Development of Equations for Differential and Integral Enthalpy

ARTICLE pubs.acs.org/Langmuir

Development of Equations for Differential and Integral Enthalpy Change of Adsorption for Simulation Studies D. D. Do,* D. Nicholson, and Chunyan Fan School of Chemical Engineering, University of Queensland, St. Lucia, Queensland 4072, Australia ABSTRACT: We present equations to calculate the differential and integral enthalpy changes of adsorption for their use in Monte Carlo simulation. Adsorption of a system of N molecules, subject to an external potential energy, is viewed as one of transferring these molecules from a reference gas phase (state 1) to the adsorption system (state 2) at the same temperature and equilibrium pressure (same chemical potential). The excess amount adsorbed is the difference between N and the hypothetical amount of gas occupying the accessible volume of the system at the same density as the reference gas. The enthalpy change is a state function, which is defined as the difference between the enthalpies of state 2 and state 1, and the isosteric heat is defined as the negative of the derivative of this enthalpy change with respect to the excess amount of adsorption. It is suitable to determine how the system behaves for a differential increment in the excess phase adsorbed under subcritical conditions. For supercritical conditions, use of the integral enthalpy of adsorption per particle is recommended since the isosteric heat becomes infinite at the maximum excess concentration. With these unambiguous definitions we derive equations which are applicable for a general case of adsorption and demonstrate how they can be used in a Monte Carlo simulation. We apply the new equations to argon adsorption at various temperatures on a graphite surface to illustrate the need to use the correct equation to describe isosteric heat of adsorption.

1. INTRODUCTION The term “heat of adsorption” is used to quantify the temperature dependence of adsorption and its use in designing a heat exchanger. Several different heats have been defined in the literature; the most commonly used one in practice is the isosteric heat.110 The description “heat” has been criticized by some workers8,10 who suggested that a more thermodynamically appropriate term would be the differential enthalpy change of adsorption. The subject of this article is to standardize the way in which we define the thermodynamic variables; to achieve this goal we begin by defining the adsorbed phase (by confining all excess molecules on a dividing surface) and the excess variables of adsorption, and this allows us to develop equations for the enthalpy change of adsorption in both differential and integral forms. This sequence of steps is necessary because the equation for enthalpy change depends on how the adsorbed phase, the accessible volume, and the excess amount adsorbed are defined. The equations developed here are presented in a suitable form for use in molecular simulation of adsorption because of the increasing use of simulation, especially the Grand Canonical Monte Carlo (GCMC), in the description of many adsorption systems.1116 2. THEORY In the adsorption process we are interested in the enthalpy change between two states which can be defined as follows (Figure 1). r 2011 American Chemical Society

1 State 1 (the reference state): N molecules of bulk homogeneous gas at T0, V0, and p0 (we use the superscript 0 to denote the reference system). The reference state could be chosen in a number of different ways; for example, it can be chosen as the usual standard state of 273 K and 1 bar. However, in the context of adsorption, it is more convenient to choose this to be the same state as the gas phase in equilibrium with the adsorption system, i.e., same temperature and pressure, T0 = T and p0 = p, where p is the equilibrium pressure of a gas phase in equilibrium with the adsorbate (i.e., the chemical potential is constant between the adsorption system and the gas phase). This state is shown as the first box on the left in Figure 1. 2 State 2: The same N molecules in the presence of the adsorbent of total accessible volume V in equilibrium with the gas phase (the middle box in Figure 1). The pressure is that of the external bulk surrounding gas at constant chemical potential. The accessible volume is defined as the volume in which the adsorbentadsorbate (SF) potential between a molecule and the adsorbent is nonpositive as the region where the SF potential is positive is essentially inaccessible to molecules, except of course at extremely high pressures or temperatures. Received: September 8, 2011 Revised: October 16, 2011 Published: October 20, 2011 14290

dx.doi.org/10.1021/la203531j | Langmuir 2011, 27, 14290–14299

Langmuir

ARTICLE

Figure 1. Schematic diagram of two states for the enthalpy change and decomposition of the adsorption system into the adsorbed (excess) phase and the hypothetical gas phase occupying the accessible volume.

We shall adopt the accessible volume as the basis on which to calculate the excess amount. Since all interfaces are diffuse at T > 0, it is not possible to locate a geometrical surface that can be physically described as ‘the interface’. Gibbs overcame this problem by the proposal that a geometrical interface can be defined as a surface, such that the deficit in concentration on one side of the interface for a component in the system exactly balances the excess of that component on the other side of the interface. The concentration of any other component in the system with respect to this interface is then referred to as the ‘surface excess’ of that component. The choice of the first component with zero surface excess is, of course, entirely arbitrary. In a system containing a solid adsorbent and an adsorbate the obvious location for the dividing surface is the plane where the excess of the solid is zero. Since the surface atoms vibrate about fixed lattice points, which can sensibly be identified with the mean position of the nuclei of the surface atoms, it involves no serious approximation to take the dividing surface as passing through these lattice points. In many simulation studies of adsorption, for example, the solid adsorbent is modeled as an array of immobile atoms and this interface is located as a surface passing through the atom centers at the adsorbent boundary. The excess of the adsorbate component would then be measured with respect to this surface, often referred to as defining the physical volume of the system. Unfortunately, this physical volume is difficult to locate accurately by experiment, and often this boundary is chosen to be the surface where the excess amount of helium is zero. However, the He volume has neither geometrical nor physical significance; its appeal lies in the fact that it can be linked to an experimental procedure to define the reference interface. Figure 2 shows the helium void volume, physical volume, Connolly volume (defined as the volume enclosed by a sphere rolling over the solid atoms), and accessible volume as functions

of temperature and confirms that the helium void volume has no geometrical significance and does not agree with any of these volumes. In Figure 3 we illustrate the concept of accessible volume and the dividing surface by considering the singlet density distribution at a flat surface. The horizontal dotted line in the figure is the density of the gas in equilibrium with the system, and the vertical dashed line is the dividing surface defined by the accessible volume. The local density oscillates around the bulk gas density and can be either positive (e.g., point A) or negative (point B) with respect to the gas density as zero. There are two particular positions where the local excess density is zero: one is the position where the solidfluid potential energy is zero (at the dividing surface) and the other at positions far away from the surface where the SF potential energy is effectively zero. When the adsorbate is confined in a pore, the local density is similar to that shown in Figure 4. In this case there is no position inside the pore space and away from the repulsive wall, where the SF potential goes to zero; the horizontal dotted line corresponds to the density in the external gas phase. Here, we have two dividing surfaces (vertical dashed lines), one at each wall of the pore. The local excess density is also either greater or smaller than the gas-phase density, and there are two positions close to the adsorbent where the SF potential energy passes through zero and the local density is equal to the gas density. The accessible volume is defined by the region between the two dividing surfaces. R RThe total number of molecules in the system is N = VF(r)dr = A LF(z)dz in these examples, i.e., the area under the singlet density distribution. The hypothetical number in the accessible volume VG is NG = FGVG, and the adsorption excess is Nex ¼ N NG ¼ N FG VG 14291

ð1Þ

dx.doi.org/10.1021/la203531j |Langmuir 2011, 27, 14290–14299

Langmuir

ARTICLE

Figure 2. He void volume as a function of temperature in a graphitic slit pore of width 1.5 nm and lengths of 4.6 and 4.4 nm. The physical volume is the volume that passes through the centers of solid atoms on the outermost layer of the pore walls. The Connolly surface is smaller than the physical volume by a thickness of one-half a collision diameter, and the accessible volume is the volume in which the SF potential energy is nonpositive.

Figure 3. Local density profile from the surface. The dashed line is the dividing surface where the SF potential energy is zero, and the dotted line is the concentration of the bulk gas that is in equilibrium with the adsorption system. The accessible volume is the volume of the region from the dividing surface. Point A has positive local excess concentration, and point B has a negative local excess concentration.

where VG is the accessible volume and FG is the density of the bulk gas. The surface excess density based on the area of the boundary at the zero of the SF potential, Aex, is Γex ¼

Nex Aex

ð2Þ

We can define adsorbate density in the pore in at least two ways based on the accessible volume. One is the absolute

pore density F¼

N VG

ð3aÞ

and the other is the excess pore density Fex ¼

Nex VG

ð3bÞ

Using the accessible volume gives a better measure of the density of the actual confined fluid. 14292


Langmuir

ARTICLE

Figure 4. Local density profile in a slit pore. The dashed lines are the dividing surfaces where the SF potential energy is zero, and the dotted line is the concentration of the bulk gas that is in equilibrium with the adsorption system, i.e., equal chemical potential between the external gas phase and the adsorption system. The accessible volume is the volume between the two dividing surfaces.

We summarize the following definitions which will be important in the subsequent development of the enthalpy of adsorption. terminology

a surface at which the SF potential energy is zero

accessible volume, VG

volume in which the SF potential energy is

inaccessible volume,

volume in which the SF potential energy is

nonpositive VS Aex

V ¼ VG þ VS

definition

dividing surface

dividing surface area,

definitions of these volumes that are separated by the boundary where the SF potential energy is zero

We shall identify the inaccessible volume as the volume occupied by the solid adsorbent, but note that the volume of the solid adsorbent is arbitrary and includes all the solid atoms outside the space defined by the Gibbs dividing surface (GDS). The excess volume is defined as Vex ¼ V VG VS

positive area of the boundary separating the accessible volume and the inaccessible volume

hypothetical amount of gas, NG

number of particles occupying the accessible volume at the density of the bulk gas which

excess amount, Nex

difference between the number in the system

surface excess density,

excess amount per unit dividing surface area

is in equilibrium with the adsorption system and the hypothetical amount, NG γex absolute pore density,

amount in the system per unit accessible

F excess pore density,

volume excess amount per unit accessible volume

Uex ¼ U UG US reference gas of N molecules having the temperature and chemical potential with the adsorption system

We use the subscripts G, ex, and S for the gas phase in the accessible volume, the excess (adsorbed phase), and the solid phase, respectively, of the adsorption system. We use the superscript 0 to denote the reference gas. The volume of the adsorption system, V, is the sum of the accessible volume and the inaccessible volume by virtue of the

ð5Þ

in which we used eq 4. The zero excess volume arises because of the geometrical and energetic (based on SF potential energy) definitions of the accessible volume, VG, and the inaccessible (solid) volume, VS. This means that we confine all excess (adsorbed) quantities to the dividing surface separating the accessible and inaccessible volumes. Thus, the excess phase has zero volume (eq 5) but a surface area of Aex. Note that for a surface with curvature this surface area is not the same as the Connolly surface area or the surface passing through the centers of all solid atoms on the outermost layer (which is the boundary of the physical volume). Similarly, we can define the excess internal energy as

Fex reference state, state 1

ð4Þ

ð6aÞ

Here, U is the internal energy of the whole system, UG is the internal energy of NG molecules of adsorptive in the gas phase that would occupy the accessible volume at the same density as the equilibrium bulk gas, and US is the internal energy of the solid. It is important to note that this excess energy is a calculated variable and depends on the way that we define the reference system, which is the hypothetical gas occupying the accessible volume, having an internal energy UG. If the solid is inert to the presence of adsorbate, we can set US = 0. Likewise, we can define the excess enthalpy, the excess 14293


Langmuir

ARTICLE

Helmholtz free energy, and the excess entropy as follows

equation of state, i.e.

Hex ¼ H HG HS

ð6bÞ

Fex ¼ F FG FS

ð6cÞ

Sex ¼ S SG SS

ð6dÞ

FG ¼

N NG ¼ ¼ gðp, TÞ 0 V VG

ð11Þ

Substituting eq 10 into eq 9 we obtain the change in the internal energy as ΔU ¼ Uex ðNex Þ ½UG0 ðNÞ UG ðNG Þ

ð12Þ

When N molecules are brought into contact with the surface in an adsorption system of (1) constant accessible volume, VG, (2) constant dividing surface area, Aex, and (3) constant temperature, T, the Helmholtz free energy of the system must be minimized, or in other words the change of the free energy with respect to the change of the excess amount must be zero at constant N. ∂F ¼0 ð7Þ ∂Nex T, VG , Aex , N

where we clearly see that the change of the internal energy between state 1 and state 2 is simply the change on moving Nex particles from the gas phase at T and p to the adsorbed (excess) phase, since the bracketted term on the right-hand side of eq 12 is the energy of Nex particles in the gas phase. Similarly, the change of the enthalpy by transferring N molecules from state 1 to state 2 is

Assuming that the Helmholtz free energy of the solid is unchanged (see above), eq 7 will become ∂Fex ∂FG ¼ ð8Þ ∂Nex T, Aex ∂NG T, VG

Once again using the concept of the excess phase we can write the enthalpy of state 2 as the sum of the excess enthalpy and the enthalpy of the hypothetical gas, i.e.

in which we used dN = dNex + dNG = 0 at constant N. The lefthand side of eq 8, the differential of the excess Helmholtz free energy of the excess phase of zero volume, is the chemical potential in the excess phase, and the right-hand side is the chemical potential in the gas phase. Equation 8 is a statement of the equilibrium between the gas phase and the excess phase. The surface excess density Γex = Nex/Aex is a function of the concentration of the hypothetical gas occupying the accessible volume VG, FG = NG/VG, which is the same as the density of the reference gas. The change in the internal energy on transferring N molecules from state 1, which is the reference gas of N molecules, to state 2, which is the adsorption system, is ΔU ¼ UðNÞ UG0 ðNÞ

HðNÞ ¼ Hex ðNex Þ þ HG ðNG Þ

ð10Þ

Here, UG is the internal energy of the hypothetical gas (of NG particles) which is in equilibrium with the excess phase. Equation 10 is the definition of the internal energy of the adsorbed (excess) phase. Note that the sum of the number of particles in the adsorbed (excess) phase, Nex, and the hypothetical gas phase, NG, is equal to N, N = Nex + NG. Since the reference state is chosen to have the same pressure and temperature as the equilibrium gas in equilibrium with the adsorbate, these two states will have the same density and the relationship between p, T, and N follows the

ð13Þ

ð14Þ

Since the excess volume is zero, the excess enthalpy Hex = Uex + pVex is equal to the excess internal energy, and by substituting eq 14 into eq 13 we get the following equation for the change of the enthalpy in transferring N molecules from state 1 to state 2 ΔH ¼ Uex ðNex Þ ½HG0 ðNÞ HG ðNG Þ

ð15Þ

The bracketed term on the right-hand side of the above equation is the enthalpy of Nex particles in the gas phase at T and p. Combining eqs 12 and 15 to get the expression for the enthalpy change in terms of the change in the internal energy and pV terms ΔHðNÞ ¼ ΔUðNÞ þ pðVG V 0 Þ

ð9Þ

where U0G(N) is the internal energy of the reference bulk gas of N molecules (state 1) and U(N) is the internal energy of the adsorption system (state 2), which can be decomposed into two parts: (1) the adsorbed (excess) phase and (2) the hypothetical gas phase that is in equilibrium with the adsorbed phase, occupying the accessible volume of the system. The density and pressure of the gas phase are FG and p, respectively. These two parts are shown in the right part of Figure 1. From eq 6a with Us = 0 we can write the internal energy of the adsorption system, U(N), as the sum of the internal energies of the excess phase and the accessible volume UðNÞ ¼ Uex ðNex Þ þ UG ðNG Þ

ΔH ¼ HðNÞ HG0 ðNÞ

ð16Þ

because H0G(N) = U0G(N) + pV0 and HG(NG) = UG(NG) + pVG. This is the enthalpy change in bringing N molecules of gas from a bulk gas state of p, V0, and T to a system containing an adsorbent. Adsorption is viewed as the process of moving one molecule from the gas phase to the excess phase, and in this sense the excess phase is an open system with mass exchange; therefore, enthalpy is the correct thermodynamic variable to study excess adsorption at the dividing surface. Let λ be the reduced chemical potential, λ = μ/kT, where μ is the equilibrium chemical potential of the reference gas and the adsorption system. A small increment in the reduced chemical potential, δλ, that results in a change in the number of particles in both states is N + δN, and the number of particles of the hypothetical gas is NG + δNG and in the number of particles in the excess phase Nex + δNex. The following equations must be satisfied (by virtue of mass balance and the same density between the reference gas and the hypothetical gas in the adsorption system)

14294

δN ¼ δNG þ δNex

ð17aÞ

V0 N N þ δN ¼ ¼ NG NG þ δNG VG

ð17bÞ


Langmuir

ARTICLE

The latter equation leads to δN N V0 ¼ ¼ δNG NG VG

ð18Þ

for a grand canonical ensemble,17 the denominator is δNG/δλ = f(NG,NG)1, where f(A,B) = ÆA Bæ ÆAæÆBæ. Substituting these into eq 24, we get the appropriate form for I1 suitable for a GCMC simulation

The increment in the change of the enthalpy between state 1 and state 2 associated with the incremental change in the chemical potential is δðΔHÞ ¼ ΔHðN þ δNÞ ΔHðNÞ

ð19aÞ

δðΔHÞ ¼ ΔUðN þ δNÞ ΔUðNÞ þ ðVG V 0 Þδp

ð19bÞ

or

Because of the small increment of the chemical potential, the number of particles in the excess phase of the adsorption system is increased by δNex and we can define the isosteric heat as the negative of the increment in the enthalpy change with respect to the increment number of particles entering the adsorbed (excess) phase,6,8 i.e. qst ¼

δðΔHÞ δNex

ð20Þ

Substituting eq 19b into the above equation gives qst as the sum of three differential terms qst ¼ J3 þ J2 þ J1

ð21Þ

where the three terms on the right-hand side are given by δp J1 ¼ ðV 0 VG Þ ð22aÞ δNex 1 " J2 ¼ J3 ¼

U ðN þ δNÞ U ðNÞ δNex 0

0

UðN þ δNÞ UðNÞ δNex

J1 ¼

ð22bÞ

ð22cÞ 2

We use subscripts 1 and 2 in the right-hand side of eqs 22a22c to denote the reference gas and the adsorption system, respectively. The first term in eqs 22a22c is the incremental change of the pressure in the reference gas with an increase in δNex particles in that phase. This increase in δNex particles is associated with an increase of δN particles in the reference box and δNG particles in the accessible volume of the adsorption system. It can be written as δp δp 0 ¼ VG ð23Þ J1 ¼ ðV VG Þ δN δNG 1 δNG 1 in which we used eqs 17a, 17b, and 18. The increment in the pressure and in NG is brought about by the increment in the chemical potential; therefore, we can write eq 23 to reflect the increments due to an increment in the chemical potential as follows δp=δλ ð24Þ J1 ¼ VG δNG =δλ 1

ð25aÞ

The factor in front of kT on the right-hand side of the above equation is calculated by carrying out a GCMC simulation of the bulk gas phase at the same chemical potential as the adsorption system with some arbitrary volume Vb large enough to contain a sufficient number of particles to ensure good statistics. From this simulation we obtain ÆNbæ1 and f(Nb,Nb)1 and eq 25a can be replaced by J1 ¼

ÆNb æ1 kT f ðNb , Nb Þ1

ð25bÞ

The second term in eq 22b is the change of the internal energy of the reference gas due to an increase in δNex in the number of particles. This increase is associated with an increase of δN particles in the reference box and δNG particles in the accessible volume of the adsorption system. It can be written as " # U 0 ðN þ δNÞ U 0 ðNÞ J2 ¼ δN δNG 1

"

# V0 U 0 ðN þ δNÞ U 0 ðNÞ ¼ 0 δN V VG

# 1

ÆNG æ kT f ðNG , NG Þ

ð26Þ 1

in which we used eqs 17a, 17b, and 18. The changes in the numerator and denominator of the above equation are brought about by the increase in the chemical potential. We write ( ) V0 ½U 0 ðN þ δNÞ U 0 ðNÞ=δλ J2 ¼ 0 ð27Þ δN=δλ V VG 1

Applying the fluctuation formula applicable for the bulk gas at constant chemical potential we get J2 ¼

V0

V0 f ðU 0 , NÞ1 VG f ðN, NÞ1

ð28aÞ

From the GCMC simulation results of the bulk gas of volume Vb we can rewrite the above equation in the form which is suitable for computation J2 ¼

V0 f ðUb , Nb Þ1 0 V VG f ðNb , Nb Þ1

ð28bÞ

The last term, eq 22c, is the change in the internal energy of the adsorption system when the number in the system is increased by δN, associated with an increase of δN G and δN ex in the accessible volume and the excess phase, respectively.

From the GibbsDuhem equation, Vdp = Ndμ, at constant T, it follows that (∂p/∂λ)T = FkT. Applying the fluctuation formula

J3 ¼ 14295

UðN þ δNÞ UðNÞ δNex

¼ 2

½UðN þ δNÞ UðNÞ2 ðδNÞ2 ðδNG Þ1

ð29Þ


Langmuir

ARTICLE

Note that the increment in the number of gas particles in the accessible volume is related to the increment in the reference gas. Therefore, for an increment in the chemical potential, we can write the above equation as J3 ¼

½UðN þ δNÞ UðNÞ2 =δλ ðδNÞ2 =δλ ðδNG Þ1 =δλ

f ðU, NÞ2 f ðN, NÞ2 f ðNG , NG Þ1

ð31aÞ

Once again, we use the results from the GCMC simulation of a bulk phase of volume V b and the above equation becomes f ðU, NÞ2 J3 ¼ f ðN, NÞ2 ðVG =Vb Þf ðNb , Nb Þ1

ÆNb æ1 kT þ f ðNb , Nb Þ1

This equation differs from eq 32. The difference between these equations is the molecular internal energy of the gas, which was missing in our earlier work. This term is practically negligible for subcritical adsorptions, but it can be very significant for supercritical conditions and subcritical conditions close to the critical point. We will discuss this when presenting simulation results under supercritical conditions. The extended equation (eq 32) must also be compared with the following equation that frequently appears in the literature qst ¼ kT

ÆNæ ÆNb æ

ð32Þ

ð33Þ

The significance of our equation is seen as follows. The first term with its factor in front of kT describes the nonideality of the gas. For an ideal gas the coefficient of kT is unity. The second term has two factors; the first is the volume term and the second is the molecular internal energy of the gas. By rearranging eq 32 in the following way qst ¼

ÆNb æ1 kT f ðNb , Nb Þ1 "

ÆNb æ1 ¼1 f ðNb , Nb Þ1

ð38aÞ

lim

f ðUb , Nb Þ1 ¼0 f ðNb , Nb Þ1

ð38bÞ

f ðN, NÞ2 > f ðNG , NG Þ1

ð38cÞ

Nb f 0

The volume V 0 of the reference gas is calculated from V 0 ¼ Vb

f ðU, NÞ2 ÆNG æub þ ub f ðN, NÞ2 ðVG =Vb Þf ðNb , Nb Þ1 ÆNex æ

ð37Þ

lim

Nb f 0

V0 f ðUb , Nb Þ1 VG f ðNb , Nb Þ1

f ðU, NÞ2 f ðN, NÞ2 ðVG =Vb Þf ðNb , Nb Þ1

f ðU, NÞ f ðN, NÞ

which is only valid for subcritical conditions far away from the critical point where

!

V0

ÆNb æ1 f ðU, NÞ2 f ðNb , Nb Þ1 f ðN, NÞ2 ðVG =Vb Þf ðNb , Nb Þ1 ð36Þ

ð31bÞ

Finally, we substitute eqs 25b, 28b, and 31b into eq 21 and obtain the working equation for the isosteric heat suitable for a grand canonical ensemble qst ¼

qst ¼ kT

ð30Þ

Applying the fluctuation formulas to the grand canonical ensemble of the adsorption system (2) and the bulk gas (1) we can write J3 ¼

Recently, Do et al.18 developed the equation for isosteric heat given below

2.1. Supercritical Adsorption. Under supercritical conditions (T > Tc), adsorption isotherms, expressed in terms of the excess density versus pressure, exhibit a maximum at a pressure, Pm, which is greater than the critical pressure. At this pressure, the number fluctuation in the adsorption system is the same as that in the accessible gas space, i.e., f(N,N) = f(NG,NG). This means that the isosteric heat is infinite at this point, resulting from the mathematical definition of the isosteric heat. At this point the change in particle number in the system is the same as that in the

#

ð34aÞ where u b is the molecular internal energy of the gas given by f ðUb , Nb Þ1 ub ¼ ð34bÞ f ðNb , Nb Þ1 Our equation is consistent with the following equation developed by Rouquerol et al. 8 (p 38) ∂Uex qst ¼ kT þ ub ð35Þ ∂Nex T, Aex where the three terms on the right-hand side correspond to the three terms in our eq 34a, but eq 35 is only valid for an ideal gas adsorptive.

Figure 5. Plots of the isosteric heats versus the excess density for argon adsorption on a graphite surface at 87.3 K. 14296


Langmuir

ARTICLE

accessible gas. Indeed, from eq 1 dNex dN dNG ¼0 ¼ dp dp dp

ð39Þ

number of particles in the excess phase. Starting from eq 19b we take its derivative with respect to the reduced chemical potential

Pm

To avoid an infinite isosteric heat under supercritical conditions we consider instead the molar integral enthalpy of adsorption, which is the change of the enthalpy of adsorption from vacuum conditions to a given pressure per unit

δðΔHÞ ΔUðN þ δNÞ ΔUðNÞ δp ¼ þ ðVG V 0 Þ δλ δλ δλ

ð40Þ

δðΔHÞ ¼ J1 þ J2 þ J3 δλ

ð41Þ

or

where the three terms on the right-hand side are given by δp 0 J1 ¼ ðV VG Þ ¼ ðV 0 VG ÞFG kT ð42aÞ δλ 1 "

U 0 ðN þ δNÞ U 0 ðNÞ J2 ¼ δλ

# ¼ f ðU 0 , NÞ1 1

¼ ðV =Vb Þf ðUb , Nb Þ1

ð42bÞ

0

UðN þ δNÞ UðNÞ J3 ¼ δλ

Figure 6. Plots of the contributions of the three terms in eq 32 for argon adsorption on a graphite surface at 87.3 K.

¼ f ðU, NÞ2

ð42cÞ

2

We use subscripts 1 and 2 in the right-hand side of eqs 42a, 42b, and 42c to denote the reference gas and the adsorption system, respectively. Therefore, the overall change of the

Table 1. Various Equations for the Calculation of Isosteric Heat in Grand Canonical equation

formula

eq no.

f ðU, NÞ f ðN, NÞ

37

traditional equation

qst ¼ kT

Do et al.’s18

ÆNb æ1 f ðU, NÞ2 kT f ðNb , Nb Þ1 f ðN, NÞ2 ðVG =Vb Þf ðNb , Nb Þ1 ! ÆNb æ1 V f ðUb , Nb Þ1 f ðU, NÞ2 kT þ qst ¼ f ðNb , Nb Þ1 V VG f ðNb , Nb Þ1 f ðN, NÞ2 ðVG =Vb Þf ðNb , Nb Þ1

This paper

qst ¼

36 32

Figure 7. Plots of the isosteric heats versus the excess density for argon adsorption on a graphite surface at 130 K. (Left) Comparison between the three heat equations, and (right) contributions of the three terms of the new heat equation (eq 32) developed in this paper. 14297


Langmuir

ARTICLE

work on adsorption on a graphite surface19 and in cylindrical carbon nanotubes.20

Figure 8. Effects of the box size on the isosteric heat calculated with the traditional equation. Lz’s of the small, medium, and large boxes are 10σ, 20σ, and 100σ, respectively.

Figure 9. Effects of the box size on the isosteric heat calculated with equation (eq 32). Lz’s of the small, medium, and large boxes are 10σ, 20σ, and 100σ, respectively.

change in the enthalpy from ∞ (i.e., zero pressure) to λ is ΔðΔHÞ ¼

Z λ ∞

ðJ1 þ J2 þ J3Þdλ

ð43Þ

The excess number of particles at the chemical potential λ is ÆN ex æ; therefore, we can define the integral molar enthalpy of adsorption based on unit excess number of particle as ΔðΔHÞ 1 Z λ ¼ ðJ1 þ J2 þ J3Þdλ ÆNex æ ÆNex æ ∞

ð44Þ

For a given chemical potential, eqs 42a, 42b, and 42c can be computed from a GCMC simulation at a series of chemical potentials. The molar integral enthalpy of adsorption can then be obtained from the numerical integration of eq 44. This equation does not suffer from the singularity encountered in the isosteric heat when maximum excess density occurs under supercritical conditions. This has been illustrated in our previous

3. RESULTS AND DISCUSSION Figure 5 shows plots of isosteric heat versus excess density for argon adsorption on a graphite surface at 87.3 K for the three equations for isosteric heat, summarized in Table 1. The dimensions of the simulation box are Lx = Ly = 10σ and Lz = 10σ, where σ is the collision diameter of argon, σ = 0.3405 nm. The well depth of the interaction energy of argon is ε/k = 119.8 K. One surface of the box is a graphite plane, and the opposite surface is a hard wall. It is no surprise that the three equations give the same results for this system, because at this temperature the adsorptive is sufficiently rarefied to behave like an ideal gas, for which the molecular internal energy is negligibly small compared to qst (dashed-dotted line in Figure 6) and the coefficient of kT in eqs 36 and 32 is unity. To show the importance of the nonideality and molecular internal energy of the bulk gas, we present in Figure 7 the plots of the isosteric heat of argon adsorption at 130 K. Here, we see a significant difference between our equations and eq 37. The discrepancy between the two equations lies in the molecular internal energy of the adsorptive gas and the contribution of the second term in the denominator of the third term in eq 32. To further show the importance of the number of particles in the accessible volume we plot in Figure 8 the isosteric heat versus loading using eq 37 for argon adsorption at 130 K for three different simulation box heights. The greater the box height, the smaller the isosteric heat, which is incorrect because the heat of adsorption should not depend on the size of the gas space above the surface. On the other hand, when we use the correct equation for the isosteric heat (eq 32), the isosteric heat does not depend on the box size, as one would expect physically. This is shown in Figure 9. 4. CONCLUSIONS We presented a new equation for calculating the isosteric heat of adsorption in a simulation that accounts for the nonideality of the gas phase, the molar internal energy, and the contribution of the gas space in the simulation box. The equation is presented in a form suitable for GCMC simulation, and we illustrate its use with simulations of argon adsorption on a graphitic surface at a range of temperatures. The equation most commonly used for calculation of isosteric heats is shown to fail for nonideal adsorptives and to show an unphysical dependence on the size of the simulation box. ’ AUTHOR INFORMATION Corresponding Author

*Fax: +61-7-3365-2789. E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by the Australian Research Council. ’ REFERENCES (1) Hill, T. Statistical mechanics of adsorption. IX. J. Chem. Phys. 1950, 18, 246. 14298


Langmuir

ARTICLE

(2) Young, D.; Crowell, A. D. Physical adsorption of gases; Butterworths: London, 1962. (3) Ross, S.; Olivier, J. On physical adsorption; Interscience Publishers: New York, 1964. (4) Everett, D. H. Manual of symbols and terminology for physical quantities and units. Pure Appl. Chem. 1972, 31, 579. (5) Steele, W. A. The interaction of gases with solid surfaces; Pergamon Press: New York, 1974. (6) Rouquerol, J.; Everett, D. H. Gas solid interactions. General derivation of reaction enthalpies from the data of isothermal microcalorimetry. Thermochim. Acta 1980, 41, 311. (7) Lopatkin, A. A. Theoretical Foundations of Physical Adsorption (in Russian); Moscow University Press: Moscow, 1983. (8) Rouquerol, F.; Rouquerol, J.; Sing, K. S. W. Adsorption by powders and porous solidsPrinciples, methodology and applications; Academic Press: New York, 1998. (9) Myers, A. L. Thermodynamics of adsorption in porous materials. AIChE J. 2002, 48, 145. (10) Myers, A. L.; Monson, P. Adsorption in porous materials at high pressure: Theory and experiment. Langmuir 2002, 18, 10261. (11) Ohba, T.; Kanoh, H.; Kaneko, K. Cluster growth induced water adsorption in hydrophobic carbon nanopores. J. Phys. Chem. B 2004, 108, 14964. (12) Ohba, T.; Kaneko, K. Internal surface area evaluation of carbon nanotube with GCMC simulation assisted N2 adsorption. J. Phys. Chem. B 2002, 106, 7171. (13) Babarao, R.; Jiang, J.; Sandler, S. Molecular Simulations for Adsorptive Separation of CO2/CH4 Mixture in Metal-Exposed, Catenated, and Charged Metal-Organic Frameworks. Langmuir 2009, 25, 5239. (14) Furmaniak, S.; Terzyk, A.; Gauden, P.; Wesolowski, R.; Kowalczyk, P. Ar, CCl4, and C6H6 adsorption outside and inside of the bundles of multi-walled carbon nanotubes—simulation study. Phys. Chem. Chem. Phys. 2009, 11, 4982. (15) Mota, J.; Esteves, I. Simplified gauge-cell method and its application to the study of capillary phase transition of propane in carbon nanotubes, Adsorption 2007, 13, 21. (16) Ravikovitch, P.; Vishnyakov, A.; Neimark, A.; Carrott, M.; Russo, P.; Carrott, P. Characterization of Micro-Mesoporous Materials from Nitrogen and Toluene Adsorption: Experiment and Modeling. Langmuir 2006, 22, 513. (17) Nicholson, D.; Parsonage, N. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (18) Do, D. D.; Do, H. D.; Nicholson, D. Molecular Simulation of Excess Isotherm and Excess Enthalpy Change in Gas-Phase Adsorption. J. Phys. Chem. B 2009, 113, 1030. (19) Fan, C. Y.; Do, D. D.; Nicholson, D.; Li, Z. Computer Simulation of Argon Adsorption on Graphite Surface from Subcritical to Supercritical Conditions: The Behaviour of Differential and Integral Molar Enthalpies of Adsorption. Langmuir 2010, 26, 15852. (20) Liu, Z.; Do, D. D.; Nicholson, D. Effects of Confinement on the Molar Enthalpy of Argon Adsorption in Graphitic Cylindrical Pores: A Grand Canonical Monte Carlo (GCMC) Simulation Study. J. Colloid Interface Sci. 2011, 361, 278.

14299


Development of Equations for Differential and Integral Enthalpy

Recommend Documents