Example: Entropy and heat flow

Understanding the situation

By now we have talked a lot about the idea of entropy and the second law of thermodynamics. The key concept is the way energy moves spontaneously in a system where things are driven by randomness. The result is a form of the second law of thermodynamics:

Energy in a thermal system is dominated by randomness will tend to move so that it is equally shared among all places where energy can reside (degrees of freedom).

So far, our way of describing this is in terms of entropy — the (logarithm of the) number of microstates corresponding to a given macrostate; or, in plain English, the logarithm of the number of different ways energy can be arranged for a given state.

This basically says that things spontaneously move so that energy is spread out — and no longer useful.

The counting definition of entropy is formally valuable and can be used (using a lot of heavy mathematical lifting) to prove lots of useful results, but it doesn't really help us build a conceptual picture of entropy. Let's see if we can help a little, by doing what we love to do in a physics class: considering the simplest possible example that shows the phenomenon (even if it doesn't have a lot of direct practical relevance). For entropy, that example is the flow of thermal energy (heat) between two blocks of different temperatures.

Presenting a sample problem

Consider a system consisting of two blocks of identical matter at different temperatures: $T_h$ (hot) and $T_c$ (cold). They are surrounded by a Styrofoam insulator that, in the time we are watching, will let no significant amount of heat energy flow in or out of the system. 

Calculate the change in entropy if the system moves from the above, uneven distribution of temperature, to one where the temperatures are equal (to $T_m$).

Explain physically what is happening to lead to this result.

Solving this problem

Let's begin with the last part of the question so we have a story to tell about what's happening.

We know intuitively what will happen physically. Thermal energy is continually being exchanged between the two blocks, but as long as the temperature of the block on the left is higher, more energy will flow to the cold block than to the hot one. The hot block will get colder and the cold block will get hotter until they reach the same (medium) temperature.

When that has happened, we expect that the amount of heat flowing from the right to left and from left to right are equal (and NOT that there is no heat flow).

This is the intuitive result we want to quantify with the statement:

Energy will spontaneously move so that the total entropy of the universe increases.

What the first law of thermodynamics tells us is that however much energy leaves the block on the left, that's how much energy will enter the block on the right. If the amount of energy that flows is $Q$ (the standard notation for thermal energy flow), then we can say

$$Q_{L \rightarrow R} = -Q_{R \rightarrow L}$$

That is, the amount of heat energy flowing out of the block on the left is equal in magnitude to the amount flowing into the block on the left. We use a negative sign to indicate that the heat is entering one of the blocks and leaving the other. 

This of course doesn't tell us which is positive and which is negative — only that they are opposite. To nail down the correspondence between the direction of the flow of heat energy and the sign of the heat flow, we use the convention that heat entering an object is positive.

A useful way of determining which way the heat will flow is to consider the quantity that is the heat energy transferred divided by the temperature of the object. It turns out that this is, for this situation, the change in the object's entropy. (Showing that this is the case requires breaking the thermal energy up into small bits and counting the number of ways it can be arranged — a bit beyond the level a detail we want to go into here.)

So for the transfer of thermal energy into or out of an object of temperature $T$,

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

For the hot object, a certain amount of heat energy has left it. If the magnitude of heat leaving is $Q$, then the entropy change for the hot object is

$$\Delta S_h = -\frac{Q}{T_h}$$

We have a minus sign since the heat is leaving.

For the cold object, that same amount of heat has entered it, so the entropy change for it is

$$\Delta S_c = \frac{Q}{T_c}$$

So the total change in entropy for the universe is

$$\Delta S = \Delta S_c + \Delta S_h = \frac{Q}{T_c} - \frac{Q}{T_h}$$

This tells us that as long as $T_h$ is bigger than $T_c$, the second term will be smaller than the first and the entropy change of the universe will be positive.

So this relation says whenever heat is exchanged between two objects at different temperatures, the condition that the entropy of the universe must increase is equivalent to the statement that the energy will flow from the hotter object to the colder one — the correct intuitive result.

Notice that the entropy of the hot object giving up the energy decreases! But it decreases by less than the cold object receiving the energy increases. So the second law is satisfied.

Now heat flow is not where living organisms typically use energy to do work. And often in biological systems we will have to use the counting definition of entropy rather than the energy/temperature form. But this simple example does illustrate how the condition, "the entropy of the universe increases", can show us the preferred direction of energy flow.

Joe Redish 1/31/12