Temperature of two moles of a monoatomic gas is increased by 600 K in a given process. Find change in internal energy of the gas.

Explanation

To find the change in internal energy of a monoatomic gas when its temperature is increased, we can use the formula for the change in internal energy:

\[ \Delta U = n C_v \Delta T \]

Where:
- \(\Delta U\) is the change in internal energy,
- \(n\) is the number of moles of the gas,
- \(C_v\) is the molar specific heat at constant volume,
- \(\Delta T\) is the change in temperature.

**Step 1: Identify the values given in the problem.**
- Number of moles (\(n\)) = 2 moles
- Change in temperature (\(\Delta T\)) = 600 K

**Step 2: Determine the value of \(C_v\) for a monoatomic gas.**
For a monoatomic gas, the molar specific heat at constant volume (\(C_v\)) is given by:

\[ C_v = \frac{3}{2} R \]

Where \(R\) is the universal gas constant (approximately \(8.314 \, \text{J/(mol K)}\)).

**Step 3: Substitute the values into the internal energy formula.**
Now, we can substitute the values into the formula for \(\Delta U\):

\[ \Delta U = n C_v \Delta T \]

Substituting the known values:

\[ \Delta U = 2 \left(\frac{3}{2} R\right) (600) \]

**Step 4: Simplify the equation.**
Calculating the expression:

\[ \Delta U = 2 \cdot \frac{3}{2} \cdot R \cdot 600 \]

The \(2\) and \(\frac{3}{2}\) cancel out:

\[ \Delta U = 3 \cdot R \cdot 600 \]

\[ \Delta U = 1800 R \]

**Step 5: Final result.**
Thus, the change in internal energy of the gas is:

\[ \Delta U = 1800 R \]

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