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

Step 1: Define the coefficient of volume expansion.

The coefficient of volume expansion ($\alpha_v$) is defined as the fractional change in volume per degree Celsius (or Kelvin) change in temperature at constant pressure. Mathematically, it is given by:
$\alpha_v = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P$
where $V$ is the volume, $T$ is the temperature, and $P$ is the constant pressure.

Step 2: Use the ideal gas equation.

For an ideal gas, the equation of state is:
$PV = nRT$
where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the ideal gas constant, and $T$ is temperature.

Step 3: Express volume in terms of temperature at constant pressure.

At constant pressure $P$ and for a fixed amount of gas (constant $n$), we can write:
$V = \left( \frac{nR}{P} \right) T$
Let $C = \frac{nR}{P}$. Since $n, R, P$ are constants, $C$ is also a constant.
Thus, $V = CT$.

Step 4: Calculate the partial derivative of volume with respect to temperature at constant pressure.

$\left( \frac{\partial V}{\partial T} \right)_P = \frac{\partial}{\partial T} (CT) = C$

Step 5: Substitute the derivative into the formula for $\alpha_v$.

$\alpha_v = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P = \frac{1}{V} \times C = \frac{C}{V}$

Step 6: Express $\alpha_v$ in terms of temperature T.

Substitute $V = CT$ into the expression for $\alpha_v$:
$\alpha_v = \frac{C}{CT} = \frac{1}{T}$

Step 7: Analyze the relationship between $\alpha_v$ and T and choose the correct graph.

The relationship $\alpha_v = \frac{1}{T}$ shows that the coefficient of volume expansion is inversely proportional to the temperature. As temperature $T$ increases, $\alpha_v$ decreases, and vice versa.
- Option (A) and (D) show $\alpha_v$ increasing linearly with $T$, which is incorrect.
- Option (C) shows $\alpha_v$ as constant, which is incorrect.
- Option (B) shows $\alpha_v$ decreasing as $T$ increases, which is consistent with the inverse relationship $\alpha_v = \frac{1}{T}$. 

Final Answer: 
The curve in option (B) resembles a hyperbola, which is the graph of $y = \frac{1}{x}$.

Answer 2

Step 1: Recall the relationship between the coefficient of volume expansion and temperature
The coefficient of volume expansion of an ideal gas is related to the temperature at constant pressure. The general expression for the coefficient of volume expansion ($\alpha_v$) is given by:
$\alpha_v = \frac{1}{T}$
This implies that as the temperature increases, the coefficient of volume expansion decreases in an inverse manner.

Step 2: Analyze the options
We need to find the curve that best represents the inverse relationship of $\alpha_v$ with temperature.

Step 3: Examine the curves
- Option (A) shows a linear increase of $\alpha_v$ with temperature, which is not correct for an ideal gas.
- Option (D) also shows a linear increase of $\alpha_v$, which is incorrect.
- Option (C) shows a constant value for $\alpha_v$, which is not valid as it ignores temperature dependence.
- Option (B) shows the expected inverse relationship ($\alpha_v \sim \frac{1}{T}$), which is correct for an ideal gas at constant pressure.

Final Answer: 
Option (B) represents the correct variation of the coefficient of volume expansion of an ideal gas at constant pressure.