He is kept in a rigid container of volume `67.2` ltr at `STP`. The heat supplied to the gas to increases its temperature by `20^(@)C` is :

Options

A. `780J`
B. `748J`
C. `718J`
D. `680J`

Answer

To solve the problem of calculating the heat supplied to a rigid container of helium gas to increase its temperature by \(20^\circ C\), we can follow these steps:

### Step 1: Understand the formula for heat supplied
The heat supplied to a gas at constant volume can be calculated using the formula:
\[ Q = N C_V \Delta T \]
where:
- \(Q\) is the heat supplied,
- \(N\) is the number of moles of the gas,
- \(C_V\) is the specific heat at constant volume,
- \(\Delta T\) is the change in temperature.

### Step 2: Calculate the number of moles \(N\)
To find the number of moles \(N\), we use the ideal gas law. At standard temperature and pressure (STP), 1 mole of an ideal gas occupies 22.4 liters. Therefore, the number of moles can be calculated as:
\[ N = \frac{\text{Volume of gas}}{\text{Volume per mole at STP}} = \frac{67.2 \, \text{liters}}{22.4 \, \text{liters/mole}} = 3 \, \text{moles} \]

### Step 3: Determine the specific heat at constant volume \(C_V\)
For a monoatomic ideal gas like helium, the specific heat at constant volume is given by:
\[ C_V = \frac{3}{2} R \]
where \(R\) is the universal gas constant, approximately \(8.314 \, \text{J/(mol K)}\). Thus,
\[ C_V = \frac{3}{2} \times 8.314 \, \text{J/(mol K)} = 12.471 \, \text{J/(mol K)} \]

### Step 4: Calculate the change in temperature \(\Delta T\)
The change in temperature is given as:
\[ \Delta T = 20^\circ C \]
Note that the change in temperature in Celsius is equivalent to the change in Kelvin.

### Step 5: Substitute values into the heat equation
Now, we can substitute the values into the heat equation:
\[ Q = N C_V \Delta T = 3 \, \text{moles} \times 12.471 \, \text{J/(mol K)} \times 20 \, \text{K} \]

### Step 6: Perform the calculation
Calculating \(Q\):
\[ Q = 3 \times 12.471 \times 20 = 748.26 \, \text{J} \]
We can round this to:
\[ Q \approx 748 \, \text{J} \]

### Final Answer
The heat supplied to the gas to increase its temperature by \(20^\circ C\) is approximately **748 Joules**.

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