A vessel contains a mixture of helium and argon gas in equilibrium at 423K. Calculate the average kinetic energy of each gas molecule.

Explanation

To calculate the average kinetic energy of each gas molecule in a mixture of helium and argon at a temperature of 423 K, we can use the formula for the average kinetic energy of a gas molecule:

### Step 1: Write down the formula for average kinetic energy
The average kinetic energy (KE) of a gas molecule is given by the formula:
\[ KE = \frac{3}{2} k T \]
where:
- \( k \) is the Boltzmann constant (\(1.38 \times 10^{-23} \, \text{J/K}\))
- \( T \) is the absolute temperature in Kelvin

### Step 2: Substitute the values into the formula
Given that the temperature \( T = 423 \, \text{K} \), we can substitute the values into the formula:
\[ KE = \frac{3}{2} \times (1.38 \times 10^{-23} \, \text{J/K}) \times (423 \, \text{K}) \]

### Step 3: Calculate the average kinetic energy
Now, we perform the multiplication:
1. Calculate \( \frac{3}{2} \times 1.38 \):
\[ \frac{3}{2} \times 1.38 = 2.07 \]
2. Now, multiply this result by the temperature:
\[ KE = 2.07 \times 423 \approx 875.01 \times 10^{-23} \, \text{J} \]
3. Finally, we convert this to scientific notation:
\[ KE \approx 8.76 \times 10^{-21} \, \text{J} \]

### Step 4: State the final result
Thus, the average kinetic energy of each gas molecule in the mixture is:
\[ KE \approx 8.76 \times 10^{-21} \, \text{J} \]