The energy balance in differential form for a simple adsorption process can be written as

$$\mathrm{d}U=h^\mathrm{in}\delta n^\mathrm{in}-h^\mathrm{b}\delta n^\mathrm{out}+\delta Q$$ (eqn:energy_balance)

Here the balance is chosen to only include the fluid in the porous medium. The molar enthalpy $h^\mathrm{b}$ of the (bulk) fluid that leaves the adsorber is at a state that is in equilibrium with the porous medium. In contrast, the incoming stream can be at any condition. Analogously, the component balance is

$$\mathrm{d}N_i=x_i^\mathrm{in}\delta n^\mathrm{in}-x_i\delta n^\mathrm{out}$$ (eqn:mass_balance)

The differential of the internal energy can be replaced with the total differential in its variables temperature $T$ and number of particles $N_i$. The volume of the adsorber is fixed and thus not considered as a variable.

$$\mathrm{d}U=\left(\frac{\partial U}{\partial T}\right){N_k}\mathrm{d}T+\sum_i\left(\frac{\partial U}{\partial N_i}\right){T,N_j}\mathrm{d}N_i$$ (eqn:U_differential)

Eqs. {eq}eqn:energy_balance, {eq}eqn:mass_balance and {eq}eqn:U_differential can be combined into an expression for the heat of adsorption $\delta Q$

$$\delta Q=\left(\frac{\partial U}{\partial T}\right)_{N_k}\mathrm{d}T-\left(h^\mathrm{in}-\sum_ix_i^\mathrm{in}\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}\right)\delta n^\mathrm{in}+\left(h^\mathrm{b}-\sum_ix_i\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}\right)\delta n^\mathrm{out}$$

The heat of adsorption can thus be split into a sensible part that depends on the change in temperature, and a latent part that depends on the change in loading. The expression can be simplified by using the definitions of the isochoric heat capacity $C_v=\left(\frac{\partial U}{\partial T}\right)_{N_k}$ and the partial molar enthalpy of adsorption

$$\Delta h_i^\mathrm{ads}=h_i^\mathrm{b}-\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}$$

yielding

$$\delta Q=C_v\mathrm{d}T-\sum_ix_i^\mathrm{in}\left(h_i^\mathrm{in}-h_i^\mathrm{b}+\Delta h_i^\mathrm{ads}\right)\delta n^\mathrm{in}+\sum_ix_i\Delta h_i^\mathrm{ads}\delta n^\mathrm{out}$$

or

$$\delta Q=C_v\mathrm{d}T-\sum_ix_i^\mathrm{in}\left(h_i^\mathrm{in}-h_i^\mathrm{b}+\Delta h_i^\mathrm{ads}\right)\delta n^\mathrm{in}+\Delta h^\mathrm{ads}\delta n^\mathrm{out}$$

with the enthalpy of adsorption

$$\Delta h^\mathrm{ads}=\sum_ix_i\Delta h_i^\mathrm{ads}=h^\mathrm{b}-\sum_ix_i\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}$$

For pure components the balance equations simplify to

$$\delta Q=C_v\mathrm{d}T-\left(h^\mathrm{in}-h^\mathrm{b}\right)\delta n^\mathrm{in}-\Delta h^\mathrm{ads}\mathrm{d}N$$

The Clausius-Clapeyron relation relates the $p-T$ slope of a pure component phase transition line to the corresponding enthalpy of phase change. For a vapor-liquid phase transition, the exact relation is

$$\frac{\mathrm{d}p^\mathrm{sat}}{\mathrm{d}T}=\frac{s^\mathrm{V}-s^\mathrm{L}}{v^\mathrm{V}-v^\mathrm{L}}=\frac{h^\mathrm{V}-h^\mathrm{L}}{T\left(v^\mathrm{V}-v^\mathrm{L}\right)}$$ (eqn:temp_dep_press)

In this expression, the enthalpy of vaporization $\Delta h^\mathrm{vap}=h^\mathrm{V}-h^\mathrm{L}$ can be identified. The molar volumes $v$ of the two phases can be replaced by the compressibility factor $Z=\frac{pv}{RT}$. Then, eq. {eq}eqn:temp_dep_press simplifies to

$$\frac{\mathrm{d}p^\mathrm{sat}}{\mathrm{d}T}=\frac{p}{R T^2}\frac{\Delta h^\mathrm{vap}}{Z^\mathrm{V}-Z^\mathrm{L}}$$

which can be compactly written as

$$\frac{\mathrm{d}\ln p^\mathrm{sat}}{\mathrm{d}\frac{1}{RT}}=-\frac{\Delta h^\mathrm{vap}}{Z^\mathrm{V}-Z^\mathrm{L}}$$ (eqn:Clausius_Clapeyron_exact)

Eq. {eq}eqn:Clausius_Clapeyron_exact is still an exact expression. In practice, the volume (and hence the compressibility factor) of the liquid phase can often be neglected compared to the volume of the gas phase. Additionally assuming an ideal gas phase ($Z^\mathrm{V}\approx1$), leads to the expression commonly referred to as Clausius-Clapeyron relation:

$$\frac{\mathrm{d}\ln p^\mathrm{sat}}{\mathrm{d}\frac{1}{RT}}=-\Delta h^\mathrm{vap}$$ (eqn:Clausius_Clapeyron)

A similar relation can be derived for fluids adsorbed in a porous medium that is in equilibrium with a bulk phase. At this point it is important to clarify which variables describe the system

• The adsorbed fluid and the bulk phase are in equilibrium. Therefore, the temperature $T$ and chemical potentials $\mu_i$ are the same for both phases.
• The density profiles and hence the number of particles $N_i$ in the porous medium is determined by $T$ and $\mu_i$. The volume of the porous medium is not considered as a thermodynamic variable but rather as a (constant) property of the adsorbent.
• All intensive properties of the bulk phase are fully determined by $T$ and $\mu_i$. In practice it can be useful to relate these properties to measurable properties like the pressure $p$ and the composition $x_i$.

To find an expression of the slope of an isostere (constant $N_i$), the pressure, which is only defined for the bulk phase, has to be related to properties of the adsorbed fluid.

$$\frac{\mathrm{d}\ln p}{\mathrm{d}\frac{1}{RT}}=-\frac{RT^2}{p}\frac{\mathrm{d}p}{\mathrm{d}T}$$

First, the pressure can be replaced using the Gibbs-Duhem relation for the bulk phase (index $\mathrm{b}$)

$$\frac{\mathrm{d}\ln p}{\mathrm{d}\frac{1}{RT}}=-\frac{RT^2}{pv^\mathrm{b}}\left(s^\mathrm{b}+\sum_ix_i\left(\frac{\partial\mu_i}{\partial T}\right)_{N_k}\right)$$ (eqn:clausius_clapeyron_intermediate)

Here the directional derivative $\frac{\mathrm{d}\mu_i}{\mathrm{d}T}$ could be replaced with a partial derivative amongst the variables describing the adsorbed fluid. The partial derivative can then be replaced using a Maxwell relation based on the Helmholtz energy $F$ as follows

$$\left(\frac{\partial\mu_i}{\partial T}\right)_{N_k}=\left(\frac{\partial^2 F}{\partial T\partial N_i}\right)=-\left(\frac{\partial S}{\partial N_i}\right)_{T,N_j}$$

Using the Maxwell relation together with the compressibility factor of the bulk phase $Z^\mathrm{b}=\frac{pv^\mathrm{b}}{RT}$ in eq. {eq}eqn:clausius_clapeyron_intermediate results in

$$\frac{\mathrm{d}\ln p}{\mathrm{d}\frac{1}{RT}}=-\frac{T}{Z^\mathrm{b}}\left(s^\mathrm{b}-\sum_ix_i\left(\frac{\partial S}{\partial N_i}\right)_{T,N_j}\right)$$

Finally, using $h^\mathrm{b}=Ts^\mathrm{b}+\sum_ix_i\mu_i$ and $\mathrm{d}U=T\mathrm{d}S+\sum_i\mu_i\mathrm{d}N_i$ leads to

$$\frac{\mathrm{d}\ln p}{\mathrm{d}\frac{1}{RT}}=-\frac{1}{Z^\mathrm{b}}\left(h^\mathrm{b}-\sum_ix_i\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}\right)=-\frac{\Delta h^\mathrm{ads}}{Z^\mathrm{b}}$$ (eqn:deriv_relation_hads)

The relation is exact and valid for an arbitrary number of components in the fluid phase.

In a DFT context, the introduction of entropies and internal energies are just unnecessary complications. The most useful definition of the (partial molar) enthalpy of adsorption is

$$\Delta h_i^\mathrm{ads}=T\left(s_i^\mathrm{b}+\left(\frac{\partial\mu_i}{\partial T}\right)_{N_k}\right)$$

The derivative at constant number of particles is problematic and has to be replaced. This is done starting from the total differential of the number of particles

$$\mathrm{d}N_i=\sum_j\left(\frac{\partial N_i}{\partial\mu_j}\right)T\mathrm{d}\mu_j+\left(\frac{\partial N_i}{\partial T}\right){\mu_k}\mathrm{d}T$$ (eqn:dn)

Calculating the derivative with respect to $T$ at constant $N_i$ leads to

$$0=\sum_j\left(\frac{\partial N_i}{\partial\mu_j}\right)T\left(\frac{\partial\mu_j}{\partial T}\right){N_k}+\left(\frac{\partial N_i}{\partial T}\right)_{\mu_k}$$ (eqn:dndt_1)

from which the unknown derivative $\left(\frac{\partial\mu_i}{\partial T}\right){N_k}$ can be calculated. In practice the expression has the disadvantage that $\left(\frac{\partial N_i}{\partial T}\right){\mu_k}$ depends on the (sometimes unknown) thermal de Broglie wavelength which cancels later with $s_i^\mathrm{b}$. This can be remedied by first calculating the derivative of eq. {eq}eqn:dn with respect to $T$ at constant (bulk) pressure and composition.

$$\left(\frac{\partial N_i}{\partial T}\right){p,x_k}=\sum_j\left(\frac{\partial N_i}{\partial\mu_j}\right)T\left(\frac{\partial\mu_j}{\partial T}\right){p,x_k}+\left(\frac{\partial N_i}{\partial T}\right){\mu_k}$$ (eqn:dndt_2)

From classical bulk thermodynamics we know $\left(\frac{\partial\mu_j}{\partial T}\right)_{p,x_k}=-s_j^\mathrm{b}$ and therefore, eq. {eq}eqn:dndt_2 can be used in eq. {eq}eqn:dndt_1 to give

$$0=\sum_j\left(\frac{\partial N_i}{\partial\mu_j}\right)_T\left(s_j^\mathrm{b}+\left(\frac{\partial\mu_j}{\partial T}\right)_{N_k}\right)+\left(\frac{\partial N_i}{\partial T}\right)_{p,x_k}$$

After multiplying with $T$, the following elegant expression remains

$$0=\sum_j\left(\frac{\partial N_i}{\partial\mu_j}\right)_T\Delta h_j^\mathrm{ads}+T\left(\frac{\partial N_i}{\partial T}\right)_{p,x_k}$$

which is a symmetric linear system of equations due to $\left(\frac{\partial N_i}{\partial\mu_j}\right)_T=-\left(\frac{\partial^2\Omega}{\partial\mu_i\partial\mu_j}\right)_T$. The derivatives of the particle numbers are obtained by integrating over the respective derivatives of the density profiles which were discussed previously.