A monoatomic gas of n-moles is heated temperature `T_1` to `T_2` under two different conditions
(i) at constant volume and
(ii) At constant pressure The change in internal energy of the gas is

Options

A. More for (i)
B. More for (ii)
C. Same in both cases
D. Independent of number of moles

Explanation

To solve the problem, we need to calculate the change in internal energy of a monoatomic gas when it is heated from temperature \( T_1 \) to \( T_2 \) under two different conditions: constant volume and constant pressure.

### Step 1: Understand the Change in Internal Energy
The change in internal energy (\( \Delta U \)) of a gas is given by the formula:
\[ \Delta U = n C_v \Delta T \]
where:
- \( n \) = number of moles of the gas
- \( C_v \) = molar specific heat capacity at constant volume
- \( \Delta T = T_2 - T_1 \) = change in temperature

### Step 2: Case 1 - Constant Volume
When the gas is heated at constant volume:
- The work done (\( \Delta W \)) is zero because there is no change in volume (\( \Delta V = 0 \)).
- According to the first law of thermodynamics:
\[ \Delta U = \Delta Q + \Delta W \]
Since \( \Delta W = 0 \), we have:
\[ \Delta U = \Delta Q \]
The heat added (\( \Delta Q \)) can be expressed as:
\[ \Delta Q = n C_v \Delta T \]
Thus, the change in internal energy at constant volume is:
\[ \Delta U = n C_v (T_2 - T_1) \]

### Step 3: Case 2 - Constant Pressure
When the gas is heated at constant pressure:
- The change in internal energy is still given by the same formula:
\[ \Delta U = n C_v \Delta T \]
This is because the change in internal energy depends only on the change in temperature and the specific heat capacity at constant volume, which does not change with the process type.

### Conclusion
In both cases (constant volume and constant pressure), the change in internal energy is given by:
\[ \Delta U = n C_v (T_2 - T_1) \]

### Final Answer
The change in internal energy of the gas is:
\[ \Delta U = n C_v (T_2 - T_1) \]