One kg of a diatomic gas is at pressure of `8xx10^4N//m^2`. The density of the gas is `4kg//m^3`. What is the energy of the gas due to its thermal motion?

Options

A. `5xx10^4J`
B. `6xx10^4J`
C. `7xx10^4J`
D. `3xx10^4J`

Explanation

To find the energy of the gas due to its thermal motion, we can use the formula for the kinetic energy of a diatomic gas. The kinetic energy (KE) can be expressed in terms of pressure (P) and volume (V) as follows:

### Step 1: Calculate the Volume of the Gas
We know the mass (m) of the gas and its density (ρ). We can use the formula for density to find the volume (V):

\[ V = \frac{m}{\rho} \]

Given:
- Mass, \( m = 1 \, \text{kg} \)
- Density, \( \rho = 4 \, \text{kg/m}^3 \)

Substituting the values:

\[ V = \frac{1 \, \text{kg}}{4 \, \text{kg/m}^3} = \frac{1}{4} \, \text{m}^3 = 0.25 \, \text{m}^3 \]

### Step 2: Use the Kinetic Energy Formula
The kinetic energy of the gas can be calculated using the formula:

\[ KE = \frac{5}{2} P V \]

Where:
- \( P = 8 \times 10^4 \, \text{N/m}^2 \)
- \( V = 0.25 \, \text{m}^3 \)

Substituting the values into the formula:

\[ KE = \frac{5}{2} \times (8 \times 10^4) \times (0.25) \]

### Step 3: Calculate the Kinetic Energy
Now we can perform the calculations:

\[ KE = \frac{5}{2} \times 8 \times 10^4 \times 0.25 \]

Calculating \( 8 \times 0.25 \):

\[ 8 \times 0.25 = 2 \]

Now substituting back:

\[ KE = \frac{5}{2} \times 2 \times 10^4 = 5 \times 10^4 \, \text{J} \]

### Final Answer
Thus, the energy of the gas due to its thermal motion is:

\[ KE = 5 \times 10^4 \, \text{J} \]