Question

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

Answer 1

Hint: One can use the van der Waals equation for gas and get the value of ${\text{Z}}$ in the equation. There is no mention of the value of ${\text{n}}$ which is the number of moles in any given option so one can take the value of the number of moles as one while getting the equation for compressibility factor.

Complete step by step answer:
1) First of all we will write the van der Waals equation for gas,
$\left( {P + \dfrac{{{n^2}a}}{{{V^2}}}} \right)\left( {V - nb} \right) = nRT$
As there is no mention of the value of the number of moles in any one of the options we can take the value of ${\text{n}}$ as ${\text{1}}$ and we get the equation as,
$\left( {P + \dfrac{a}{{{V^2}}}} \right)\left( {V - b} \right) = RT$
2) Now the condition of low pressure has been given which means the volume is also very negligible. Hence, we can say that $V > > > > b$ where ${\text{b}}$ is the volume factor in the van der Waals equation. We get the equation as below,
$\left( {P + \dfrac{a}{{{V^2}}}} \right)\left( V \right) = RT$
Now by rearranging the above equation we get,
$PV + \dfrac{a}{V} = RT$ ( As the value of ${\text{V}}$ cancels out in denominator and numerator)
Now, let's take the value of ${\text{PV}}$ on one side we get,
$PV = RT - \dfrac{a}{V}$
3) Now to compare the equation for the value of the compressibility factor we need to divide both sides by ${\text{RT}}$ we get,
$\dfrac{{PV}}{{RT}} = \dfrac{{RT}}{{RT}} - \dfrac{a}{{VRT}}$
By simplifying we get,
$\dfrac{{PV}}{{RT}} = 1 - \dfrac{a}{{VRT}}$
Now as we know the value of $\dfrac{{PV}}{{RT}}$ is equal to ${\text{Z}}$ which is the compressibility factor we get,
$Z = 1 - \dfrac{a}{{VRT}}$

Therefore, the compressibility factor value at low pressure is $Z = 1 - \dfrac{\alpha }{{VRT}}$ which shows option B as the correct choice.

Note:
While finding the equation for a particular term by using another equation one needs to put in the relevant possible terms and neglect some common terms according to the conditions given in the question in order to get the desired answer and select the correct choice.