Chemical Equilibrium – Free Energy

In chemical systems, the Gibbs free energy \( G \) changes as a function of pressure \( p \) and temperature \( T \). The behavior of the Gibbs free energy differs for different states of matter (solid, liquid, gas), especially in terms of how their volumes respond to changes in pressure.

Change of \( G \) with Pressure

At constant temperature, the Gibbs free energy of gases is more sensitive to changes in pressure than liquids or solids due to the significant change in volume of gases:

\[ \left( \frac{\partial G}{\partial p} \right)_T = V \]

\[ \int_{p^\circ}^{p} dG = G(T,p) - G(T,p^\circ) = \int_{p^\circ}^{p} V dp \]

  • the volume of liquids and solids is relatively constant, thus:

\[ V (p - p^\circ) \implies \Delta G = V \Delta p \]

  • gasses undergoe significant chagne in volume, thus:

\[ \int_{p^\circ}^{p} \frac{nRT}{p} dp = nRT \ln \left( \frac{p}{p^\circ} \right) \]

\[ \Delta G = nRT \ln \left( \frac{p}{p^\circ} \right) \]

\[ \Delta G = \int_{p_1}^{p_2} V dp \]

 

Temperature Dependence of Gibbs Free Energy

At constant pressure, the temperature dependence of the Gibbs free energy is given by the Gibbs-Helmholtz equation:

\[ \left( \frac{\partial (G/T)}{\partial T} \right)_p = \frac{1}{T} \left( \frac{\partial G}{\partial T} \right)_p + G \frac{\partial (1/T)}{\partial T} = \frac{1}{T} \left( \frac{\partial G}{\partial T} \right)_p - \frac{G}{T^2}  = - \frac{S}{T} - \frac{G}{T^2} \]

for the euqation is used the relationship $\left( \frac{\partial G}{\partial T} \right)_p = -S $

Therefore, the differential form of the Gibbs-Helmholtz equation is:

\[ \left( \frac{\partial (G/T)}{\partial T} \right)_p = - \frac{G + TS}{T^2}= - \frac{H}{T^2} \]

The equation can further be reformulated as:

\[ \left( \frac{\partial (G/T)}{\partial (1/T)} \right)_p = H \]

We can then adapt it to calcualte the change in the fee energy for a given reaction (i.e. double difference, the temperature depdnece of the change in the Gibbs free energy):

\[ \left( \frac{\partial (G/T)}{\partial (1/T)} \right)_p = H \implies \left( \frac{\partial (\Delta G/T)}{\partial (1/T)} \right)_p = \Delta H \]

The Gibbs-Helmholtz euqation is crucial for understanding the equilibrium constant \( K \) in thermodynamic systems.

 

Chemical Potential

The chemical potential \( \mu \) represents the change in Gibbs free energy with respect to the change in the number of particles in a system at constant pressure and temperature:

\[ dG_T = \left( \frac{\partial G_T}{\partial T} \right)_{p,n_1} dT + \left( \frac{\partial G_T}{\partial p} \right)_{T,n_2} dp + \left( \frac{\partial G_T}{\partial n_1} \right)_{p,T,n_2} dn_1 + \left( \frac{\partial G_T}{\partial n_2} \right)_{p,T,n_1} dn_2 + \dots \]

\[ \implies \mu_i = \left( \frac{\partial G}{\partial n_i} \right)_{p,T,n_{j \neq i}} \]

Therefore it is equivalent to the molar Gibbs free energy $\mu_i = G_{m,i}$

The Gbbs free energy can now be expanded to inlcude the chemical potential

\[ dG = \left( \frac{\partial G}{\partial T} \right)_{p,n_i \neq n_j} dT + \left( \frac{\partial G}{\partial p} \right)_{T,n_j \neq n_i} dp + \sum_i \mu_i dn_i \]

\[ dG = -S dT + V dp + \sum_i \mu_i dn_i \]

For constat T and p, we get:

\[ \Delta G = \int_0^G dG = \sum_i \mu_i \int_0^{n_i} dn_i \]

\[ \Delta G = \sum_i n_i \mu_i \]

We can express the chemical potential also in terms of other thermodynamics functions:

\[ \mu_i = \left( \frac{\partial G}{\partial n_i} \right)_{p,T,n_{j \neq i}} \]

\[ \mu_i = \left( \frac{\partial U}{\partial n_i} \right)_{S,V,n_{j \neq i}} = \left( \frac{\partial H}{\partial n_i} \right)_{S,p,n_{j \neq i}} = \left( \frac{\partial A}{\partial n_i} \right)_{T,V,n_{j \neq i}} \]

The chemical potential measures the tendency of a substance to undergo a physical or chemical transformation. In a mixture, equilibrium is achieved when the chemical potentials of all species are equal. Chemical potential allows us to understand chemical reactions and chemical equilibrium.

 

Fugacity – Deviation from Ideal Gas Behavior

Fugacity is a corrected pressure that accounts for the non-ideal behavior of real gases. It is related to the chemical potential and behaves logarithmically with pressure. For an ideal gas:

\[ \mu = \mu^\circ + RT \ln \left( \frac{p}{p^\circ} \right) \]

For real gases, fugacity replaces pressure (i.e. it has dimension of pressure) in the equation for chemical potential:

\[ \mu = \mu^\circ + RT \ln \left( \frac{f}{p^\circ} \right) \]

Fugacity Coefficient

The fugacity coefficient \( \phi \) describes how far a real gas deviates from ideal behavior:

\[ \phi = \frac{f}{p} \]

Experimental Determination of Fugacity

Fugacity can be experimentally determined using an integral expression derived from a Maxwell relation:

From the expression for the Gibbs free energy ($ dG_T = -S dT + V dp + \sum_i \mu_i dn_i $, considering $ T = const.$ and using the Maxwell relation $\left( \frac{\partial \mu}{\partial p} \right)_T = V_m $, we obtain:

\[ d\mu = V_m dp \]

Taking the difference between the chemical potentials of ideal and real gas leads to an expression for fugacity:

\[ d\mu_{\text{real}} - d\mu_{\text{ideal}} = (V_m - V_{m, \text{ideal}}) dp \]

\[ \mu_{\text{real}} - \mu_{\text{ideal}} = \int_{p^\circ}^{p} V_m dp - \int_{p^\circ}^{p} V_{m, \text{ideal}} dp \]

\[ \mu^\circ + RT \ln \frac{f}{p^\circ} - \left( \mu^\circ + RT \ln \frac{p}{p^\circ} \right) = RT \ln \frac{f}{p^\circ} - RT \ln \frac{p}{p^\circ} = RT \ln \frac{f}{p} = RT \ln \phi \]

\[ \ln \phi = \frac{1}{RT} \left( \int_{0}^{p} V_m dp - \int_{0}^{p} V_{m, \text{ideal}} dp \right) \]

Here, the fugacity can be calculated as The difference of the areas of the $Vdp$ functions for ideal and real gas.

 

Compression Factor \( Z \)

The compression factor \( Z \) is a dimensionless quantity that provides further simplificaiton to the calcuation of the deviation of the real gas behavior from that of ideal gas. It is derived as follows:

Taking the ideal gas equation $ V_m = \frac{RT}{p}$ and the real gas equation in the form of $ V_m = \frac{ZRT}{p}$, we can write:

\[ \ln \phi = \frac{1}{RT} \int_0^p \left( \frac{ZRT}{p} - \frac{RT}{p} \right) dp = \int_0^p \left( \frac{Z}{p} - \frac{1}{p} \right) dp = \int_0^p \left( \frac{Z - 1}{p} \right) dp \]

Thus, the fugacity is related to the compression factor \( Z \) via:

\[ \ln \left( \frac{f}{p} \right) = \int_{0}^{p} \left( \frac{Z - 1}{p} \right) dp \]

\[ Z = \frac{V_m}{V_{m,\text{ideal}}} \]

For an ideal gas, \( Z = 1 \), but for real gases, \( Z \) can be greater or less than 1, depending on the conditions.