An activity-based formulation for Langmuir adsorption isotherm

Abstract

This work presents an activity-based formulation for Langmuir adsorption isotherm. Treating adsorption as a chemical reaction between the gas molecule and the adsorption vacant site, the classical Langmuir isotherm model expresses the reaction in terms of the species concentrations. Designed to capture the surface heterogeneity, the proposed thermodynamic Langmuir isotherm model substitutes the species concentrations with the species activities and calculates the species activity coefficients with the adsorption non-random two-liquid activity coefficient model. The resulting isotherm model accurately represents pure component adsorption isotherms for gases with wide varieties of adsorbents including silica gels, activated carbons, zeolites and metal organic frameworks at various temperatures. With three physically meaningful parameters, the model outperforms the classical Langmuir isotherm model for the 98 isotherms of 33 systems examined.

1 Introduction

Adsorption is widely practiced in industrial processes for molecule separations by taking advantages of the difference in adsorbate molecule affinity to adsorbents (Li et al. 2009). To support process research and development, many researchers have pursued development of empirical or semi-empirical engineering correlations or models for both pure component and mixed-gas adsorption equilibria (Myers and Prausnitz 1965; Mathias et al. 1996; Myers 2005; Talu and Zwiebel 1986; Walton and Sholl 2015). Successful engineering models for adsorption equilibria are expected to (1) be thermodynamically consistent, (2) require few adjustable model parameters, (3) be applicable to both pure component adsorption isotherms and mixed-gas adsorption isotherms, and (4) calculate mixed-gas adsorption isotherms from pure component adsorption isotherms (Sircar 1991). While adsorption equilibria of a single gas on an adsorbent represents the simplest case of adsorption processes, accurate correlation of pure component adsorption isotherms remains a challenge due to adsorbent surface heterogeneity (Sircar 1991).

The classical Langmuir isotherm model (Langmuir 1918) is considered the first scientifically sound expression for pure component adsorption isotherms:

$$n_{i} = n_{i}^{0} \frac{KP}{1 + KP}$$

(1)

where $n_{i}$ is the adsorption amount of gas component i; $n_{i}^{0}$ is the adsorption maximum amount; $P$ is the gas vapor pressure. Indicative of the affinity between adsorbate and adsorbent, $K$ is the apparent adsorption equilibrium constant. The Langmuir isotherm has been successfully used to describe adsorption behavior of many systems such as adsorption of non-polar gases on activated carbons and zeolites. Ignoring the surface heterogeneity and the van der Waals interactions between adsorbates and adsorbents (Sreńscek-Nazzal et al. 2015; Foo and Hameed 2010), the Langmuir isotherm may be inadequate in describing pure component adsorption isotherms especially at low temperature and high pressure regions (Benard and Chahine 1997). (see Supporting Information. Figs. S1 and S2 for examples).

Among the many efforts (Sips 1948, 1950; Toth 1971) to improve upon the classical Langmuir isotherm model, the empirical Sips isotherm model (Sips 1948, 1950) probably is the most successful one. Following Freundlich isotherm (Freundlich 1907), Sips introduced an empirical “heterogeneity” parameter m, which is usually less than unity (Pakseresht et al. 2002), to the Langmuir isotherm. Shown in Eq. 2, the resulting Sips isotherm expression is much more flexible in representing adsorption isotherm data.

$$n_{i} = n_{i}^{0} \frac{{\left( {KP} \right)^{m} }}{{1 + \left( {KP} \right)^{m} }}$$

(2)

With three adjustable parameters ($n_{i}^{0}$, $K$ and $m)$, the Sips isotherm expression and other similar empirical expressions are capable of correlating pure component adsorption isotherm data much better than the Langmuir isotherm could achieve with two adjustable parameters ($n_{i}^{0}$ and $K$). However, the introduction of empirical heterogeneity parameter $m$ distorts the theoretical basis of the classical Langmuir isotherm and the physical significance of the Langmuir isotherm parameters ($n_{i}^{0}$ and K) is lost.

Instead of pursuing empirical corrections of the classical Langmuir isotherm to address the issue of surface heterogeneity, this work re-examines the theoretical basis of the Langmuir isotherm and proposes an activity-based formulation for the isotherm. The reformulation is achieved by substituting the concentrations of both the vacant sites and the occupied sites with the site activities. Specifically, the surface heterogeneity is treated as a departure from ideal adsorbate phase solution. The reference state for the vacant sites is at zero surface coverage while the reference state for the occupied sites is at full surface coverage. The site activities are further calculated with the adsorption non-random two-liquid (aNRTL) activity coefficient model (Kaur et al. 2019). Derived from the two fluid theory (Renon and Prausnitz 1968; Ravichandran et al. 2018) and the assumption that the adsorbate phase nonideality is dominated by adsorbate-adsorbent interactions, the aNRTL model has been shown to successfully correlate and predict wide varieties of mixed-gas adsorption isotherms with a single binary interaction parameter per adsorbate–adsorbate pair. The resulting activity-based Langmuir isotherm, called “thermodynamic Langmuir isotherm” in this work, should represent a theoretically rigorous refinement of the classical Langmuir isotherm. The model parameters include $n_{i}^{0}$, the adsorption maximum, $K^\circ$, the thermodynamic adsorption equilibrium constant, and τ, the aNRTL binary interaction parameter.

The subsequent sections present the formulation of the thermodynamic Langmuir isotherm, the adsorption NRTL activity coefficient model, and the model results for 98 pure component adsorption isotherms for adsorbents including silica gels, activated carbons, zeolites and metal organic frameworks (MOFs). Also presented are the results with the classical Langmuir isotherm and the Sips isotherm. Lastly, the physical interpretation of the thermodynamic Langmuir isotherm model parameters is discussed.

2 Theory

2.1 Thermodynamic Langmuir isotherm

The classical Langmuir adsorption isotherm equation is derived from reaction kinetics (Sohn and Kim 2005). Suppose there is an adsorption and desorption reaction of pure gas $A$:

$$A_{\left( g \right)} + S \leftrightarrow AS$$

(3)

where $S$ stands for the vacant sites and $AS$ the occupied sites with gas $A$. When this reaction reaches chemical equilibrium state at pressure $P$, the rates of adsorption and desorption are the same.

$$k_{a} P\left[ S \right] = k_{d} \left[ {AS} \right]$$

(4)

where $k_{a}$ is the rate constant of adsorption, $k_{d}$ is the rate constant of desorption, $\left[ S \right]$ is the vacant site concentration, and [$AS$] is the occupied site concentration. The apparent chemical equilibrium constant, $K,$ can be written as:

$$K = \frac{{k_{a} }}{{k_{d} }} = \frac{{\left[ {AS} \right]}}{P\left[ S \right]} = \frac{{n_{1} }}{{\left( {n_{1}^{0} - n_{1} } \right)P}} = \frac{{x_{1} }}{{\left( {1 - x_{1} } \right)P}}$$

(5)

where $n_{1}$ stands for the adsorption amount of adsorbed gas component 1, $n_{1}^{0}$ stands for the adsorption maximum, and $x_{1}$ stands for the adsorption extent, i.e., the ratio of $n_{1}$ and $n_{1}^{0}$. Langmuir isotherm equation, Eq. 1, can be obtained after solving for $x_{1}$. Note that here we denote gas $A$ and gas component 1 interchangeably.

The Langmuir isotherm assumes the adsorption and desorption rates are proportional to the concentrations of vacant sites and occupied sites respectively. In other words, the model ignores the “heterogeneity” of the adsorption sites and the apparent chemical equilibrium constant, $K$, should be a function of the surface coverage, or the adsorption extent, $x_{1}$. To account for the “heterogeneity” of the adsorption sites and to achieve a rigorous thermodynamic formulation of Langmuir isotherm, this work substitutes the site concentrations in Eq. 5 with the site activities, i.e., the product of site concentration and site activity coefficient. See Eq. 6.

$$K^\circ = \frac{{k_{a} }}{{k_{d} }} = \frac{{a_{AS} }}{{Pa_{S} }} = \frac{{\gamma_{1} x_{1} }}{{\gamma_{\phi } \left( {1 - x_{1} } \right)P}}$$

(6)

here $K^\circ$ is the thermodynamic adsorption equilibrium constant, $a_{AS}$ is the activity of the occupied sites with adsorbed gas $A$, $a_{S}$ is the activity of the vacant sites, and $\gamma_{1}$ and $\gamma_{\phi }$ are the activity coefficient of the occupied sites with adsorbed gas component 1 and the activity coefficient of the vacant sites, respectively. The reference state for the occupied sites with adsorbed gas component 1 is chosen to be at full surface coverage, i.e., saturated adsorption state with $x_{1} = 1$. The reference state for the vacant sites is chosen to be at zero surface coverage, i.e., the vacant adsorption state with $x_{1} = 0$. In other words, $\gamma_{1} = 1$ at $x_{1} = 1,$ and $\gamma_{\phi } = 1$ at $x_{1} = 0.$

Reformulating Eq. 6, one obtains the following implicit adsorption isotherm expression.

$$n_{1} = n_{1}^{0} \frac{{K^\circ \gamma_{\phi } P}}{{\gamma_{1} + K^\circ \gamma_{\phi } P}}$$

(7)

here $\gamma_{1}$ and $\gamma_{\phi }$ are functions of $x_{1}$. The relationship between the thermodynamic adsorption equilibrium constant $K^\circ$ and the apparent adsorption equilibrium constant $K$ is shown in Eq. 8.

$$K\left( {x_{1} } \right) = K^\circ \frac{{\gamma_{\phi } \left( {x_{1} } \right)}}{{\gamma_{1} \left( {x_{1} } \right)}}$$

(8)

Equation 7 is referred to as the “thermodynamic Langmuir isotherm model”. The classical Langmuir isotherm is recovered if both the activity coefficients of the occupied sites and the vacant sites are unity. However, the surface heterogeneity suggests there are vacant sites with stronger adsorption potential and vacant sites with weaker adsorption potential. It is expected that the vacant sites with stronger adsorption potential should be occupied before the sites with weaker adsorption potential. Therefore, the activity coefficient of vacant sites should start with unity at zero surface coverage (reference state) and decline and deviate from unity as the adsorption extent increases. To the contrary, the activity coefficient of occupied sites should increase and approach unity as the adsorption proceeds to full surface coverage (reference state). In other words, we expect negative deviations from ideal solution behavior for both the vacant sites and the occupied sites.

2.2 The adsorption NRTL activity coefficient model

The aNRTL model activity coefficient expressions (Kaur et al. 2019) for two competing adsorbate components 1 and 2 on the adsorbate phase are as follows.

$$\ln \gamma_{1} = x_{2}^{2} \left[ {\tau_{12} \frac{{\left( {G_{12} - 1} \right)}}{{\left( {x_{2} + x_{1} G_{12} } \right)^{2} }}} \right]$$

(9a)

$$\ln \gamma_{2} = x_{1}^{2} \left[ {\tau_{21} \frac{{\left( {G_{21} - 1} \right)}}{{\left( {x_{1} + x_{2} G_{21} } \right)^{2} }}} \right]$$

(9b)

with

$$G_{12} = \exp \left( { - \alpha \tau_{12} } \right)$$

(10a)

$$G_{21} = \exp \left( { - \alpha \tau_{21} } \right)$$

(10b)

and

$$\tau_{12} = - \tau_{21} = \frac{{g_{10} - g_{20} }}{RT}$$

(11)

where $g_{10}$ is the interaction potential between adsorbate 1 and adsorbent 0, $g_{20}$ is the interaction potential between adsorbate 2 and adsorbent 0, $R$ is gas constant, $T$ is temperature, and $\alpha$ is the non-randomness parameter. Following the convention of NRTL model (Renon and Prausnitz 1968), $\alpha$ is fixed at 0.3 in this study. $\tau_{12}$ is the binary interaction parameter for the pair of adsorbates 1 and 2.

To apply the adsorption NRTL model, we follow the concept of “competition” between two adsorbate components 1 and 2 in mixed-gas adsorption equilibria. Specifically, we consider pure component adsorption equilibria as a “competition” between adsorbate component 1 and a phantom molecule $\phi$. In other words, while the occupied sites are covered with adsorbate component 1, the vacant sites are “covered” with phantom molecule $\phi$. Therefore, the adsorption NRTL model becomes

$$\ln \gamma_{1} = x_{\phi }^{2} \left[ {\tau_{1\phi } \frac{{\left( {G_{1\phi } - 1} \right)}}{{\left( {x_{\phi } + x_{1} G_{1\phi } } \right)^{2} }}} \right]$$

(12a)

$$\ln \gamma_{\phi } = x_{1}^{2} \left[ {\tau_{\phi 1} \frac{{\left( {G_{\phi 1} - 1} \right)}}{{\left( {x_{1} + x_{\phi } G_{\phi 1} } \right)^{2} }}} \right]$$

(12b)

with

$$G_{1\phi } = \exp \left( { - \alpha \tau_{1\phi } } \right)$$

(13a)

$$G_{\phi 1} = \exp \left( { - \alpha \tau_{\phi 1} } \right)$$

(13b)

and

$$\tau_{1\phi } = - \tau_{\phi 1} = \frac{{g_{10} - g_{\phi 0} }}{RT}$$

(14)

where $x_{\phi } = 1 - x_{1} ,$ and $g_{10}$ and $g_{\phi 0}$ are the interaction potential between component 1 and adsorbent 0 and the “interaction potential” between phantom molecule $\phi$ and adsorbent 0, respectively. Conceptually, $g_{\phi 0}$ may be considered as the potential field for the vacant sites.

As to be shown later, the binary interaction parameter $\tau_{1\phi }$ are found to be in the range of 0 to − 5 for our test systems. The activity coefficients show negative deviation from ideality and the negative deviation increases as $\tau_{1\phi }$ becomes more negative, suggesting stronger attractive interaction between the adsorbate and the adsorbent (i.e., more negative $g_{10}$). Figure 1 illustrates the variations in activity coefficients with the adsorption extent as functions of $\tau_{1\phi }$. $\gamma_{1}$ shows negative deviation from unity in the beginning of adsorption process (weaker desorption potential) and approaches unity when the adsorption reaches saturation (reference state for the occupied sites). $\gamma_{\phi }$ shows an opposite trend from that of the occupied sites. $\gamma_{\phi }$ is unity in the beginning of adsorption process (reference state for the vacant cites) and then exhibits negative deviation from unity as the adsorption extent approaches saturation (weaker adsorption potential).

Fig. 1

Site activity coefficients as functions of adsorption extent with different $\tau_{1\phi }$ ($\alpha =$ 0.3): $\tau_{1\phi } =$-1 (dashed line), $\tau_{1\phi } =$-2 (dotted dashed line) and $\tau_{1\phi } =$-3 (solid line); blue lines stand for activity coefficients of occupied sites with adsorbate gas ‘1’ and red lines stand for activity coefficients of vacant sites with phantom molecule ‘φ’

3 Results and discussion

We examine the model performance in correlating data for 98 selected pure component adsorption isotherms with the classical Langmuir isotherm model, the semi-empirical Sips isotherm model, and the thermodynamic Langmuir isotherm model. There are two adjustable parameters ($n_{i}^{0}$ and $K$) with the Langmuir isotherm, three adjustable parameters ($n_{i}^{0}$, $K$ and $m)$ with the Sips isotherm, and three adjustable parameters ($n_{i}^{0} , K^\circ$ and $\tau_{1\phi } )$ with the thermodynamic Langmuir isotherm.

The Maximum Likelihood Objective Function (Britt and Luecke 1973) is adopted in the regression of adsorption isotherm data. Specifically, the sum of square of the ratio of the difference between calculated $n_{i}$ and experimental $n_{i}$ to the expected standard deviation $\sigma^{expt}$ (set to 0.05 mmol/g in this study) is minimized by adjusting the corresponding isotherm parameters.

$$Obj = \mathop \sum \limits_{i} \left( \frac{n_{i}^{calc} - n_{i}^{expt}} {\sigma^{expt} } \right)^{2}$$

(15)

where $Obj$ is the objective function; superscripts $calc$ and $expt$ stand for calculated value and experimental data, respectively.

We use root mean square error ($RMS$) to evaluate the performance of the three isotherm models. The $RMS$ is defined as following:

$$RMS = \sqrt {\frac{{\mathop \sum \nolimits_{i} \left( {n_{i}^{calc} - n_{i}^{expt} } \right)^{2} }}{N}}$$

(16)

where $N$ is the number of data points for the adsorption isotherm.

Table 1 shows the corresponding $RMS$ values with the models. Figure 2a and b show the $RMS$ values for the isotherms with the new model plotted against those with the Langmuir isotherm and those with the Sips isotherm respectively. The results with the new model are superior to those with the Langmuir isotherm as all of the $RMS$ data points are located in the lower right half corner of Fig. 2a. The new model is comparable to the Sips isotherm as Fig. 2b shows the $RMS$ data points are mostly centered around the 45° line.

Table 1 Comparison of root mean square errors among Langmuir, Sips and thermodynamic Langumir

Fig. 2

Comparison of RMS with different models: a thermodynamic Langmuir compared to Langmuir, b thermodynamic Langmuir compared to Sips

Figures 3a to c present the model results for CO₂, CH₄ and N₂ in zeolite 5A, respectively. Figure 3a shows the Langmuir isotherm fails to accurately describe the CO₂-zeolite 5A isotherm at 348.15 K (Wang and LeVan 2009) while the Sips isotherm and the new model fit the experimental data very well. All three models are able to fit the experimental data accurately for CH₄ and N₂ adsorption isotherms with zeolite 5A (Bakhtyari and Mofarahi 2014), as shown in Figs. 3b and c respectively. Figure 3d further shows the Langmuir isotherm fails to describe the CH₄ adsorption isotherm with activated carbon (Reich et al. 1980) while the isotherm is well represented with both the Sips isotherm and the thermodynamic Langmuir isotherm.

Fig. 3

Comparison of adsorption isotherms with different models: a CO₂/zeolite 5A (Wang and LeVan 2009) at 348.15 K, b CH₄/zeolite 5A (Bakhtyari and Mofarahi 2014) at 343 K, c N₂/zeolite 5A (Bakhtyari and Mofarahi 2014) at 343 K and d CH₄/activated carbon (Reich et al. 1980) at 212.7 K; experimental data (open circle), Langmuir (dotted blue line), Sips (dashed red line), and thermodynamic Langmuir (green solid line)

Tables S1 to S3 of Supporting Information report the regressed model parameters for Langmuir, Sips and the new model respectively. From the regressed parameters for Langmuir and for Sips, it becomes obvious that the Langmuir $n_{i}^{0}$ and $K$ parameters can be distorted significantly when the “heterogeneity” parameter $m$ is introduced in the Sips isotherm. The distortion is particularly pronounced when $m$ is far from unity. Take CO₂ adsorption with activated carbon (Zhang et al. 2013) as an example, with $m \approx$ 0.85, the Sips $n_{i}^{0}$ values are 4 to 6 times of the Langmuir $n_{i}^{0}$ values while the Sips $K$ values are one order of magnitude less than that of the Langmuir K values.

By contrast, the thermodynamic Langmuir $n_{i}^{0}$ and $K^\circ$ remain in line with the Langmuir $n_{i}^{0}$ and $K$. In fact, the thermodynamic Langmuir $K^\circ$ is an intrinsic quantity and it is related to the Langmuir $K$ with Eq. 8. Figure 4a and b show comparisons of the thermodynamic Langmuir $\ln K^\circ$ and the Langmuir $\ln K$ for N₂ adsorption with zeolite 5A (Bakhtyari and Mofarahi 2014) and CH₄ adsorption with activated carbon (Reich et al. 1980) respectively. While the thermodynamic Langmuir $\ln K^\circ$ remains constant at a given temperature, the Langmuir $\ln K$ decreases with the adsorption extent. It is worth noting that the ${{\tau }}_{1\phi }$ is near zero for the N₂/zeolite 5A system, the Langmuir $\ln K$ deviates only slightly from the thermodynamic Langmuir $\ln K^\circ$, and the classical Langmuir should be able to capture the isotherm data well. To the contrary, the absolute value of $\tau_{1\phi }$ is significantly larger for the CH₄/activated carbon system, the Langmuir $\ln K$ deviates significantly from the thermodynamic Langmuir $\ln K^\circ$, and the classical Langmuir would fail to describe the adsorption isotherm.

Fig. 4

$\ln \left ( {{K}/{K^\circ }} \right)$ of a N₂/zeolite 5A (Bakhtyari and Mofarahi 2014) at 273 K (solid line), 303 K (thin dashed line), and 343 K (dotted line); b CH₄/activated carbon (Reich et al. 1980) at 212.7 K (thick dashed line), 260.2 K (thick dashed line with dot), and 301.4 K (thick dashed line with double dots)

Given the thermodynamic Langmuir $n_{i}^{0}$ and $K^\circ$, one may define a thermodynamic driving force for adsorption, or adsorption strength $\eta$, as the product of $n_{i}^{0}$ and $K^\circ$.

$$\eta = n_{i}^{0} K^\circ$$

(16)

Figure 5a to c show the adsorption strength for CH₄, CO₂ and N₂ in various adsorbents respectively. The adsorption strength declines as temperature increases. $\eta$ could be an effective measure to select adsorbents for a given separation task since the unit of $\eta$ is adsorption amount per adsorbent unit mass per unit pressure. In other words, $\eta$ has the same unit as the Henry’s constant $H$. The relation between the Henry’s constant $H$ and the adsorption strength $\eta$ can be obtained from Eq. 7 when the pressure is approaching zero:

$$H = \frac{\eta }{{\gamma_{1} \left( {P \to 0} \right)}} = \frac{\eta }{{\gamma_{1}^{\infty } }}$$

(17)

where $\gamma_{1}^{\infty }$ is the infinite dilution activity coefficient and always less than or equal to unity. Different from the Henry’s constant, the adsorption strength is a measure of the intrinsic adsorption potential of the isotherm while the Henry’s constant is a measure valid only at low pressures. Given $\eta$, for example, zeolite 5A is the strongest of the adsorbents shown in Fig. 5b for CO₂ adsorption.

Fig. 5

Adsorption strength of a CH₄, b CO₂, and c N₂ in different adsorbents; silica gel (pink asterisk), activated carbon (green times), zeolite 5A (red open diamond), zeolite 13X (orange open square), Cu-BTC (purple open triangle), UiO-66 (blue open circle), and Zn-MOF (plus)

While the new model is successful in capturing adsorption behavior of most systems, Table 1 shows that the thermodynamic Langmuir isotherm is not able to capture well the experimental data for systems with Cu-BTC MOF (Ferreira et al. 2011; Al-Janabi et al. 2015). The identified ${\tau }_{1\phi }$’s for these systems are all around zero, suggesting ideal solution behavior. The Sips isotherm is able to correlate the data slightly better, albeit with the Sips parameter $m$ greater than unity. Figures 6a and b present the isotherms for C₃H₈ and i-C₄H₁₀ adsorption with Cu-BTC (Ferreira et al. 2011) respectively. These isotherms show near step change behavior in reaching saturation. For these systems, the thermodynamic Langmuir isotherm initially overpredicts and then underpredicts $n_{i}$ at the low adsorption region, and it predicts relatively well at the high adsorption region. To the contrary, the Sips isotherm predicts well at the low adsorption region but underpredicts at the high adsorption region.

Fig. 6

Adsorption isotherm of a C₃H₈, b i-C₄H₁₀ in Cu-BTC at 348 K; experimental data (open circle), Langmuir (dotted blue line), Sips (dashed red line), thermodynamic Langmuir (solid green line). Note that the Langmuir line and the thermodynamic Langmuir line overlap each other

Figure 7a and b show the ratio of the observed apparent adsorption equilibrium constant to the thermodynamic adsorption equilibrium constant, $\ln K/K^\circ$, calculated from each of the isotherm data point for C₃H₈ and i-C₄H₁₀ systems in Cu-BTC (Ferreira et al. 2011) respectively. For both systems, the observed $\ln K/K^\circ$ values jump in the beginning of adsorption and then quickly reach a constant value of 0 as pressure increases. One plausible explanation is that the adsorption data points at low pressure (< 0.1 bar) may be subject to higher relative uncertainty although the literature did not report the corresponding uncertainty. If the first adsorption data point at very low pressure is removed, the thermodynamic Langmuir clearly captures the isotherm data of Cu-BTC systems very well.

Fig. 7

Ratio of thermodynamic adsorption equilibrium constant and observed apparent adsorption equilibrium constant for a C₃H₈ and b i-C₄H₁₀ adsorption with Cu-BTC at 348 K (Ferreira et al. 2011)

4 Conclusion

A thermodynamic Langmuir isotherm model is proposed by introducing the concept of activity and activity coefficient to the classical Langmuir isotherm. With three physically meaningful parameters, i.e., adsorption maximum amount $n_{i}^{0}$, thermodynamic adsorption equilibrium constant $K^\circ ,$ and binary interaction parameter $\tau_{1\phi }$, the model accurately describes the 98 isotherms of 33 tested adsorption systems. Based on these three parameters, we further propose an adsorption strength, the product of $n_{i}^{0}$ and $K^\circ ,$ as an intrinsic measure for selecting adsorbents for a given gas adsorption task. The proposed model is superior to the classical Langmuir isotherm model and should be very useful in accurate correlation of pure component adsorption isotherms and subsequent correlation and prediction of mixed-gas adsorption isotherms. Future work will report predictions on enthalpy of adsorption and mixed-gas adsorption equilibria from pure component adsorption isotherms represented with the thermodynamic Langmuir isotherm model.