Question

4.0 g of a gas occupies 22.4 litres at NTP. The specific heat capacity of the gas at constant volume is 5.0 $JK^{-1} \, mol^{-1}$. If the speed of sound in this gas at NTP is 952 $ms^{-1}$, then the heat capacity at constant pressure is (Take gas constant R = 8.3 $JK^{-1} \, mol^{-1}$ )

• A. $7.0 \, JK^{-1} \, mol^{-1}$
• B. $8.5 \, JK^{-1} \, mol^{-1}$
• C. $8.0 \, JK^{-1} \, mol^{-1}$
• D. $7.5 \, JK^{-1} \, mol^{-1}$

Answer 1

The Correct Option is C

To determine the heat capacity at constant pressure, we can use the basic principles of thermodynamics related to ideal gases. Given data includes:

• Mass of the gas = 4.0 g
• Volume of the gas at NTP = 22.4 L
• Specific heat capacity at constant volume, C_v = 5.0 \, JK^{-1} \, mol^{-1}
• Speed of sound in the gas at NTP, v = 952 \, ms^{-1}
• Gas constant, R = 8.3 \, JK^{-1} \, mol^{-1}

First, we need to determine the molar mass of the gas. At Normal Temperature and Pressure (NTP), 1 mole of an ideal gas occupies 22.4 L. Hence, the molar mass of the gas can be calculated as:

\text{Molar mass} = \frac{\text{mass of gas}}{\text{moles of gas}} = \frac{4.0 \, g}{1 \, mol} = 4.0 \, g/mol.

Next, we use the relation between the speed of sound, the adiabatic index (γ), and the specific heat capacities:

v = \sqrt{\frac{\gamma \cdot R \cdot T}{M}}

where γ is the adiabatic index, given by \gamma = \frac{C_p}{C_v}, and M is the molar mass in kg/mol.

Rearranging the speed of sound formula:

v^2 = \frac{\gamma \cdot R \cdot T}{M}

NTP conditions give us the temperature T = 273 \, K and the molar mass M = 0.004 \, kg/mol.

Using the speed of sound formula:

\gamma = \frac{v^2 \cdot M}{R \cdot T}

Insert the values:

\gamma = \frac{(952)^2 \cdot 0.004}{8.3 \times 273}

Calculating γ:

\gamma = \frac{(952)^2 \cdot 0.004}{8.3 \times 273} \approx 1.6

Now, using the relationship between gamma, C_p, and C_v:

\gamma = \frac{C_p}{C_v}

C_p = \gamma \cdot C_v

Substitute γ and C_v to find C_p:

C_p = 1.6 \times 5.0 \, JK^{-1} \, mol^{-1} = 8.0 \, JK^{-1} \, mol^{-1}

Therefore, the heat capacity at constant pressure is 8.0 \, JK^{-1} \, mol^{-1}, which matches the correct answer option.