Question

At a moderate pressure, the van der Waals equation is written as:
$\left[ {{\text{P + }}\dfrac{{\text{a}}}{{{{\text{V}}^{\text{2}}}}}} \right]{\text{V = RT}}$
The compressibility factor is equal to:
A) $\left( {{\text{1 - }}\dfrac{{\text{a}}}{{{\text{RTV}}}}} \right)$
B) $\left( {{\text{1 + }}\dfrac{{\text{a}}}{{{\text{RTV}}}}} \right)$
C) $\left( {{\text{1 - }}\dfrac{{{\text{RTV}}}}{{\text{a}}}} \right)$
D) $\left( {{\text{1 + }}\dfrac{{{\text{RTV}}}}{{\text{a}}}} \right)$

Answer 1

Hint: The dominant forces existing among the gases are related to the compressibility factor. The factor that describes deviation of a gas from the ideal gas behaviour is known as the compressibility factor. The compressibility factor is denoted by ${\text{Z}}$. The compressibility factor is also known as the compression factor or the gas deviation factor.

Formula Used:
${\text{Z}} = \dfrac{{{\text{PV}}}}{{{\text{RT}}}}$

Complete answer:
We know the equation for the compressibility factor is as follows:
${\text{Z}} = \dfrac{{{\text{PV}}}}{{{\text{RT}}}}$ …… (1)
Where Z is the compressibility factor,
P is the pressure of the gas,
V is the volume of the gas,
 R is the universal gas constant,
T is the temperature.
The van der Waals equation at a moderate pressure is as follows:
$\left[ {{\text{P + }}\dfrac{{\text{a}}}{{{{\text{V}}^{\text{2}}}}}} \right]{\text{V = RT}}$
Rearrange the van der Waals equation for the compressibility factor as follows:
${\text{PV + }}\dfrac{{{\text{aV}}}}{{{{\text{V}}^2}}}{\text{ = RT}}$
Divide the equation by ${\text{RT}}$ as follows:
$\dfrac{{{\text{PV}}}}{{{\text{RT}}}}{\text{ + }}\dfrac{{\text{a}}}{{{\text{RTV}}}}{\text{ = 1}}$
From equation (1), substitute $\dfrac{{{\text{PV}}}}{{{\text{RT}}}}{\text{ = Z}}$. Thus,
${\text{Z + }}\dfrac{{\text{a}}}{{{\text{RTV}}}}{\text{ = 1}}$
Rearrange the equation for the compressibility factor as follows:
${\text{Z = 1 - }}\dfrac{{\text{a}}}{{{\text{RTV}}}}$
Thus, the compressibility factor is equal to $\left( {{\text{1 - }}\dfrac{{\text{a}}}{{{\text{RTV}}}}} \right)$.

Thus, the correct option is (A) $\left( {{\text{1 - }}\dfrac{{\text{a}}}{{{\text{RTV}}}}} \right)$.

Note: Do not get confused between the compressibility factor $\left( {\text{Z}} \right)$ and the compressibility. Compressibility is also known as the coefficient of compressibility of isothermal compressibility. The compressibility is the measure of relative change in volume of a solid or a fluid with change in pressure.
When the compressibility factor is greater than one, the gas is more compressible and thus, the forces existing are repulsive. When the compressibility factor is less than one, the gas is more expandable and thus, the forces existing are attractive. For an ideal gas, compressibility factor is always equal to one.