The pressure (P) and temperature (T) relationship of an ideal gas obeys the equation The volume expansion coefficient of the gas will be [2023]

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Solution

Q.
The pressure (P) and temperature (T) relationship of an ideal gas obeys the equation $P T^{2} = constant .$ The volume expansion coefficient of the gas will be [2023]

1 $3 T^{2}$

2 $\frac{3}{T^{2}}$

3 $\frac{3}{T^{3}}$

4 $\frac{3}{T}$

Ans.
(4)

$P T^{2} = constant$ , Using $P V = n R T$

$P = \frac{n R T}{V}$

$P T^{2} = \frac{n R T}{V} \times T^{2} = constant = K$

$\Rightarrow T^{3} = K V$

So, $\frac{d}{d T} (K V) = 3 T^{2}$

$\Rightarrow \frac{K d V}{d T} = 3 T^{2}$

$\Rightarrow d V = \frac{3 T^{2}}{K} d T = V γ d T$

$d V = V γ d T$

$\Rightarrow γ V = \frac{3 T^{2}}{K}$

$\Rightarrow γ = \frac{3 T^{2}}{K V}$ $= \frac{3 T^{2}}{T^{3}} = \frac{3}{T}$

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