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Thermodynamics Part 3

Section: Physics

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Question : 70 of 100

Marks: +1, -0

An ideal gas is expanding such that

p T^{3} = \text{constant}

. The coefficient of volume expansion of the gas is

[27 Aug 2021 Shift 1]

$\frac{1}{T}$
$\frac{2}{T}$
$\frac{4}{T}$
$\frac{3}{T}$

Solution:

Given,

p T^{3} = \text{constant}

...(i)

From ideal gas equation, we have

pV = nRT

\Rightarrow p = \frac{nRT}{V}

From Eq. (i), we get

\therefore \left(\frac{nRT}{V}\right) T^{3} = \text{constant}

\Rightarrow n R T^{4} V^{-1} = \text{constant}

Differentiate above expression, we get

nR\left[4 T^{3} V^{-1} dT - \frac{T^{4}}{V^{2}} dV\right] = 0

\Rightarrow 4 dT = \frac{T}{V} dV

...(ii)

We know that,

dV = \gamma V dT

Here, γ = coefficient of volume expansion.

\Rightarrow \gamma = \frac{dV}{V dT}

...(iii)

From Eq. (ii),

\frac{4}{T} = \frac{dV}{V dT}

...(iv)

On comparing Eqs. (iii) and (iv), we get

\gamma = \frac{4}{T}

Thus, the coefficient of volume expansion is

\frac{4}{T}

.

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