One mole of an ideal monoatomic gas is taken at a temperature of `300 K`. Its volume is doubled keeping its pressure constant. Find the change in internal energy.

Explanation

To solve the problem step by step, we will follow the reasoning presented in the video transcript while providing detailed explanations for each step.

### Step 1: Understand the Initial Conditions
We have one mole of an ideal monoatomic gas at an initial temperature \( T_i = 300 \, K \). The volume of the gas is doubled while keeping the pressure constant.

### Step 2: Use the Ideal Gas Law
The ideal gas law is given by:
\[ PV = nRT \]
Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = universal gas constant
- \( T \) = temperature

Since the pressure is constant and the volume is doubled, we can relate the initial and final states of the gas.

### Step 3: Relate Initial and Final Temperatures
Let the initial volume be \( V_i \) and the final volume be \( V_f = 2V_i \). Since pressure is constant, we can write:
\[ \frac{V_f}{V_i} = \frac{T_f}{T_i} \]
Substituting the known values:
\[ \frac{2V_i}{V_i} = \frac{T_f}{300} \]
This simplifies to:
\[ 2 = \frac{T_f}{300} \]
From which we can find \( T_f \):
\[ T_f = 2 \times 300 = 600 \, K \]

### Step 4: Calculate the Change in Temperature
The change in temperature \( \Delta T \) is given by:
\[ \Delta T = T_f - T_i = 600 \, K - 300 \, K = 300 \, K \]

### Step 5: Calculate the Change in Internal Energy
The change in internal energy \( \Delta U \) for an ideal monoatomic gas can be calculated using the formula:
\[ \Delta U = \frac{3}{2} n R \Delta T \]
Where:
- \( n = 1 \) mole (given)
- \( R \) is the universal gas constant (approximately \( 8.314 \, J/(mol \cdot K) \))

Substituting the values:
\[ \Delta U = \frac{3}{2} \times 1 \times R \times 300 \]
\[ \Delta U = \frac{3}{2} \times 8.314 \times 300 \]
Calculating this gives:
\[ \Delta U = 3 \times 8.314 \times 150 = 1247.1 \, J \]

### Final Answer
Thus, the change in internal energy is:
\[ \Delta U = 1247.1 \, J \]