In the previous module, we saw that the Second Law of Thermodynamics provides a criterion for spontaneous change:
\[ \Delta S_{\mathrm{univ}} > 0 \]
While this criterion is extremely powerful, it is somewhat inconvenient. To determine whether a process is spontaneous, we must calculate not only the entropy change of the system, but also the entropy change of the surroundings.
Chemists would much rather work with a criterion that depends only on properties of the system itself.
Fortunately, such criteria exist.
Consider a process occurring at constant temperature and constant volume.
For the surroundings,
\[ \Delta S_{\mathrm{surr}} = -\frac{q_{\mathrm{sys}}}{T} \]
At constant volume, no expansion work is performed, so
\[ q_{\mathrm{sys}} = \Delta U \]
Substituting into
\[ \Delta S_{\mathrm{univ}} = \Delta S_{\mathrm{sys}} + \Delta S_{\mathrm{surr}} \]
gives
\[ \Delta S_{\mathrm{univ}} = \Delta S_{\mathrm{sys}} - \frac{\Delta U}{T} \]
Multiplying by \(-T\),
\[ -T\Delta S_{\mathrm{univ}} = \Delta U - T\Delta S_{\mathrm{sys}} \]
This motivates the definition of the Helmholtz function:
\[ \boxed{ A = U - TS } \]
At constant temperature and volume, spontaneous processes occur when
\[ \boxed{ \Delta A < 0 } \]
Most chemical reactions, however, occur at constant temperature and constant pressure rather than constant volume.
In this case,
\[ q_{\mathrm{sys}} = \Delta H \]
and therefore
\[ \Delta S_{\mathrm{surr}} = -\frac{\Delta H}{T} \]
Substituting into the entropy criterion:
\[ \Delta S_{\mathrm{univ}} = \Delta S_{\mathrm{sys}} - \frac{\Delta H}{T} \]
Multiplying by \(-T\) gives
\[ -T\Delta S_{\mathrm{univ}} = \Delta H - T\Delta S_{\mathrm{sys}} \]
This motivates the definition of the Gibbs function (or Gibbs Free Energy):
\[ \boxed{ G = H - TS } \]
At constant temperature and pressure, spontaneous processes occur when
\[ \boxed{ \Delta G < 0 } \]
Because most chemical reactions occur at approximately constant temperature and pressure, the Gibbs function is the free-energy function most frequently used by chemists.
The change in Gibbs free energy (at constant temperature and pressure) is given by
\[ \boxed{ \Delta G = \Delta H - T\Delta S } \]
This expression shows that spontaneity is determined by a competition between enthalpy and entropy. A process can be driven by a favorable enthalpy change, a favorable entropy change, or both.
Big picture: The Helmholtz and Gibbs functions allow spontaneity to be predicted using only properties of the system. For chemical processes occurring at constant temperature and pressure, the Gibbs free energy provides the most useful criterion: \(\Delta G < 0\) indicates a spontaneous process.
A reaction is endothermic, with
\[ \Delta H = +45.0\ \mathrm{kJ\,mol^{-1}} \]
and has a positive entropy change:
\[ \Delta S = +125\ \mathrm{J\,mol^{-1}\,K^{-1}} \]
Calculate the temperature above which the reaction becomes spontaneous.
At constant temperature and pressure, spontaneity is determined by
\[ \Delta G = \Delta H - T\Delta S \]
The process becomes spontaneous when
\[ \Delta G < 0 \]
The boundary between spontaneous and non-spontaneous behavior occurs when
\[ \Delta G = 0 \]
Set \(\Delta G=0\):
\[ 0 = \Delta H - T\Delta S \]
\[ T\Delta S = \Delta H \]
\[ T = \frac{\Delta H}{\Delta S} \]
Before substituting, convert \(\Delta H\) to joules so the units match \(\Delta S\):
\[ \Delta H = 45.0\ \mathrm{kJ\,mol^{-1}} = 4.50\times10^4\ \mathrm{J\,mol^{-1}} \]
Now substitute:
\[ T = \frac{4.50\times10^4\ \mathrm{J\,mol^{-1}}} {125\ \mathrm{J\,mol^{-1}\,K^{-1}}} \]
\[ T = 360\ \mathrm{K} \]
\[ \boxed{ T > 360\ \mathrm{K} } \]
Interpretation: Because the reaction is endothermic, \(\Delta H\) opposes spontaneity. However, the positive entropy change favors spontaneity through the \(-T\Delta S\) term. At sufficiently high temperature, the entropy term becomes large enough to overcome the unfavorable enthalpy term, so the reaction becomes spontaneous above \(360\ \mathrm{K}\).
A reaction will be generated with random values of \(\Delta H\) and \(\Delta S\). Use \[ \Delta G = \Delta H - T\Delta S \] to determine the temperature conditions under which the reaction is spontaneous.
Big picture: The Gibbs and Helmholtz functions allow spontaneity to be predicted using only properties of the system. For chemical processes at constant temperature and pressure, the Gibbs free energy provides a simple criterion for spontaneity: processes proceed naturally in the direction of decreasing \(G\).