Question

At very high pressures the compressibility factor of one mole of a gas is given by:

• A. $1+\frac{\mathrm{pb}}{\mathrm{RT}}$
• B. $1-\frac{\mathrm{pb}}{\mathrm{RT}}$
• C. $1-\frac{\mathrm{b}}{\mathrm{VRT}}$
• D. $\frac{\mathrm{pb}}{\mathrm{RT}}$

Answer 1

The correct option is A

$1+\frac{\mathrm{pb}}{\mathrm{RT}}$

Explanation for correct option:

(A) $1+\frac{\mathrm{pb}}{\mathrm{RT}}$

• Compressibility Factor is also often defined as “the ratio of molar volume of gas to that of an ideal gas.”

Step 1: Vanderwaals equation

• The van der Waals equation for $1$ mole of gas is :
  $$\begin{gathered} \left(\mathrm{p}+\frac{a}{V^{\mathit{2}}}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT} \\ \mathrm{p}-\mathrm{pressure} \\ \mathrm{V}-\mathrm{volume} \\ \mathrm{T}-\mathrm{temperature} \\ \mathrm{R}-\mathrm{Universal}\mathrm{gas}\mathrm{constant} \end{gathered}$$

Step 2: High pressure neglection factor

• At very high pressure, the ratio $\frac{\mathrm{a}}{{\mathrm{V}}^2}$ can be neglected in comparison to $\mathrm{p}$, and the equation becomes :
  $$\begin{gathered} \mathrm{p}(\mathrm{V}-\mathrm{b})=\mathrm{RT} \\ \mathrm{pV}-\mathrm{pb}=\mathrm{RT} \\ \mathrm{pV}=\mathrm{RT}+\mathrm{Pb} \end{gathered}$$

Step 3: Dividing the equation by $\mathrm{RT}$

• Dividing by $\mathrm{RT}$ in both LHS and RHS, we get,

$$\begin{gathered} \frac{\mathrm{pV}}{\mathrm{RT}}=\frac{\mathrm{RT}}{\mathrm{RT}}+\frac{\mathrm{pb}}{\mathrm{RT}} \\ \mathrm{Since},\mathrm{Z}=\frac{\mathrm{pV}}{\mathrm{RT}} \\ \frac{\mathrm{pV}}{\mathrm{RT}}=1+\frac{\mathrm{pb}}{\mathrm{RT}} \end{gathered}$$

​Hence, this is the correct option.

Explanation for incorrect option:

(B) $1-\frac{\mathrm{pb}}{\mathrm{RT}}$

• This equation does not match the above equation of very high pressure. so this is an incorrect option.

(C) $1-\frac{\mathrm{b}}{\mathrm{VRT}}$

• This equation does not match the above equation of very high pressure. so this is an incorrect option.

(D) $\frac{\mathrm{pb}}{\mathrm{RT}}$

• This equation does not match the above equation of very high pressure. so this is an incorrect option.

Hence, the correct option is (A) $1+\frac{\mathrm{pb}}{\mathrm{RT}}$