Calculate the average and total kinetic energy of 0.5 mole of an ideal gas at `0^@C`

Answer

To calculate the average and total kinetic energy of 0.5 moles of an ideal gas at \(0^\circ C\), we can follow these steps:

### Step 1: Convert Temperature to Kelvin
The first step is to convert the temperature from Celsius to Kelvin. The conversion formula is:
\[ T(K) = T(°C) + 273.15 \]
For \(0^\circ C\):
\[ T = 0 + 273.15 = 273.15 \, K \]

### Step 2: Calculate Average Kinetic Energy
The average kinetic energy (\(KE_{avg}\)) of one molecule of an ideal gas is given by the formula:
\[ KE_{avg} = \frac{3}{2} k_B T \]
where \(k_B\) is the Boltzmann constant, approximately \(1.38 \times 10^{-23} \, J/K\).

Substituting the values:
\[ KE_{avg} = \frac{3}{2} \times (1.38 \times 10^{-23} \, J/K) \times (273.15 \, K) \]
Calculating this gives:
\[ KE_{avg} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 273.15 \approx 5.65 \times 10^{-21} \, J \]

### Step 3: Calculate Total Kinetic Energy
The total kinetic energy (\(KE_{total}\)) for \(N\) moles of gas can be calculated using the formula:
\[ KE_{total} = \frac{3}{2} N R T \]
where \(N\) is the number of moles, \(R\) is the universal gas constant (\(8.314 \, J/(mol \cdot K)\)), and \(T\) is the temperature in Kelvin.

For \(0.5\) moles:
\[ KE_{total} = \frac{3}{2} \times 0.5 \times 8.314 \, J/(mol \cdot K) \times 273.15 \, K \]
Calculating this gives:
\[ KE_{total} = \frac{3}{2} \times 0.5 \times 8.314 \times 273.15 \approx 1702.3 \, J \]

### Final Results
- Average Kinetic Energy: \(KE_{avg} \approx 5.65 \times 10^{-21} \, J\)
- Total Kinetic Energy: \(KE_{total} \approx 1702.3 \, J\)

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