Theory — Entropy Change of the Universe (Second Law)
Spontaneity at a temperature \(T\) is judged by the entropy change of the
universe (system + surroundings). The Second Law states that a process is
spontaneous if the total entropy increases.
\[
\begin{aligned}
\Delta S_{\text{univ}}
&= \Delta S_{\text{sys}} + \Delta S_{\text{surr}} \\
&> 0 \quad \text{(spontaneous)}
\end{aligned}
\]
Surroundings entropy from heat exchange
If the surroundings are a large thermal reservoir at temperature \(T\),
their entropy change is related to the heat that flows into the surroundings.
With the sign convention \(q_{\text{sys}}>0\) when the system absorbs heat,
\[
\begin{aligned}
\Delta S_{\text{surr}}
&= -\,\dfrac{q_{\text{sys}}}{T}
\end{aligned}
\]
Under constant pressure with only \(pV\) work, the heat exchanged by the
system equals the enthalpy change: \(q_{\text{sys}}=\Delta H\).
Therefore,
\[
\begin{aligned}
\Delta S_{\text{univ}}
&= \Delta S_{\text{sys}} + \Delta S_{\text{surr}} \\
&= \Delta S_{\text{sys}} - \dfrac{\Delta H}{T}
\end{aligned}
\]
How to use in problems
- Determine or estimate the system entropy change \(\Delta S_{\text{sys}}\)
(e.g., from heating/cooling, phase change, or tabulated values).
- Use the process temperature \(T\) (in kelvin) and the reaction enthalpy
\(\Delta H\) (per mole of reaction as written) to compute
\(\Delta S_{\text{surr}}=-\Delta H/T\).
- Add: \(\Delta S_{\text{univ}}=\Delta S_{\text{sys}}+\Delta S_{\text{surr}}\).
- Decide spontaneity:
- \(\Delta S_{\text{univ}}>0\): spontaneous
- \(\Delta S_{\text{univ}}=0\): reversible (ideal limit)
- \(\Delta S_{\text{univ}}<0\): nonspontaneous (reverse is spontaneous)
Worked template
\[
\begin{aligned}
T &= \text{(given temperature in kelvins)}\ \mathrm{K}, \\[6pt]
\Delta H &= \text{(given enthalpy change)}\ \mathrm{J\,mol^{-1}}, \\[6pt]
\Delta S_{\text{sys}} &= \text{(given entropy change of the system)}\ \mathrm{J\,mol^{-1}\,K^{-1}}, \\[8pt]
\Delta S_{\text{surr}} &= -\,\dfrac{\Delta H}{T}
\quad \text{(entropy change of the surroundings)}, \\[8pt]
\Delta S_{\text{univ}} &= \Delta S_{\text{sys}} + \Delta S_{\text{surr}}
\quad \text{(total entropy change of the universe).}
\end{aligned}
\]
Notes & pitfalls
- Units: report entropies in \(\mathrm{J\,mol^{-1}\,K^{-1}}\).
Convert \(\Delta H\) to \(\mathrm{J\,mol^{-1}}\) and use \(T\) in kelvin.
- \(\Delta H\) and \(\Delta S_{\text{sys}}\) must correspond to the same
chemical equation (per mole of reaction as written).
- The relation \(\Delta S_{\text{surr}}=-\Delta H/T\) assumes the surroundings’
temperature does not change appreciably (large reservoir).
Physical insight
Processes that release heat to cooler surroundings (\(\Delta H<0\)) make the
surroundings more disordered (\(\Delta S_{\text{surr}}>0\)), helping
\(\Delta S_{\text{univ}}\) be positive even when the system becomes more ordered
(e.g., freezing). Conversely, strongly endothermic steps require a sufficiently
positive \(\Delta S_{\text{sys}}\) to be spontaneous.
