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. `3xx10^(4) J`
B. `5xx10^(4) J`
C. `6xx10^(4) J`
D. `4xx10^(4) J`

Explanation

To find the energy of the gas due to its thermal motion, we can use the formula for the thermal energy of a diatomic gas. The energy \( U \) can be expressed as:

\[ U = \frac{5}{2} nRT \]

However, since we don't have the temperature \( T \) directly, we can use the ideal gas law, which states:

\[ PV = nRT \]

From this, we can rearrange the formula for thermal energy in terms of pressure \( P \) and volume \( V \):

\[ U = \frac{5}{2} PV \]

### Step 1: Calculate the Volume of the Gas

We know the mass \( m \) of the gas and its density \( \rho \). The volume \( V \) can be calculated using the formula:

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

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: Substitute Values into the Thermal Energy Formula

Now we can substitute the values of pressure \( P \) and volume \( V \) into the thermal energy formula:

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

Substituting these into the equation:

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

### Step 3: Calculate the Energy

Now we perform the calculation:

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

Calculating step-by-step:

1. Calculate \( 8 \times 0.25 = 2 \)
2. Then, \( \frac{5}{2} \times 2 = 5 \)
3. Finally, \( U = 5 \times 10^4 = 50000 \, \text{Joules} \)

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

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

### Final Answer

The energy of the gas due to its thermal motion is \( 50000 \, \text{J} \).

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