A diatomic gas of moleculer weight 30 gm/mole is filled in a container at `27^(@)C.` It is moving at a velocity 100 m/s. If it is suddenly stopped, the rise in temperature of gas is :

Options

A. 60/R
B. `(600)/(R )`
C. `(6 xx 10^(4))/(R )`
D. `(6xx 10^(5))/(R )`

Answer

To solve the problem of finding the rise in temperature of a diatomic gas when it is suddenly stopped, we can follow these steps:

### Step 1: Identify the given data
- Molecular weight of the gas, \( M = 30 \, \text{g/mol} \)
- Initial temperature, \( T_i = 27^\circ C = 300 \, \text{K} \) (since \( T(K) = T(°C) + 273 \))
- Velocity of the gas, \( v = 100 \, \text{m/s} \)

### Step 2: Calculate the kinetic energy of the gas
The kinetic energy (KE) of the gas can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
Where \( m \) is the mass of the gas. To find the mass, we need to convert the molecular weight into kilograms:
\[ M = 30 \, \text{g/mol} = 0.030 \, \text{kg/mol} \]
Assuming we have 1 mole of gas, the mass \( m = 0.030 \, \text{kg} \). Thus,
\[ KE = \frac{1}{2} \times 0.030 \, \text{kg} \times (100 \, \text{m/s})^2 = \frac{1}{2} \times 0.030 \times 10000 = 150 \, \text{J} \]

### Step 3: Relate kinetic energy to temperature change
The kinetic energy lost by the gas will convert into internal energy, which will cause a rise in temperature. For a diatomic gas, the molar heat capacity at constant volume \( C_V \) is given by:
\[ C_V = \frac{5}{2} R \]
Where \( R \) (the universal gas constant) is approximately \( 8.314 \, \text{J/(mol K)} \). Thus,
\[ C_V = \frac{5}{2} \times 8.314 = 20.785 \, \text{J/(mol K)} \]

### Step 4: Use the formula for temperature change
The change in internal energy is given by:
\[ \Delta U = n C_V \Delta T \]
Where \( n \) is the number of moles (which is 1 mole in this case). Setting the kinetic energy equal to the change in internal energy:
\[ 150 \, \text{J} = 1 \times 20.785 \, \Delta T \]
Solving for \( \Delta T \):
\[ \Delta T = \frac{150}{20.785} \approx 7.22 \, \text{K} \]

### Step 5: Conclusion
The rise in temperature of the gas when it is suddenly stopped is approximately \( 7.22 \, \text{K} \).

---