6.1 Entropy changes in various processes

Entropy Changes in Thermodynamic Processes

Entropy quantifies the dispersal of energy within a system, and tracking how it changes during different processes is central to applying the second law of thermodynamics. Because entropy is a state function, you can calculate its change between two states using any convenient reversible path, regardless of the actual process. This section covers the key formulas and reasoning for the most common thermodynamic processes.

Entropy Changes by Process Type

Each standard thermodynamic process holds one variable constant, which simplifies the entropy calculation. For all of these, assume an ideal gas with $n$ moles.

Isothermal process (constant temperature):

$\Delta S = nR \ln \frac{V_2}{V_1} = nR \ln \frac{P_1}{P_2}$

Because temperature is fixed, the only thing driving entropy change is the redistribution of molecules over a larger or smaller volume. Entropy increases during expansion ($V_2 > V_1$) and decreases during compression.

Isobaric process (constant pressure):

$\Delta S = nC_p \ln \frac{T_2}{T_1}$

Here you use $C_p$ (heat capacity at constant pressure) because the system can do expansion work as it heats up. Heating raises entropy; cooling lowers it.

Isochoric process (constant volume):

$\Delta S = nC_v \ln \frac{T_2}{T_1}$

With volume locked, no expansion work occurs, so you use $C_v$ (heat capacity at constant volume). Again, heating increases entropy and cooling decreases it.

Adiabatic process (no heat transfer):

• Reversible (isentropic): $\Delta S = 0$. No heat crosses the boundary, and no irreversibilities exist, so entropy stays constant.
• Irreversible: $\Delta S > 0$. Even though no heat enters, internal friction, turbulence, or rapid expansion generates entropy within the system.

General Entropy Change for an Ideal Gas

When both temperature and volume change simultaneously, you need the general expression:

$\Delta S = nC_v \ln \frac{T_2}{T_1} + nR \ln \frac{V_2}{V_1}$

The first term captures the entropy change due to temperature, and the second captures the change due to volume. An equivalent form using pressure instead of volume is:

$\Delta S = nC_p \ln \frac{T_2}{T_1} - nR \ln \frac{P_2}{P_1}$

Both forms give the same result for the same initial and final states. You pick whichever is more convenient based on the variables you're given. This is a direct consequence of entropy being a state function: the answer depends only on the endpoints, not on how the gas got there.

System and Surroundings Entropy Analysis

The total entropy change of the universe is the sum of the system and surroundings contributions:

$\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}}$

This single equation is how you determine whether a process is possible:

• Reversible process: $\Delta S_{\text{universe}} = 0$. The system's entropy gain is exactly offset by the surroundings' entropy loss (or vice versa), so $\Delta S_{\text{system}} = -\Delta S_{\text{surroundings}}$.
• Irreversible process: $\Delta S_{\text{universe}} > 0$. There is a net production of entropy. Every real process falls into this category.
• Impossible process: $\Delta S_{\text{universe}} < 0$. This violates the second law and cannot occur spontaneously.

A common way to find $\Delta S_{\text{surroundings}}$: if the surroundings act as a large thermal reservoir at constant temperature $T_{\text{surr}}$, then $\Delta S_{\text{surroundings}} = -\frac{Q_{\text{system}}}{T_{\text{surr}}}$, where $Q_{\text{system}}$ is the heat absorbed by the system.

Entropy as a State Function

Two key consequences follow from entropy being a state function:

1. Cyclic processes: After a system completes a full cycle and returns to its initial state, $\oint dS = 0$. The system's entropy is back where it started. (The surroundings' entropy, however, may have increased if the cycle involved irreversibilities.)

2. Path independence: You can compute $\Delta S$ between any two states by inventing a convenient reversible path between them, even if the actual process was irreversible. For example, to find the entropy change of a free expansion (irreversible, no work, no heat), you replace it with a reversible isothermal expansion between the same initial and final states and use that calculation.

Entropy and the Second Law of Thermodynamics

Relationship Between Entropy and the Second Law

The second law of thermodynamics states that the entropy of an isolated system can never decrease. In equation form: $\Delta S_{\text{universe}} \geq 0$ for any real process, with equality holding only in the idealized reversible limit.

This has direct physical meaning:

• Heat flows spontaneously from hot objects to cold objects. The reverse never happens on its own because it would require $\Delta S_{\text{universe}} < 0$.
• Every spontaneous process (mixing of gases, friction converting kinetic energy to heat, chemical reactions proceeding toward equilibrium) increases the universe's entropy.
• The second law establishes an arrow of time: entropy tells you which direction a process will naturally proceed. A process that decreases universal entropy is not forbidden by energy conservation, but it is forbidden by the second law.