Find the change in internal energy of `2` moles of an ideal gas when its temperature is increased by `75K`
(a) Keeping the pressure constant.
(b) Keeping the volume constant and
(c ) adiabatically. `(gamma = (C_(P))/(C_(V)) =(7)/(5)) [R = (25)/(3)J//mol-K]`

Answer

To find the change in internal energy of 2 moles of an ideal gas when its temperature is increased by 75 K under different conditions, we can follow these steps:

### Given Data:
- Number of moles (n) = 2 moles
- Change in temperature (ΔT) = 75 K
- \( \gamma = \frac{C_P}{C_V} = \frac{7}{5} \)
- Universal gas constant (R) = \( \frac{25}{3} \) J/mol-K

### Step 1: Find \( C_V \)
We know the relationship between \( C_P \) and \( C_V \):
\[ C_P - C_V = R \]
Using the given ratio \( \frac{C_P}{C_V} = \frac{7}{5} \), we can express \( C_P \) in terms of \( C_V \):
\[ C_P = \frac{7}{5} C_V \]
Substituting this into the equation \( C_P - C_V = R \):
\[ \frac{7}{5} C_V - C_V = R \]
\[ \frac{7}{5} C_V - \frac{5}{5} C_V = R \]
\[ \frac{2}{5} C_V = R \]
\[ C_V = \frac{5}{2} R \]

### Step 2: Substitute the value of R
Now substitute \( R = \frac{25}{3} \) J/mol-K into the equation for \( C_V \):
\[ C_V = \frac{5}{2} \times \frac{25}{3} = \frac{125}{6} \text{ J/mol-K} \]

### Step 3: Calculate the change in internal energy (ΔU)
The change in internal energy for an ideal gas can be calculated using the formula:
\[ \Delta U = n C_V \Delta T \]
Substituting the known values:
\[ \Delta U = 2 \times \frac{125}{6} \times 75 \]

### Step 4: Simplify the expression
Calculating \( \Delta U \):
\[ \Delta U = 2 \times \frac{125 \times 75}{6} \]
\[ = \frac{2 \times 125 \times 75}{6} \]
\[ = \frac{18750}{6} = 3125 \text{ J} \]

### Conclusion
The change in internal energy of the gas when its temperature is increased by 75 K is:
\[ \Delta U = 3125 \text{ J} \]