A tank contains 135 moles of the monatomic gas argon at a temperature of `15.3^@ C`. How much energy must be added to the gas to increase its temperature to `45.0^@ `C?

Options

A. `2.50 xx 10^3 J`
B. `5.00 xx 10^4 J`
C. `3.33 xx 10^4 J`
D. `5.70 xx 10^5 J`

Answer

To solve the problem of how much energy must be added to the gas to increase its temperature, we can follow these steps:

### Step 1: Convert the temperatures from Celsius to Kelvin
The initial temperature \( T_1 \) is given as \( 15.3^\circ C \). To convert this to Kelvin:
\[ T_1 = 15.3 + 273 = 288.3 \, K \]

The final temperature \( T_2 \) is given as \( 45.0^\circ C \). To convert this to Kelvin:
\[ T_2 = 45.0 + 273 = 318.0 \, K \]

### Step 2: Calculate the change in temperature (\( \Delta T \))
The change in temperature is calculated as:
\[ \Delta T = T_2 - T_1 = 318.0 \, K - 288.3 \, K = 29.7 \, K \]

### Step 3: Determine the formula for the change in internal energy (\( \Delta U \))
For a monatomic ideal gas, the change in internal energy can be calculated using the formula:
\[ \Delta U = \frac{3}{2} n R \Delta T \]
where:
- \( n \) is the number of moles,
- \( R \) is the universal gas constant (\( R = 8.314 \, J/(mol \cdot K) \)),
- \( \Delta T \) is the change in temperature.

### Step 4: Substitute the known values into the formula
Here, \( n = 135 \) moles, \( R = 8.314 \, J/(mol \cdot K) \), and \( \Delta T = 29.7 \, K \):
\[ \Delta U = \frac{3}{2} \times 135 \times 8.314 \times 29.7 \]

### Step 5: Calculate the change in internal energy
First, calculate \( \frac{3}{2} \times 135 \):
\[ \frac{3}{2} \times 135 = 202.5 \]

Now, calculate \( 202.5 \times 8.314 \):
\[ 202.5 \times 8.314 \approx 1688.4 \]

Finally, multiply this result by \( 29.7 \):
\[ \Delta U \approx 1688.4 \times 29.7 \approx 50000.0 \, J \]

### Conclusion
The energy that must be added to the gas to increase its temperature from \( 15.3^\circ C \) to \( 45.0^\circ C \) is approximately \( 5.0 \times 10^4 \, J \).

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