A cylinder witih 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 ________ J. [Given that R = 8.31 `J mol^(-1) K^(-1)]`

Explanation

To solve the problem, we need to calculate the amount of heat required to raise the temperature of helium gas in a fixed volume cylinder. Here are the steps to arrive at the solution:

### Step 1: Determine the number of moles of helium gas in the cylinder.

Given:
- Volume of the cylinder = 67.2 L
- At STP (Standard Temperature and Pressure), 1 mole of gas occupies 22.4 L.

Using the formula:
\[ \text{Number of moles (n)} = \frac{\text{Volume}}{\text{Molar volume at STP}} = \frac{67.2 \, \text{L}}{22.4 \, \text{L/mol}} = 3 \, \text{moles} \]

### Step 2: Identify the specific heat capacity at constant volume (C_v) for helium.

Helium is a monatomic gas, and for monatomic gases:
\[ C_v = \frac{3}{2}R \]
Where \( R = 8.31 \, \text{J/(mol K)} \).

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

### Step 3: Calculate the heat required to raise the temperature by 20°C.

The formula for heat (Q) in a constant volume process is given by:
\[ Q = n C_v \Delta T \]
Where:
- \( n = 3 \, \text{moles} \)
- \( C_v = 12.465 \, \text{J/(mol K)} \)
- \( \Delta T = 20 \, \text{°C} = 20 \, \text{K} \) (since the change in temperature in Celsius is the same as in Kelvin)

Substituting the values:
\[ Q = 3 \, \text{moles} \times 12.465 \, \text{J/(mol K)} \times 20 \, \text{K} \]
\[ Q = 3 \times 12.465 \times 20 = 746.1 \, \text{J} \]

### Final Answer:
The amount of heat needed to raise the temperature of the gas by \( 20^\circ C \) is **746.1 J**.

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