$$

It has been determined, as stated in the second law of thermodynamics, that the entropy is a state function: It depends only upon the state that the system is in, and not what path the system took to get there. This is in contrast to the amount of energy added as heat $\delta Q$ and as work $\delta W$, which may vary depending on the path. In a cyclic process, therefore, the entropy of the system at the beginning of the cycle must equal the entropy at the end of the cycle. In the irreversible case, entropy will be created in the system, and more entropy must be extracted than was added $\Delta S<0$ in order to return the system to its original state. In the reversible case, no entropy is created and the amount of entropy added is equal to the amount extracted $\Delta S=0$.

Then should be clear that is the Clausius theorem that gives sense to the entropy (actually was Clausius the first to call entropy with its name). The connection between entropy and the disorder of the system came only after with the genius of Boltzmann...

For your interest, I strongly suggest this amazing lecture at MIT on Statistical Mechanics

I don't know how it relates to the microstate. But in my opinion the entropy is not always positive, it's just that the probability of entropy being positive is very high and of being negative is low. Because no. of microstate also include some microstate where they entropy is negative. But those microstate are very less.