Question

Which of the following curves represent the variation of coefficient of volume expansion of an ideal gas at constant pressure?

• A.
• B.
• C.
• D.

Answer 1

The Correct Option is B

The coefficient of volume expansion ($\alpha_V$) for an ideal gas is defined as the fractional change in volume per unit change in temperature at constant pressure:

$\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P$

For an ideal gas, the equation of state is given by $PV = nRT$, where:

• $P$ is the pressure,
• $V$ is the volume,
• $n$ is the number of moles,
• $R$ is the ideal gas constant, and
• $T$ is the absolute temperature.

At constant pressure, we can write:

$V = \frac{nR}{P}T$

Taking the derivative with respect to temperature:

$\left(\frac{\partial V}{\partial T}\right)_P = \frac{nR}{P}$

Substituting this back into the expression for $\alpha_V$:

$\alpha_V = \frac{1}{V} \left(\frac{nR}{P}\right) = \frac{nR}{PV} = \frac{nRT}{PV \cdot T} = \frac{1}{T}$

Therefore, $\alpha_V = \frac{1}{T}$.

Since the x-axis of the graphs is $\frac{1}{T}$, and $\alpha_V = \frac{1}{T}$, the graph of $\alpha_V$ vs. $\frac{1}{T}$ should be a straight line with a slope of 1, passing through the origin.

The correct answer is (B)