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 \( 747.9 \, \mathrm{J} \).