Explanation
Hint: Using ideal gas equation rearrange the given equation and then differentiate.
Where P - Pressure, T - Temperature
Consider the ideal gas equation and substitute the value of pressure in the given equation.
Explanation of Correct Option:
Step 1: Consider the formulas mentioned
Expression of is given by,
Volume expansion can be expressed as:
$$V={{V}_{0}}(1+\gamma t)$$
Where V is the volume at temperature t
$${{V}_{0}}$$ Is the volume at temperature t = 0
$$\gamma $$ is the coefficient of volume expansion and t is temperature.
Step 2: Consider the ideal gas expansion
Ideal gas is expansion is given as
$$P{{T}^{2}}=\text{constant}$$….. (1)
Step 3: Consider the ideal gas equation,
$$PV=nRT $$
$$\dfrac{PV}{T}=\text{constant}$$
$$P=\dfrac{\text{constant}\times T}{V}......\left( 2 \right)$$
Coefficient of volume expansion can be calculated by the values of gas denoted by $$\gamma$$
Step 4: Coefficient of volume expansion of gas is given by
$$V={{V}_{0}}(1+\gamma t).......\left( 3 \right)$$
Substitute the equation (2) in equation (1),
$$\dfrac{\text{constant}\times T}{V}\times {{T}^{2}}=\text{constant}$$
$$ \dfrac{{{T}^{3}}}{V}=\text{constant=k}$$
$$ V=\dfrac{1}{k}{{T}^{3}}={{k}^{1}}{{T}^{3}}.....\left( 4 \right)\text{ }\left( \because {{k}^{1}}=\dfrac{1}{k} \right)$$
Step 5: Differentiate equation (3) with respect to t
On differentiating with respect to t
$$ \dfrac{dV}{dt}={{V}_{0}}(\gamma .1) $$
$$\dfrac{dV}{dt}={{V}_{0}}\gamma .......\left( 5 \right) $$
Dedifferentiate equation (4) with respect to T,
$$\dfrac{dV}{dT}={{k}^{1}}3{{T}^{2}}$$
Put values of $${{k}^{1}}$$ from equation (4)
$$ \dfrac{dV}{dT}=\dfrac{V}{{{T}^{3}}}\times 3{{T}^{2}} $$
$$\dfrac{dV}{dT}=\dfrac{3V}{T} $$
$$\therefore \dfrac{dV}{V}=3\dfrac{dT}{T} $$
$$dV=\dfrac{3}{T}VdT......(6)$$
Volume expansion is given by,
$$\Delta V=\gamma {{V}_{0}}\Delta T......\left( 7 \right)$$
Step 6: Compare the equations (6) and (7)
$$dV=\Delta V$$
$$\dfrac{3}{T}VdT = \gamma {{V}_{0}}\Delta T......\left( 7 \right)$$
$$\gamma =\dfrac{3}{T}$$
Hence the coefficient of volume expansion of the gas is $$\gamma =\dfrac{3}{T}$$.
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