One way of writing the equation of state for real gases is $P ¯ ¯¯ ¯ V = R T [1 + \frac{B}{¯ ¯¯ ¯ V} + . . . .]$ , where B is a constant. An approximate expression for B in terms of the Van der Waals' constant 'a' and 'b' is $b - \frac{n a}{R T}$ . Then, the value of n is _____.

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Solution

$[P + \frac{a}{V^{2}}] [V - b] = R T$
$P = \frac{R T}{(V - b)} - \frac{a}{V^{2}}$
Multiply by [V], then
$P V = \frac{R T V}{V - b} - \frac{a \times V}{V^{2}}$
$P V = R T [\frac{V}{(V - b)} - \frac{a}{V R T}]$
$P V = R T [{(1 - \frac{b}{V})}^{- 1} - \frac{a}{V R T}]$
Now, ${[1 - \frac{b}{V}]}^{- 1} = 1 + \frac{b}{V} + {[\frac{b}{V}]}^{2} + {[\frac{b}{V}]}^{3} + . . . . .$
$∴ P V = R T [1 + \frac{b}{V} - \frac{a}{V R T} + {(\frac{b}{V})}^{2} . . . . .]$
$P V = R T [1 + (b - \frac{a}{R T}) \cdot \frac{1}{V} + {(\frac{b}{V})}^{2} + . . . . .]$
Thus, $B = b - \frac{a}{R T}$
so n is 1(one).

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