A sample of helium gas is at a temperature of `300 K` and a pressure of `0.5 atm`. What is the average kinetic energy of a molecule of a gas ?

Explanation

To find the average kinetic energy of a molecule of helium gas at a temperature of 300 K and a pressure of 0.5 atm, we can follow these steps:

### Step 1: Identify the formula for average kinetic energy
The average kinetic energy (KE) of a molecule in a gas can be expressed using the formula:
\[ KE = \frac{f}{2} k T \]
where:
- \( f \) is the degrees of freedom,
- \( k \) is the Boltzmann constant (\( 1.38 \times 10^{-23} \, \text{J/K} \)),
- \( T \) is the absolute temperature in Kelvin.

### Step 2: Determine the degrees of freedom for helium
Helium is a monoatomic gas. For monoatomic gases, the degrees of freedom \( f \) is given by:
\[ f = 3 \]
This is because monoatomic gases can only move in three dimensions (x, y, z).

### Step 3: Substitute the values into the formula
Now, we can substitute the values of \( f \), \( k \), and \( T \) into the kinetic energy formula:
\[ KE = \frac{3}{2} \times (1.38 \times 10^{-23} \, \text{J/K}) \times (300 \, \text{K}) \]

### Step 4: Calculate the average kinetic energy
Now we perform the calculation:
\[ KE = \frac{3}{2} \times 1.38 \times 10^{-23} \times 300 \]
\[ KE = \frac{3 \times 1.38 \times 300}{2} \times 10^{-23} \]
\[ KE = \frac{1242}{2} \times 10^{-23} \]
\[ KE = 621 \times 10^{-23} \, \text{J} \]
\[ KE = 6.21 \times 10^{-21} \, \text{J} \]

### Final Result
The average kinetic energy of a molecule of helium gas at 300 K is:
\[ KE = 6.21 \times 10^{-21} \, \text{J} \]

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