A triatomic, diatomic and monoatomis gas is supplied same amount of heat at constant pressure then

Options

A. Fractional energy used to change internal energy is maximum in monatomic gas
B. Fractional energy used to change internal energy is maximum in diatomic gas
C. Fractional energy used to change internal energy is maximum is triatomic gases
D. Fractional energy used to change internal energy is same in all the three gases

Answer

To solve the problem, we need to analyze the situation where a triatomic, diatomic, and monoatomic gas are supplied the same amount of heat at constant pressure. We will determine the fractional energy used to change the internal energy for each type of gas.

### Step-by-Step Solution:

1. **Understanding the Process**:
- The process is isobaric, meaning that the pressure remains constant while heat is added to the gas.

2. **Heat Supplied**:
- Let the amount of heat supplied to each gas be \( Q \).

3. **Change in Internal Energy**:
- For an isobaric process, the change in internal energy (\( \Delta U \)) 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. **Molar Heat Capacity**:
- The relationship between the molar heat capacities at constant pressure (\( C_P \)) and constant volume (\( C_V \)) is given by:
\[ C_P = C_V + R \]
where \( R \) is the universal gas constant.

5. **Degrees of Freedom**:
- The degrees of freedom (\( F \)) for different types of gases are:
- Monoatomic gas: \( F = 3 \) (3 translational)
- Diatomic gas: \( F = 5 \) (3 translational + 2 rotational)
- Triatomic gas: \( F = 6 \) (3 translational + 3 rotational)

6. **Calculating \( C_V \)**:
- The molar heat capacity at constant volume can be expressed in terms of degrees of freedom:
\[ C_V = \frac{F}{2} R \]
- Therefore:
- For monoatomic gas: \( C_V = \frac{3}{2} R \)
- For diatomic gas: \( C_V = \frac{5}{2} R \)
- For triatomic gas: \( C_V = \frac{6}{2} R = 3R \)

7. **Fractional Energy Used to Change Internal Energy**:
- The fractional energy used to change internal energy is given by:
\[ \text{Fraction} = \frac{\Delta U}{Q} = \frac{n C_V \Delta T}{Q} \]
- Since \( Q \) is the same for all gases, we can analyze the fraction based on \( C_V \):
\[ \text{Fraction} \propto C_V \]

8. **Comparison of Fractions**:
- The proportionality of the fractions for each gas type is:
- Monoatomic: \( \frac{3}{2} R \)
- Diatomic: \( \frac{5}{2} R \)
- Triatomic: \( 3R \)

9. **Conclusion**:
- Since \( C_V \) is highest for the triatomic gas, the fractional energy used to change internal energy is maximum for the triatomic gas.

### Final Answer:
The fractional energy used to change internal energy is maximum for the triatomic gas.

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