How Do You Calculate Entropy Change (ΔS = ΔQ/T)?

HL only: B.4 Thermodynamics. Calculate entropy change by dividing the thermal energy transferred reversibly by the absolute temperature at which the transfer occurs:

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

Use joules for $\Delta Q$ and kelvin for $T$, giving $\Delta S$ in $\text{J K}^{-1}$.

The equation is exact for a reversible transfer. For an irreversible process, determine the system’s entropy change using a reversible path between the same initial and final equilibrium states. Entropy is a state function, so the calculated change does not depend on the actual path.

The Reasoning

Entropy measures how energy is distributed among the possible microscopic arrangements of a system. When thermal energy is transferred at constant temperature, the entropy change depends on both the transferred energy and the temperature.

Follow these steps:

1. Identify the thermal energy transfer $\Delta Q$ in joules.
2. Convert the temperature to kelvin using $T = \theta + 273.15$.
3. Assign the correct sign to $\Delta Q$.
4. Substitute into $\Delta S = \Delta Q/T$.

For example, a reservoir at $300\,\text{K}$ absorbs $600\,\text{J}$:

$$\Delta S = \frac{600}{300} = 2.0\,\text{J K}^{-1}$$

If that energy came from a hot reservoir at $400\,\text{K}$, its entropy change would be:

$$\Delta S_{\text{hot}} = \frac{-600}{400} = -1.5\,\text{J K}^{-1}$$

Therefore, the total entropy change is:

$$\Delta S_{\text{total}} = 2.0 - 1.5 = 0.5\,\text{J K}^{-1}$$

The positive total is consistent with the second law of thermodynamics: the entropy of an isolated system cannot decrease. Although $600\,\text{J}$ leaves and enters, the entropy changes do not cancel because transfer occurs at different temperatures. The colder reservoir gains more entropy per joule than the hotter reservoir loses.

A common misconception is using degrees Celsius directly. Temperature must be in kelvin because entropy calculations require an absolute temperature scale. Also, use the temperature at the boundary where transfer occurs, not an unrelated average temperature.

Exam Technique

For a Calculate question, show the equation, substitution, sign, and final unit. If two bodies exchange energy, calculate each entropy change separately before finding $\Delta S_{\text{total}}$; do not assume their entropy changes cancel.