Question

A cylinder of fixed capacity $67.2 \, \mathrm{L}$ contains helium gas at STP. The amount of heat needed to raise the temperature of the gas in the cylinder by $20^\circ \mathrm{C}$ is:

• A. $700.5 \, \mathrm{J}$
• B. $747.9 \, \mathrm{J}$
• C. $760.2 \, \mathrm{J}$
• D. $800.0 \, \mathrm{J}$

Hint:

For monatomic gases like helium, $C_v = \frac{3}{2} R$ and $C_p = \frac{5}{2} R$. Use $Q = n C_v \Delta T$ for calculations at constant volume.

Answer 1

The Correct Option is B

Step 1: Recall the formula for heat capacity at constant volume.
The heat required to raise the temperature of a gas is given by: $$Q = n C_v \Delta T,$$ where: $n$ is the number of moles of gas, $C_v$ is the molar heat capacity at constant volume for helium, $\Delta T$ is the temperature change. Step 2: Calculate the number of moles.
At STP (Standard Temperature and Pressure), the molar volume of an ideal gas is $22.4 \, \mathrm{L}$. The number of moles is: $$n = \frac{\text{Volume of gas}}{\text{Molar volume}} = \frac{67.2}{22.4} = 3.0 \, \mathrm{mol}.$$ Step 3: Substitute the known values.
For helium, $C_v = \frac{3}{2} R$, where $R = 8.314 \, \mathrm{J/mol \, K}$: $$C_v = \frac{3}{2} \times 8.314 = 12.471 \, \mathrm{J/mol \, K}.$$ The temperature change is $\Delta T = 20 \, \mathrm{K}$. Substitute these into the formula: $$Q = n C_v \Delta T = 3.0 \times 12.471 \times 20.$$ Step 4: Calculate the result.
$$Q = 3.0 \times 249.42 = 747.9 \, \mathrm{J}.$$ Thus, the heat required is $747.9 \, \mathrm{J}$.