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Question: At low pressures, the van der Waals equation is written as \(\left[ {P + \frac{a}{{{V^2}}}} \right]V...

At low pressures, the van der Waals equation is written as [P+aV2]V  =  RT\left[ {P + \frac{a}{{{V^2}}}} \right]V\,\, = \,\,RT
The compressibility factor is then equal to:
A. [1−aRTV]\left[ {1 - \frac{a}{{RTV}}} \right]
B. [1−RTVa]\left[ {1 - \frac{{RTV}}{a}} \right]
C. [1+aRTV]\left[ {1 + \frac{a}{{RTV}}} \right]
D. [1+RTVa]\left[ {1 + \frac{{RTV}}{a}} \right]

Explanation

Solution

To solve this question, we must first understand the concept of van der Waals equation and have some basic knowledge about compressibility factor. Then we need to assess the van der Waals equation and try to equate the compressibility factor within the equation itself and doing this will leave behind the required answer.

Complete step-by-step solution: Before we move forward with the solution of this given question, let us first understand some basic concepts:
Van der Waals equation: Van der Waals equation is an equation relating the relationship between the pressure, volume, temperature, and amount of real gases. For a real gas containing ‘n’ moles, the equation is written as:
[P+an2V2](V−nb)  =  nRT\left[ {P + \frac{{a{n^2}}}{{{V^2}}}} \right](V - nb)\,\, = \,\,nRT
Where, P, V, T, n are the pressure, volume, temperature and moles of the gas. ‘a’ and ‘b’ constants specific to each gas.
In thermodynamics, the compressibility factor (Z)(Z) , also known as the compression factor or the gas deviation factor, is a correction factor which describes the deviation of a real gas from ideal gas behaviour. It is simply defined as the ratio of the molar volume of a gas to the molar volume of an ideal gas at the same temperature and pressure.
Step 1: In this step we will use the given equation to calculate the compressibility factor:
[P+aV2]V  =  RT\left[ {P + \frac{a}{{{V^2}}}} \right]V\,\, = \,\,RT
PV  +  aV  =  RT   ⇒  PVRT  +   aRTV  =  1 ⇒ PVRT  =  1  −   aRTV  \begin{gathered} PV\,\, + \,\,\frac{a}{V}\,\, = \,\,RT\,\, \\\ \Rightarrow \,\,\frac{{PV}}{{RT}}\,\, + \,\,\,\frac{a}{{RTV}}\,\, = \,\,1 \\\ \Rightarrow \,\frac{{PV}}{{RT}}\,\, = \,\,1\,\, - \,\,\,\frac{a}{{RTV}}\, \\\ \end{gathered}
And since we know that, Z  = PVRTZ\,\, = \,\frac{{PV}}{{RT}}
∴  PVRT  =  Z  =  [1−aRTV]\therefore \,\,\frac{{PV}}{{RT}}\,\, = \,\,Z\,\, = \,\,\left[ {1 - \frac{a}{{RTV}}} \right]

So, clearly we can conclude that the correct answer is Option A.

Note: The compressibility factor should not be confused with the compressibility (also known as coefficient of compressibility or isothermal compressibility) of a material, which is the measure of the relative volume change of a fluid or solid in response to a pressure change.