The ratio of the molar heat capacities of an ideal gas is `(C_p / C_v = 7/6)`. Calculate the change in internal energy of 1.0mole of the gas when its temperature is raised by `50 K `, (a) keeping the pressure constant , (b) keeping the volume constant and (c ) adiabatically.

Explanation

To solve the problem step by step, we will calculate the change in internal energy (\( \Delta U \)) for 1 mole of an ideal gas when its temperature is raised by \( 50 \, K \) under three different conditions: (a) at constant pressure, (b) at constant volume, and (c) adiabatically.

### Given Data:
- Number of moles, \( n = 1 \, \text{mole} \)
- Temperature change, \( \Delta T = 50 \, K \)
- Ratio of molar heat capacities, \( \frac{C_p}{C_v} = \frac{7}{6} \)

### Step 1: Calculate \( C_v \) and \( C_p \)
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