Question

If Z is a compressibility factor, van der waals equation at low pressure can be written as:
(A) $Z=1-\dfrac{Pb}{RT}$
(B) $Z=1+\dfrac{Pb}{RT}$
(C) $Z=1+\dfrac{RT}{Pb}$
(D) $Z=1-\dfrac{a}{VRT}$

Answer 1

Hint: Recollect the concept of behavior of real gases. For one mole of a gas, Van der Waals equation is given as, $\left( P+\dfrac{a}{{{V}^{2}}} \right)\left( V-b \right)=RT$. Find out what the compressibility factor is. Think about the changes which will occur when pressure is low and accordingly derive the equation for compressibility factor, Z.

Complete step by step answer:
- According to Van der Waals equation for a real gas,
\[\left( P+\dfrac{{{n}^{2}}a}{{{V}^{2}}} \right)\left( V-nb \right)=nRT\] where a is the correction term for intermolecular attractive forces and b is the correction term for volume of molecules. P is the pressure, V is the volume of the gas, R is the universal gas constant and T is the temperature.
- For one mole of a real gas, Van der Waals equation is given as, $\left( P+\dfrac{a}{{{V}^{2}}} \right)\left( V-b \right)=RT$
- Compressibility factor is the correction term which shows the deviation of a real gas from an ideal gas behavior. It is represented as Z.
- We know pressure is inversely proportional to volume. So, if the pressure is low then the volume will be high.

- At low pressure, V>>>b and therefore, Van der waals equation becomes,
\[\left( P+\dfrac{a}{{{V}^{2}}} \right)V=RT\]
- Solving this equation we get,
\[\begin{align}
  & PV+\dfrac{a}{V}=RT \\
 & PV=RT-\dfrac{a}{V} \\
\end{align}\]
- Dividing the equation by RT we get,
\[\dfrac{PV}{RT}=1-\dfrac{a}{VRT}\]
- Therefore, at low pressure, compressibility factor, $Z=1-\dfrac{a}{VRT}$.
So, the correct answer is “Option D”.

Note: Remember Van der Waals equation is the equation which describes the behavior of a real gas and is given as, $\left( P+\dfrac{{{n}^{2}}a}{{{V}^{2}}} \right)\left( V-nb \right)=nRT$. Pressure is inversely proportional to volume. Therefore, at low pressure, volume will be very high and so (V-nb) term will be approximately equal to V since V>>>b.