Question 6.7.1: Balances on an Adsorption Process A 50.0-liter tank contains......

The minimum amount of activated carbon is required if adsorption equilibrium is achieved in the final state, so that the adsorbent holds as much CCl_{4} as it can. The strategy will be to determine
1. n from the ideal-gas equation of state.
2. y_{0} from the specified initial relative saturation.
3. p_{CCl_{4}} (the final partial pressure of CCl_{4}) = 0.001P.
4. X^{∗}_{ CCl_{4}} (the mass ratio of adsorbed CCl_{4} to carbon at equilibrium) from the Langmuir isotherm (Equation 6.7-2).

X^{*}_{i} = \frac{aK_{L}p_{i}}{1 + K_{L}p_{i}}     or     X^{*}_{i} = \frac{aK^{′}_{L}c_{i}}{1 + K^{′}_{L}c_{i}}            (6.7-2)

5. The mass of CCl_{4} adsorbed (m_{ads}) as the difference between the mass initially present in the gas ( = y_{0}nM_{CCl_{4}} ) and the mass present at the end  ( = 0.001nM_{CCl_{4}} ).
6. The mass of carbon from X^{∗}_{ CCl_{4}} and m_{ads} (m_{C} = m_{ads}X^{∗}_{ CCl_{4}} ).

Ideal-Gas Equation of State:    n= \frac{PV}{RT} = \frac{(1.00  atm)(50.0  L)}{\left(0.08206  \frac{L\cdot atm}{mol\cdot K} \right) (307  K)} = 1.98  mol

Initial Relative Saturation = 0.300:
From the Antoine equation (Table B.4), the vapor pressure of carbon tetrachloride at 34°C is p^{∗}_{ CCl_{4}} = 169 mm Hg. Consequently,

\frac{p_{ CCl_{4}}}{p^{∗}_{ CCl_{4}}(34°C)} = \frac{y_{0}P}{169  mm  Hg} = 0.300 \overset{P=760  mm  Hg}{ \left. \Large{\Longrightarrow } \right.} y_{0}= 0.0667  mol  CCl_{4}/mol

Langmuir Isotherm:
The final partial pressure of carbon tetrachloride is

p_{ CCl_{4}} = y_{0}P = 0.001 (760  mm  Hg) = 0.760  mm  Hg

From Equation 6.7-2,

X^{*}_{ CCl_{4}} = \frac{aK_{L}p_{ CCl_{4}}}{ 1+ K_{L}p_{ CCl_{4}}} \\ \left. \Large{\Downarrow} \right. \begin{matrix} a= 0.794  g  CCl_{4}/g  C \\ K_{L}= 0.096  (mm  Hg)^{-1} \\ p_{ CCl_{4}}= 0.760  mm  Hg\end{matrix} \\ X^{*}_{ CCl_{4}} = 0.0540  \frac{g  CCl_{4}  ads}{g  C}

Mass of CCl_{4} Adsorbed:

Mass of Carbon Required:        m_{C}= \frac{20.0  g  CCl_{4}  ads}{0.0540  g  CCl_{4}  ads/g  C} = \boxed{370  g  carbon}

More activated carbon than this would be put into the container, for several reasons. First, since the rate of adsorption approaches zero as the adsorbent approaches saturation, it would take an infinite amount of time for the mole fraction of CCl_{4} in the gas phase to reach 0.001. If more carbon is present, the target mole fraction would be reached in a finite amount of time (before the carbon becomes saturated). Second, the Langmuir isotherm is an approximate correlation with parameters obtained by fitting scattered experimental data, and so the estimated adsorption capacity of the adsorbent (X*) could be too high. Third, we have assumed that nothing but CCl_{4} is adsorbed on the carbon. If any oxygen, nitrogen, or other species that may be present in the gas is adsorbed, it could lower the amount of carbon tetrachloride adsorbed.