If a real gas is following equation $P (V - n b) = n R T$ , at low pressure then find the intercept and slope of graph between $\frac{d}{p} v / s P$ .

\frac{M R}{T}, \frac{M (R T)^{2}}{b}

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\frac{M}{R T}, - \frac{M b}{(R T)^{2}}

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\frac{M b}{R T}, - \frac{M}{(R T)^{2}}

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\frac{R T}{M}, - \frac{b}{M (R T)^{2}}

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Solution

The correct option is B $\frac{M}{R T}, - \frac{M b}{(R T)^{2}}$
$P (V - \frac{w}{M} b) = \frac{w}{M} R T$
$P [1 - \frac{d b}{M}] = \frac{d R T}{M}$
$P [M - d b] = d R T$
$\frac{d}{P} = \frac{M}{R T} \times \frac{- d b}{R T} \dots (1)$
$\frac{d}{P} = y, P = x, ∴ d = y x$
$∴ y = \frac{M}{R T} - \frac{b}{R T} (y x)$
$y [1 + \frac{b x}{R T}] = \frac{M}{R T} \dots (2)$
For y-intercept, $x = 0$
$∴ y = \frac{M}{R T} \dots (3)$
For slope, $\frac{d y}{d x} = \frac{M}{R T} ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{1}{{(1 + \frac{b x}{R T})}^{2}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (\frac{b}{R T})$
At low pressure, $P = 0 \Rightarrow x = 0$
$∴$ Slope $= \frac{d y}{d x} = \frac{- M b}{(R T)^{2}}$

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Similar questions

Column A	Column B
(A) High pressure	(i) $P V = R T + P b$
(B) Low pressure	(ii) $P V = R T - \frac{a}{V}$
(C) $P \to 0$	(iii) $P V = R T$
(D) Neither $a$ nor $b$ is negligible	(iv) $(P + \frac{a}{V^{2}}) (V - b) = R T$

P (V - b) = R T

^{'} b^{'}

P (V - b) = R T

b

P V

NTP

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