Calculate the change in internal energy of `3 .0` mol of helium gas when its temperature is increased by `2. 0 K`.

Explanation

To calculate the change in internal energy (ΔU) of 3.0 moles of helium gas when its temperature is increased by 2.0 K, we can follow these steps:

### Step 1: Identify the formula for change in internal energy
The change in internal energy (ΔU) for an ideal gas can be calculated using the formula:
\[ \Delta U = n C_v \Delta T \]
where:
- \( n \) = number of moles
- \( C_v \) = molar specific heat at constant volume
- \( \Delta T \) = change in temperature

### Step 2: Determine the values given in the problem
From the problem:
- Number of moles, \( n = 3.0 \, \text{mol} \)
- Change in temperature, \( \Delta T = 2.0 \, \text{K} \)

### Step 3: Calculate \( C_v \) for helium gas
Helium is a monoatomic gas. The degrees of freedom (F) for a monoatomic gas is 3 (only translational motion). The formula for \( C_v \) is:
\[ C_v = \frac{F}{2} R \]
Substituting the values:
\[ C_v = \frac{3}{2} R \]
where \( R \) is the universal gas constant, approximately \( 8.314 \, \text{J/(mol K)} \).

### Step 4: Substitute \( C_v \) into the ΔU formula
Now we can substitute \( C_v \) into the ΔU formula:
\[ \Delta U = n \left(\frac{3}{2} R\right) \Delta T \]

### Step 5: Plug in the values
Now substituting the known values:
\[ \Delta U = 3.0 \, \text{mol} \times \left(\frac{3}{2} \times 8.314 \, \text{J/(mol K)}\right) \times 2.0 \, \text{K} \]

### Step 6: Calculate ΔU
Calculating this step-by-step:
1. Calculate \( C_v \):
\[ C_v = \frac{3}{2} \times 8.314 \approx 12.471 \, \text{J/(mol K)} \]
2. Now substitute \( C_v \) back into the ΔU equation:
\[ \Delta U = 3.0 \times 12.471 \times 2.0 \]
3. Calculate:
\[ \Delta U = 3.0 \times 12.471 \times 2.0 \approx 74.826 \, \text{J} \]

### Step 7: Round the answer
Rounding to three significant figures, we get:
\[ \Delta U \approx 74.8 \, \text{J} \]

### Final Answer
The change in internal energy of 3.0 moles of helium gas when its temperature is increased by 2.0 K is approximately **74.8 J**.

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