For the same rise in temperature of one mole of gas at constant volume , heat required for a non - linear tratomic gas is K times that required for monoatomic gas . The value of K is

Options

A. 1
B. `0.5`
C. `2`
D. `2.5`

Answer

To solve the problem, we need to determine the value of K, which represents the ratio of heat required for a non-linear triatomic gas to that required for a monoatomic gas at constant volume for the same rise in temperature.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have one mole of gas at constant volume, and we want to find the heat required (Q) for both monoatomic and triatomic gases when the temperature is raised by the same amount (ΔT).

2. **First Law of Thermodynamics**:
According to the first law of thermodynamics, the heat added to the system (Q) is equal to the change in internal energy (ΔU) plus the work done (W):
\[ Q = \Delta U + W \]
Since the volume is constant, the work done (W) is zero. Therefore:
\[ Q = \Delta U \]

3. **Change in Internal Energy**:
The change in internal energy for one mole of gas can be expressed as:
\[ \Delta U = n C_v \Delta T \]
where \( n \) is the number of moles, \( C_v \) is the molar heat capacity at constant volume, and \( \Delta T \) is the change in temperature.

4. **Heat for Monoatomic Gas**:
For a monoatomic gas, the molar heat capacity at constant volume \( C_{v, mono} \) is given by:
\[ C_{v, mono} = \frac{3}{2} R \]
Therefore, the heat required for the monoatomic gas (Q_M) is:
\[ Q_M = n C_{v, mono} \Delta T = 1 \cdot \frac{3}{2} R \Delta T = \frac{3}{2} R \Delta T \]

5. **Heat for Triatomic Gas**:
For a non-linear triatomic gas, the molar heat capacity at constant volume \( C_{v, tri} \) is given by:
\[ C_{v, tri} = 3R \]
Therefore, the heat required for the triatomic gas (Q_T) is:
\[ Q_T = n C_{v, tri} \Delta T = 1 \cdot 3R \Delta T = 3R \Delta T \]

6. **Finding the Ratio K**:
We know that:
\[ Q_T = K \cdot Q_M \]
Substituting the expressions for \( Q_T \) and \( Q_M \):
\[ 3R \Delta T = K \cdot \left(\frac{3}{2} R \Delta T\right) \]
Dividing both sides by \( R \Delta T \) (assuming \( R \Delta T \neq 0 \)):
\[ 3 = K \cdot \frac{3}{2} \]
Solving for K:
\[ K = \frac{3}{\frac{3}{2}} = 2 \]

7. **Conclusion**:
The value of K is 2.

### Final Answer:
K = 2