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Question

At low pressures, the van der Waals equation is written as $$\left [ P+\cfrac{a}{V^{2}} \right ]V= RT$$.  The compressiblity factor of the gas is: 


A
[1aV]
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B
[1aTV]
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C
[1aRTV]
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D
[1RTaV]
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Solution

The correct option is C $$\left [ 1-\cfrac{a}{RTV} \right ]$$
Solution:- (C) $$\left[ 1 - \cfrac{a}{RTV} \right]$$
Given that at low pressure, van der Waals equation is-

$$\left[ P + \cfrac{a}{{V}^{2}} \right] V = RT$$

$$\Rightarrow PV + \cfrac{a}{V} = RT$$

$$\Rightarrow \cfrac{PV}{RT} + \cfrac{a}{RTV} = 1$$

$$\Rightarrow \cfrac{PV}{RT} = 1 - \cfrac{a}{RTV}$$

$$\Rightarrow Z = 1 - \cfrac{a}{RTV} \quad \left( \because Z = \cfrac{PV}{RT} \right)$$

Hence the compressiblity factor of the gas is $$\left[ 1 - \cfrac{a}{RTV} \right]$$.

Chemistry

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