KAHIBARO
Table of Contents
Disorder, Multiplicity, and the Direction of Change
Entropy is one of the central ideas in thermodynamics. It helps describe why some processes happen naturally in one direction and not in the reverse direction. A hot object cools down in a cold room, gases spread out to fill a container, and mixed substances do not spontaneously separate. Entropy gives a way to describe this tendency.
At an introductory level, entropy can be thought of as a measure of how spread out energy is, or how many microscopic arrangements are possible for a system. A system with more possible internal arrangements has greater entropy. This idea connects the large scale behavior we observe with the tiny scale motion of atoms and molecules.
Entropy is usually represented by the symbol $S$. Its SI unit is joule per kelvin, written as $\mathrm{J/K}$.
Entropy is a state function. This means its change depends only on the initial and final states, not on the path taken between them.
Entropy Change in Thermodynamics
In thermodynamics, the change in entropy is defined most directly for a reversible process. If a small amount of heat $dQ_{\text{rev}}$ enters a system reversibly at absolute temperature $T$, then the entropy change is
$$
dS = \frac{dQ_{\text{rev}}}{T}
$$
For a finite reversible change,
$$
\Delta S = \int \frac{dQ_{\text{rev}}}{T}
$$
If the temperature stays constant during a reversible process, this becomes
$$
\Delta S = \frac{Q_{\text{rev}}}{T}
$$
This formula is very important, but it must be used carefully. The heat must be the heat exchanged along a reversible path.
Important formula:
$$
dS = \frac{dQ_{\text{rev}}}{T}
$$
For constant temperature:
$$
\Delta S = \frac{Q_{\text{rev}}}{T}
$$
Use reversible heat transfer in the formula, even if the actual process is irreversible.
Physical Meaning of Entropy
Entropy is often introduced using the idea of disorder, but that word can be vague. A more precise picture is that entropy measures the number of microscopic ways a system can exist while still looking the same macroscopically.
For example, imagine a gas in a box. If all the gas molecules are confined to one corner, that is a very special arrangement. If they are spread throughout the box, there are many more possible molecular arrangements. The spread out state is therefore much more probable and has higher entropy.
Entropy also reflects energy spreading. When heat flows from a hot object to a cold one, the energy becomes more evenly distributed. This increases the total entropy.
Entropy and the Second Law
The second law of thermodynamics is closely tied to entropy. It states that for an isolated system, the total entropy never decreases.
If a process is reversible, the total entropy remains constant. If a process is irreversible, the total entropy increases.
Mathematically,
$$
\Delta S_{\text{total}} \geq 0
$$
where the equality holds for reversible processes and the inequality is strict for irreversible processes.
Second law in entropy form:
$$
\Delta S_{\text{total}} \geq 0
$$
For an isolated system, entropy can stay the same or increase, but it cannot decrease.
This is why natural processes have a preferred direction. A broken cup does not spontaneously reassemble, and heat does not naturally flow from cold to hot, because such changes would reduce total entropy.
Entropy in Reversible and Irreversible Processes
A reversible process is an ideal process that happens infinitely slowly and can be undone without leaving any net change in the system and surroundings. In such a case, entropy transfer is balanced perfectly.
An irreversible process is a real process, such as friction, free expansion, mixing, or heat flow across a finite temperature difference. These processes produce entropy.
For the system alone, entropy may increase, decrease, or stay the same depending on what happens. What matters for the second law is the system plus surroundings.
The table below summarizes the difference.
| Process type | System entropy change | Total entropy change |
|---|---|---|
| Reversible | can be positive or negative | $0$ |
| Irreversible | can be positive or negative | $>0$ |
Entropy and Heat Flow
Consider heat $Q$ flowing reversibly into a system at temperature $T$. Then the system entropy increases by
$$
\Delta S_{\text{system}} = \frac{Q}{T}
$$
If the same amount of heat leaves the surroundings, and the surroundings are also at temperature $T$, then
$$
\Delta S_{\text{surroundings}} = -\frac{Q}{T}
$$
So for a reversible transfer,
$$
\Delta S_{\text{total}} = 0
$$
Now suppose heat flows from a hot object at temperature $T_h$ to a cold object at temperature $T_c$, with $T_h > T_c$. If an amount of heat $Q$ leaves the hot object and enters the cold one, then
$$
\Delta S_{\text{hot}} = -\frac{Q}{T_h}
$$
and
$$
\Delta S_{\text{cold}} = \frac{Q}{T_c}
$$
Therefore,
$$
\Delta S_{\text{total}} = \frac{Q}{T_c} - \frac{Q}{T_h}
$$
Since $T_c < T_h$, we have
$$
\Delta S_{\text{total}} > 0
$$
This shows that spontaneous heat flow from hot to cold increases total entropy.
Entropy in Common Processes
Several common thermodynamic processes illustrate entropy clearly.
In free expansion of a gas into a vacuum, no work is done on the surroundings and no heat may enter or leave, yet the entropy increases. The gas has more available volume and more possible microscopic arrangements.
When two substances at different temperatures come into thermal contact, heat flows until they reach equilibrium. The total entropy increases during this process.
When a solid melts or a liquid evaporates, entropy usually increases because the particles gain more freedom of arrangement and motion.
When a gas is compressed reversibly and isothermally, its entropy decreases because the molecules are confined to a smaller volume.
Entropy Change of an Ideal Gas
For an ideal gas, entropy change can be calculated from measurable quantities. A very useful result for a change from state 1 to state 2 is
$$
\Delta S = nC_V \ln\!\left(\frac{T_2}{T_1}\right) + nR \ln\!\left(\frac{V_2}{V_1}\right)
$$
Another equivalent form is
$$
\Delta S = nC_P \ln\!\left(\frac{T_2}{T_1}\right) - nR \ln\!\left(\frac{P_2}{P_1}\right)
$$
Here, $n$ is the number of moles, $R$ is the gas constant, and $C_V$, $C_P$ are molar heat capacities.
Special cases are especially useful. For an isothermal process, $T_2 = T_1$, so
$$
\Delta S = nR \ln\!\left(\frac{V_2}{V_1}\right)
$$
For a reversible isochoric process, $V_2 = V_1$, so
$$
\Delta S = nC_V \ln\!\left(\frac{T_2}{T_1}\right)
$$
For a reversible isobaric process, using constant pressure,
$$
\Delta S = nC_P \ln\!\left(\frac{T_2}{T_1}\right)
$$
Useful ideal gas entropy formulas:
$$
\Delta S = nC_V \ln\!\left(\frac{T_2}{T_1}\right) + nR \ln\!\left(\frac{V_2}{V_1}\right)
$$
$$
\Delta S = nC_P \ln\!\left(\frac{T_2}{T_1}\right) - nR \ln\!\left(\frac{P_2}{P_1}\right)
$$
Microscopic View of Entropy
In statistical physics, entropy is related to the number of microscopic states. If $\Omega$ is the number of possible microstates consistent with the observed macrostate, then
$$
S = k_B \ln \Omega
$$
where $k_B$ is Boltzmann's constant.
This equation shows that entropy increases when the number of possible microscopic arrangements increases. A state with very many microstates is much more likely than one with very few.
For beginners, this microscopic idea is valuable because it explains why entropy increase is so common. Systems naturally move toward states that can occur in more ways.
Boltzmann relation:
$$
S = k_B \ln \Omega
$$
More possible microstates means greater entropy.
Entropy and Equilibrium
Thermodynamic equilibrium corresponds to a state of maximum entropy for an isolated system, subject to the system's constraints. This means the system naturally evolves toward the most probable state.
For example, if gas molecules begin unevenly distributed in a box, they will spread out until the distribution becomes uniform. The uniform state has much higher entropy and is the equilibrium state.
Entropy therefore provides a criterion for spontaneity. A process is spontaneous in an isolated system if it increases total entropy.
A Simple Example
Suppose $200\ \mathrm{J}$ of heat flows reversibly into a system at constant temperature $400\ \mathrm{K}$. Then
$$
\Delta S = \frac{Q_{\text{rev}}}{T} = \frac{200}{400} = 0.50\ \mathrm{J/K}
$$
So the system entropy increases by $0.50\ \mathrm{J/K}$.
Now imagine that the same $200\ \mathrm{J}$ flows from a hot reservoir at $500\ \mathrm{K}$ to a cold reservoir at $300\ \mathrm{K}$. Then
$$
\Delta S_{\text{hot}} = -\frac{200}{500} = -0.40\ \mathrm{J/K}
$$
and
$$
\Delta S_{\text{cold}} = \frac{200}{300} \approx 0.67\ \mathrm{J/K}
$$
Thus,
$$
\Delta S_{\text{total}} \approx 0.27\ \mathrm{J/K}
$$
The total entropy increases, so the process is consistent with the second law.
Visualizing Entropy Increase
The drawing below shows spontaneous heat flow from a hotter body to a colder body.
As heat moves from the hot body to the cold body, the energy becomes more evenly distributed. That is why the total entropy rises.
Key Ideas to Remember
Entropy is a thermodynamic quantity that measures the spreading of energy and the number of possible microscopic arrangements. It is a state function, and for reversible heat transfer its change is given by $dS = dQ_{\text{rev}}/T$. The second law states that the total entropy of an isolated system cannot decrease. Reversible processes leave total entropy unchanged, while irreversible processes increase it. On the microscopic level, entropy is connected to the number of microstates by $S = k_B \ln \Omega$.
Core facts:
$$
dS = \frac{dQ_{\text{rev}}}{T}, \qquad \Delta S_{\text{total}} \geq 0, \qquad S = k_B \ln \Omega
$$
Entropy explains the natural direction of thermodynamic processes.