A cylinder with fixed capacity of 67.2 lit contains helium gas at STP. The amount of heat needed to raise the temperature of the gas by `20^@C` is :[Given that R=8.31] `"mol"^(-1) K^(-1)`]

Options

A. 350 J
B. 374 J
C. 748 J
D. 700 J

Answer

To solve the problem of finding the amount of heat needed to raise the temperature of helium gas in a fixed volume cylinder by 20°C, we will follow these steps:

### Step 1: Understand the Given Information
- The volume of the cylinder is 67.2 liters.
- The gas is helium at Standard Temperature and Pressure (STP).
- The temperature change (ΔT) is 20°C.
- The gas constant (R) is given as 8.31 J/(mol·K).

### Step 2: Convert Volume to Moles
At STP, 1 mole of an ideal gas occupies 22.4 liters. Therefore, we can calculate the number of moles (n) of helium gas in the cylinder:

\[ n = \frac{\text{Volume}}{\text{Molar Volume at STP}} = \frac{67.2 \, \text{liters}}{22.4 \, \text{liters/mol}} = 3 \, \text{moles} \]

### Step 3: Determine the Heat Capacity at Constant Volume (Cv)
For a monatomic ideal gas like helium, the heat capacity at constant volume (Cv) is given by:

\[ C_v = \frac{3}{2} R \]

Substituting the value of R:

\[ C_v = \frac{3}{2} \times 8.31 \, \text{J/(mol·K)} = 12.465 \, \text{J/(mol·K)} \]

### Step 4: Calculate the Heat (Q)
The amount of heat required to raise the temperature of the gas at constant volume is given by the formula:

\[ Q = n C_v \Delta T \]

Substituting the values we have:

\[ Q = 3 \, \text{moles} \times 12.465 \, \text{J/(mol·K)} \times 20 \, \text{K} \]

Calculating this gives:

\[ Q = 3 \times 12.465 \times 20 = 748.0 \, \text{J} \]

### Step 5: Conclusion
The amount of heat needed to raise the temperature of the helium gas by 20°C is approximately **748 J**.

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