Solid adsorbent/gas interface: characteristic thermodynamic quantities of adsorption

The following definitions refer to the solid/gas interface. Changes in enthalpy and entropy associated with adsorption are usually attributed to changes in the thermodynamic state of the adsorbate only. It should be borne in mind, however, that the measured changes include contributions from the perturbation of the adsorbent.

Differential energy of adsorption

When the addition of a differential amount d $n^{\sigma}_i$ or d$n^s_i$ is effected at constant gas volume, the differential molar energy of adsorption of component $i$, ${\Delta}_aU^{\sigma}_i$ or ${\Delta}_aU^s_i$, is defined as:

$${\Delta}_aU^{\sigma}_i=U^{\sigma}_i-U^g_i,$$    or $${\Delta}_aU^s_i=U^s_i-U^g_i,$$

where the differential molar surface excess energy, $U^{\sigma}_i$, is given by ¹⁸

$$U^{\sigma}_i=\left(\frac{{\partial}U^{\sigma}}{{\partial}n^{\sigm... ...t(\frac{{\partial}U}{{\partial}n^{\sigma}_i}\right)_{T,m,V^g,p_i,n^{\sigma}_j},$$

and the differential molar interfacial energy, $U^s_i$, by

$$U^s_i=\left(\frac{{\partial}U^s}{{\partial}n^s_i}\right)_{T,m,V^s,n^s_j} =\left(\frac{{\partial}U}{{\partial}n^s_i}\right)_{T,m,V^g,V^s,p_i,n^s_j}.$$

$U^g_i$ is the differential molar energy of component $i$ in the gas phase, i.e. $({\partial}U^g/{\partial}n^g_i)_{T,V,n^g_i}$.

Differential enthalpy of adsorption

When the addition of the differential amount d $n^{\sigma}_i$ or d$n^s_i$ is effected at constant pressure $p$, the differential molar enthalpy of adsorption, ${\Delta}_aH^{\sigma}_i$ or ${\Delta}_aH^s_i$, also called the isosteric enthalpy of adsorption ( $q^$st)¹⁹ is defined as²⁰

$$\begin{array}{rclcl} {\Delta}_aH^{\sigma}_i&=&-q^{\text{st},{... ..._aH^s_i&=&-q^{\text{st},s}&=&\mathscr{H}^s_i-H^g_i; \end{array}$$

where $\mathscr{H}^s_i=({\partial}\mathscr{H}^s/{\partial}n^s_i)_{T,p,m,n^s_j}$, and $H^g_i$ is the partial molar enthalpy of component $i$ in the gas phase, i.e. $({\partial}H^g/{\partial}n^g_i)_{T,p,n^g_j}$.

${\Delta}_aU^{\sigma}_i$ and ${\Delta}_aH^{\sigma}_i$ are related by the equation:

$${\Delta}_aU^{\sigma}_i-{\Delta}_aH^{\sigma}_i = pV^g_m{\approx}RT,$$

and the same applies to the difference between ${\Delta}_aU^s_i$ and ${\Delta}_aH^s_i$ when $V^s/n^s_i$ is negligibly small compared with $V^g_m$.

When an excess $n^{\sigma}$ of a single adsorptive is adsorbed on a surface initially free of adsorbate species, the molar integral energy and molar integral enthalpy of adsorption are given by

$${\Delta}_aU^{\sigma}_m=\frac{1}{n^{\sigma}}{\int}_0^{n^{\sigma}}{\Delta}_aU^{\sigma}$$d$$n^{\sigma}$$ and $${\Delta}_aH^{\sigma}_m=\frac{1}{n^{\sigma}}{\int}_0^{n^{\sigma}}{\Delta}_aH^{\sigma}$$d$$n^{\sigma}.$$

Experimental calorimetric methods have to be analysed carefully to establish the appropriate procedure for deducing a particular energy or enthalpy of adsorption from measured data. The isosteric enthalpy of adsorption is usually calculated from adsorption isotherms measured at several temperatures by using the equation

$$\left(\frac{{\partial}\ln p}{{\partial}T}\right)_{n^{\sigma}}=\frac{q^{\text{st},{\sigma}}}{pV^g_mT} {\approx}\frac{q^{\text{st},{\sigma}}}{RT^2},$$

where $p$ is the equilibrium partial pressure of the adsorptive when an amount $n^{\sigma}$ is adsorbed at a temperature $T$.

Standard thermodynamic quantities. For different purposes it may be convenient to define standard changes of a thermodynamic quantity on adsorption in two alternative ways:

(i)

the change of the thermodynamic quantity on going from the standard gas state to the adsorbed state in equilibrium with gas at a partial pressure (fugacity) of $p_i$ $(f_i)$. Such quantities are sometimes called `half-standard' quantities.

(ii)

the change of the thermodynamic quantity on going from the standard gas state to a defined standard condition of the adsorbed state.

It must always be stated clearly which of these conventions is being followed.

The standard Gibbs energy of adsorption is thus:

$${\Delta}_a{\mu}^{\ominus}_i=RT\ln[f_i/f^{\ominus}_i],$$

where $f^{\ominus}_i$ is the fugacity of $i$ in the standard gas state and $f_i$ the fugacity of gas in equilibrium with the (standard) adsorbed state. For most practical purposes the fugacity may be replaced by the (partial) pressure.

If unit pressure is chosen for the standard gas state, this may be written

$${\Delta}_a{\mu}^{\dagger}_i=RT\ln[f_i/f^{\dagger}_i],$$

while if vapour in equilibrium with pure liquid $i$ is chosen²¹, then

$${\Delta}_a{\mu}^*_i=RT\ln[f_i/f^*_i].$$

Similarly, the standard differential molar entropy of adsorption is given by

$${\Delta}_aS^{\ominus}_i=\frac{1}{T}({\Delta}_aH^{\ominus}_i-{\Delta}_a{\mu}^{\ominus}_i),$$

where ${\Delta}_aH^{\ominus}_i$ is the standard differential molar enthalpy of adsorption.

The standard integral molar entropy of adsorption is

$${\Delta}_aS^{\ominus}_m=\frac{1}{n^{\sigma}_i}{\int}_o^{n^{\sigma}_i}{\Delta}_aS^{\ominus}_i$$d$$n^{\sigma}_i.$$

The above definitions refer to equilibrium conditions, and their applicability in regions where adsorption hysteresis occurs is open to doubt.