An ideal gas is expanding

Question:

An ideal gas is expanding such that $\mathrm{PT}^{3}=$ constant. The coefficient of volume expansion of the gas is:

  1. $\frac{1}{T}$

  2. $\frac{2}{\mathrm{~T}}$

  3. $\frac{4}{T}$

  4. $\frac{3}{\mathrm{~T}}$


Correct Option: , 3

Solution:

$\mathrm{PT}^{3}=\mathrm{constant}$

$\left(\frac{\mathrm{nRT}}{\mathrm{V}}\right) \mathrm{T}^{3}=\mathrm{constant}$

$\mathrm{T}^{4} \mathrm{~V}^{-1}=\mathrm{constant}$

$\mathrm{T}^{4}=\mathrm{kV}$

$\Rightarrow 4 \frac{\Delta \mathrm{T}}{\mathrm{T}}=\frac{\Delta \mathrm{V}}{\mathrm{V}}$...(1)

$\Delta \mathrm{V}=\mathrm{V} \gamma \Delta \mathrm{T}$...(2)

comparing (1) and (2)

we get

$\gamma=\frac{4}{\mathrm{~T}}$

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