Isotherms | Skarstrom Docs

Isotherm Models

The isotherm describes the equilibrium adsorbed-phase loading of a component, $q_i^*$ , as a function of the local gas state (temperature, pressure and composition). It is the thermodynamic ceiling that the kinetic rate model drives the actual loading $q_i$ towards.

Each component in each adsorbent material is assigned its own isotherm. In the GUI, open the Adsorbent Material window and select the Isotherms tab; for every component choose an Isotherm Type from the drop-down. The parameter fields shown below the drop-down change to match the selected model, and a live equation preview is displayed. Tick Show Isotherm Plot to visualise the resulting curve.

Partial pressure vs. concentration. Several models come in two forms. Partial-pressure models are written in terms of the partial pressure $P_i = y_i P$ . Concentration models use the molar concentration $C_i = P_i / (R T)$ , computed internally from the ideal gas law. The two forms are otherwise identical; choose whichever matches how your isotherm parameters were regressed.

Heat of adsorption. Each component also has an isosteric heat of adsorption $\Delta H_i$ , entered on the Heat of Adsorption tab of the Adsorbent Material window. It couples the isotherm to the energy balance; it is a separate input and is not part of the isotherm equation itself. The temperature-dependent parameters in the models below (the $q$ , $B_2$ , $\Delta H_0$ terms) describe how loading shifts with temperature but do not by themselves supply the heat released on adsorption.

The sections below give the exact formula implemented for each model together with the parameter names as they appear in the GUI. $R$ is the ideal gas constant and $T$ the temperature.

Inert

A non-adsorbing component. Its loading is fixed at zero, so it acts purely as a carrier gas.

q_i^* = 0

Henry

The linear (low-coverage) limit, with a temperature-dependent Henry constant.

q_i^* = k \, \exp\!\left(\frac{Q}{R T}\right) P_i

Symbol	Description	Unit
$q_i^*$	Equilibrium loading of component $i$	mol kg⁻¹
$k$	Henry coefficient	mol kg⁻¹ Pa⁻¹
$Q$	Adsorption energy parameter	J mol⁻¹
$P_i$	Partial pressure of component $i$	Pa

Single Site Langmuir

The classic monolayer model, available on a partial-pressure or concentration basis.

Partial pressure form (Single Site Langmuir):

q_i^* = \frac{m_1 \, b_1 \exp\!\left(\frac{q_1}{R T}\right) P_i}{1 + b_1 \exp\!\left(\frac{q_1}{R T}\right) P_i}

Concentration form (Single Site Langmuir (Concentration)) — replace $P_i$ with $C_i = P_i/(RT)$ :

q_i^* = \frac{m_1 \, b_1 \exp\!\left(\frac{q_1}{R T}\right) C_i}{1 + b_1 \exp\!\left(\frac{q_1}{R T}\right) C_i}

Symbol	Description	Unit (P-form / C-form)
$m_1$	Saturation capacity	mol kg⁻¹
$b_1$	Affinity pre-factor	Pa⁻¹ / m³ mol⁻¹
$q_1$	Adsorption energy	J mol⁻¹

Dual Site Langmuir

Two independent Langmuir sites, for heterogeneous surfaces. Available on a partial-pressure or concentration basis.

Partial pressure form (Dual Site Langmuir):

q_i^* = \frac{m_1 \, b_1 \exp\!\left(\frac{q_1}{R T}\right) P_i}{1 + b_1 \exp\!\left(\frac{q_1}{R T}\right) P_i} + \frac{m_2 \, b_2 \exp\!\left(\frac{q_2}{R T}\right) P_i}{1 + b_2 \exp\!\left(\frac{q_2}{R T}\right) P_i}

The concentration form (Dual Site Langmuir (Concentration)) substitutes $C_i$ for $P_i$ in both terms.

Symbol	Description	Unit
$m_1, m_2$	Saturation capacity of each site	mol kg⁻¹
$b_1, b_2$	Affinity pre-factor of each site	Pa⁻¹ (P-form)
$q_1, q_2$	Adsorption energy of each site	J mol⁻¹

Extended (Competitive) Langmuir

The Extended variants make adsorption competitive: the loading of each component is reduced by the presence of every other adsorbing component through a shared denominator. Use these for multi-component mixtures where components compete for the same sites. They are available in single-site and dual-site forms, each on a partial-pressure or concentration basis (Extended Single Site Langmuir, Extended Dual Site Langmuir, and their (Concentration) counterparts).

Extended single-site (partial-pressure), with the sum running over every adsorbing component $j$ :

q_i^* = \frac{m_{1,i} \, b_{1,i} \exp\!\left(\frac{q_{1,i}}{R T}\right) P_i}{1 + \sum_{j} b_{1,j} \exp\!\left(\frac{q_{1,j}}{R T}\right) P_j}

The dual-site extended form adds a second competitive term with its own denominator $1 + \sum_j b_{2,j}\exp(q_{2,j}/RT)\,P_j$ . Parameters are entered per component exactly as for the ordinary Langmuir models; the coupling between components is applied automatically.

Anti-Langmuir

For systems where loading accelerates with concentration (e.g. some cooperative-adsorption cases).

q_i^* = \frac{a \, P_i}{1 - b \, P_i}

Symbol	Description	Unit
$a$	Linear coefficient	mol kg⁻¹ Pa⁻¹
$b$	Non-linearity coefficient	Pa⁻¹

Freundlich

An empirical power-law model.

q_i^* = \left(a \, P_i\right)^{1/v}

Symbol	Description	Unit
$a$	Freundlich coefficient	mol kg⁻¹ Pa⁻¹
$v$	Heterogeneity exponent	-

Sips

A Langmuir–Freundlich hybrid with a heterogeneity exponent.

q_i^* = q_{sat}\,\frac{\left(b \, P_i\right)^{1/v}}{1 + \left(b \, P_i\right)^{1/v}}

Symbol	Description	Unit
$q_{sat}$	Saturation capacity	mol kg⁻¹
$b$	Affinity coefficient	Pa⁻¹
$v$	Heterogeneity exponent	-

Sips 2 (temperature-dependent)

A Sips form with explicit temperature dependence built into the capacity, affinity and exponent, referenced to a temperature $T_{ref}$ .

b_i = B_1 \exp\!\left(\frac{B_2}{R T}\right), \qquad s_i = C_1 + C_2\!\left(\frac{T}{T_{ref}} - 1\right), \qquad q_{max} = A_1 \exp\!\left[A_2\!\left(\frac{T}{T_{ref}} - 1\right)\right]

q_i^* = q_{max}\,\frac{\left(b_i P_i\right)^{s_i}}{1 + \left(b_i P_i\right)^{s_i}}

Symbol	Description	Unit
$A_1$	Saturation capacity at $T_{ref}$	mol kg⁻¹
$A_2$	Capacity temperature coefficient	-
$B_1$	Affinity pre-factor	Pa⁻¹
$B_2$	Affinity energy	J mol⁻¹
$C_1, C_2$	Exponent and its temperature coefficient	-
$T_{ref}$	Reference temperature	K

An Extended Sips 2 competitive multi-component form is also available; it shares the denominator $1 + \sum_j (b_j P_j)^{s_j}$ across components (with a small numerical regularisation applied near zero mole fraction for solver stability).

Dual Site Sips

Two independent Sips sites.

q_i^* = q_{sat,1}\,\frac{\left(b_1 P_i\right)^{1/v_1}}{1 + \left(b_1 P_i\right)^{1/v_1}} + q_{sat,2}\,\frac{\left(b_2 P_i\right)^{1/v_2}}{1 + \left(b_2 P_i\right)^{1/v_2}}

Symbol	Description	Unit
$q_{sat,1}, q_{sat,2}$	Saturation capacity of each site	mol kg⁻¹
$b_1, b_2$	Affinity of each site	Pa⁻¹
$v_1, v_2$	Heterogeneity exponent of each site	-

Redlich–Peterson

A three-parameter model bridging Henry and Langmuir behaviour.

q_i^* = \frac{a \, P_i}{1 + \left(b \, P_i\right)^{v}}

Symbol	Description	Unit
$a$	Linear coefficient	mol kg⁻¹ Pa⁻¹
$b$	Affinity coefficient	Pa⁻¹
$v$	Exponent ( $0 < v \le 1$ )	-

Toth

A widely used heterogeneous model that reduces to Langmuir when $v = 1$ .

q_i^* = \frac{q_{sat}\, b \, P_i}{\left[1 + \left(b \, P_i\right)^{v}\right]^{1/v}}

Symbol	Description	Unit
$q_{sat}$	Saturation capacity	mol kg⁻¹
$b$	Affinity coefficient	Pa⁻¹
$v$	Heterogeneity exponent	-

Toth 2 (temperature-dependent)

A Toth form in which the capacity, affinity and exponent all vary with temperature about a reference $T_0$ .

q_{s} = q_{sat}\exp\!\left[x\left(1 - \frac{T}{T_0}\right)\right], \qquad b = b_0 \exp\!\left[\frac{\Delta H_0}{R T_0}\left(\frac{T_0}{T} - 1\right)\right], \qquad w = w_0 + \alpha\left(1 - \frac{T_0}{T}\right)

q_i^* = \frac{q_{s}\, b \, P_i}{\left[1 + \left(b \, P_i\right)^{w}\right]^{1/w}}

Symbol	Description	Unit
$q_{sat}$	Saturation capacity at $T_0$	mol kg⁻¹
$x$	Capacity temperature coefficient	-
$T_0$	Reference temperature	K
$b_0$	Affinity at $T_0$	Pa⁻¹
$\Delta H_0$	Isosteric heat parameter	J mol⁻¹
$w_0$	Exponent at $T_0$	-
$\alpha$	Exponent temperature coefficient	-

Once you have chosen an isotherm, pair it with a kinetic rate model to control how quickly the loading approaches $q_i^*$ .