For a diatomic gas change in internal energy for unit change in temperature for constant pressure and constant volume is `Delta U_(1)` and `Delta U_(2)` respectively. The ratio of `Delta U_(1) : Delta U_(2)` is

Options

A. `5:3`
B. `3:5`
C. `1:1`
D. `5:7`

Explanation

To solve the problem of finding the ratio of the change in internal energy for a diatomic gas at constant pressure and constant volume, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Definitions**:
- For a diatomic gas, the internal energy change (ΔU) is related to the temperature change (ΔT) and the specific heat capacities at constant pressure (Cp) and constant volume (Cv).

2. **Formulas for Internal Energy**:
- At constant pressure:
\[ \Delta U_1 = nC_p\Delta T - W \]
where \(W\) is the work done by the gas. For constant pressure, the work done \(W\) is given by:
\[ W = P\Delta V = nR\Delta T \]
- Therefore, we can express ΔU1 as:
\[ \Delta U_1 = nC_p\Delta T - nR\Delta T = n(C_p - R)\Delta T \]

3. **At Constant Volume**:
- The change in internal energy at constant volume is given by:
\[ \Delta U_2 = nC_v\Delta T \]
- Since there is no work done at constant volume (W = 0), we have:
\[ \Delta U_2 = nC_v\Delta T \]

4. **Finding the Ratio**:
- Now, we need to find the ratio \(\frac{\Delta U_1}{\Delta U_2}\):
\[ \frac{\Delta U_1}{\Delta U_2} = \frac{n(C_p - R)\Delta T}{nC_v\Delta T} \]
- The \(n\) and \(\Delta T\) cancel out:
\[ \frac{\Delta U_1}{\Delta U_2} = \frac{C_p - R}{C_v} \]

5. **Using Mayer's Relation**:
- From Mayer's relation, we know:
\[ C_p - C_v = R \implies C_p = C_v + R \]
- Substituting this into our ratio gives:
\[ \frac{\Delta U_1}{\Delta U_2} = \frac{(C_v + R) - R}{C_v} = \frac{C_v}{C_v} = 1 \]

### Conclusion
Thus, the ratio of \(\Delta U_1 : \Delta U_2\) is:
\[ \Delta U_1 : \Delta U_2 = 1 : 1 \]