THEORY
4.5 Different Types of Amount Adsorbed
where, Mw is the molar mass of the gas, P is the pressure, T is the temperature, R is the gas constant and Bgas is the gas phase second virial coefficient. The usual temperature dependency for Bgas is taken as
1 2 33 84 95 T
B T
B T
B T B B
Bgas = + + + +
4.5 The values of Bi for the gases used in this study are tabulated in Table 4.2.
Table 4.2: Second virial coefficients for different gases [200].
Gas B1×102 B2×10-1 B3×10-5 B4×10-15 B5×10-17 m3 kmol-1 m3 kmol-1 K m3 kmol-1 K3 m3 kmol-1 K8 m3 kmol-1 K9
He 1.400 -3.540 -5.950E-06 3.610E-13 -7.940E-15
CO2 5.440 -3.635 -14.960 85.900 -139.700
CH4 5.438 -2.714 -2.135 0.920 -0.785
CO 5.122 -1.709 -0.742 0.046 -0.029
N2 4.670 -1.495 -0.611 0.081 -0.046
Ar 3.805 -1.518 -0.808 0.185 -0.110
O2 3.900 -1.554 -0.848 0.164 -0.115
C2H6 8.095 -6.171 -14.350 67.600 -97.400
C3H8 11.250 -10.000 -43.140 -18.000 -165.000
Fugacity was used instead of pressure to handle with the gas phase non-ideality at higher pressures [191]. It was obtained from the virial equation for the bulk gas phase using
=
RT P B P
f gas
ln 4.6
dividing surface. The common reference states used in adsorption literature are given in Figure 4.2 [196].
Figure 4.2: (a) Equilibrium between Adsorbate and Adsorbent; (b) Reference state for Absolute Adsorption; (c) Reference state for Excess Adsorption; (d) Reference state for net Adsorption Framework [196].
Absolute Adsorption
Figure 4.2 (b) shows the reference state for absolute adsorption. The gas molecules exist in the non-shaded “available” volume of the tank at the same conditions (temperature and pressure) as that in Figure 4.2 (a). Thus, the density of molecules in the non-shaded region of this figure is same as that of the bulk density in case (a). The shaded black region is considered to be unavailable for the gas at the reference state conditions. The difference between the number of gas molecules between the two cases (a) and (b) is called as the absolute amount adsorbed. On a normalized basis, the difference per unit mass of adsorbent is the so called specific absolute amount adsorbed.
Excess Adsorption
Figure 4.2 (c) (bottom left) shows the reference state for excess adsorption. The gas molecules exist in the non-shaded “available” volume of the tank at the same conditions (temperature and pressure) as that in Figure 4.2 (a). Thus, the density of molecules in the non-shaded region (including that inside the pores) of this Figure 4.2 (c) is same as that of the bulk density in case (equilibrium). The shaded black region is the so called “impenetrable solid volume” that is unavailable for the gas. The difference between the number of gas molecules between the two cases (a) and (c) is called as the excess amount adsorbed. Thus, excess amount is equivalent to the number of additional gas that can be accommodated in the “penetrable” region of the tank due to adsorption. By far this is the most commonly used reference state in adsorption literature.
Net Adsorption
Figure 4.2 (d) shows the reference state for net adsorption. The gas molecules exist in the total volume of the tank at the same conditions (temperature and pressure) as that in Figure 4.2 (a).
Note that in this reference state the total volume of the tank is considered available to the gas unlike in the earlier two cases. Further, the density of molecules in the tank at the reference state is same as that of the bulk density in case (a). The difference between the number of gas molecules between the two cases (a) and (d) is called as the net amount adsorbed. Thus, net amount is equivalent to the number of additional gas that can be accommodated in the total volume of the tank (as opposed to that in the penetrable volume only for case of excess) due to adsorption.
4.5.1 Determination of Buoyancy Volume for Various Reference States
In order to calculate the amount adsorbed based on any one (absolute, excess or net) reference states, it is necessary to correct the microbalance signal with appropriate buoyancy correction term as described in Eq. 4.1.
The buoyancy volume for net adsorption is the buoyancy acting on the bucket alone (excluding the sample). This is usually obtained via Eq. 4.1, by using a convenient gas to record the microbalance signal of the bucket only (in the absence of any sample) at several densities of the surrounding bulk gas. In the absence of adsorbent sample, the change in microbalance signal is solely due to buoyancy of the bucket assembly (irrespective of the gas used) and hence LHS of Eq. 4.1 is zero. The slope of microbalance signal against bulk gas density is thus directly related to the buoyancy volume (of the bucket in this case, Vbucket). Argon was used as probe gas to obtain buoyancy of the bucket for data reported in this work. In net adsorption only bucket experience buoyancy force therefore
bucket net
buyoancy
V
V
,=
4.7 In case of excess adsorption the impenetrable solid volume also experiences the buoyancy force ands bucket excess
buyoancy
V V
V
,= +
4.8 where Vs is the impenetrable solid volume. The buoyancy volume is calculated again from the change in microbalance signal on the bucket and sample assembly. However, since a porous solid sample adsorbs the probe gas, LHS of Eq. 4.1 is non-zero unlike in the previous case when only the bucket is used to measure the buoyancy volume. LHS of Eq. 4.1 is zero in presence of a sample only if the sample does not adsorb the probe gas. Hence, helium is conventionally chosen as a non-adsorbing probe gas to obtain impenetrable solid volume (and hence Vbuoyancy for excess adsorption). With non-adsorbing helium assumption the LHS of Eq. 4.1 is zero; the slope of microbalance signal against helium density is related to the buoyancy volume for excess adsorption.In case of absolute adsorption the impenetrable solid volume as well as the pores also experiences the buoyancy force. Thus the absolute amount adsorbed at a given condition is the amount of gas present in the solid adsorbent including that in the “pores”.
p s bucket absolute
buyoancy
V V V
V
,= + +
4.9 where Vp is the pore volume of the adsorbent.