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. `5 xx 10^(4)J`
B. `6 xx 10^(4)J`
C. `7 xx 10^(4)J`
D. `3 xx 10^(4)J`

Explanation

To find the energy of the diatomic gas due to its thermal motion, we can use the kinetic energy formula for an ideal gas. Here’s a step-by-step solution:

### Step 1: Understand the formula for kinetic energy of gas
The kinetic energy (KE) of a gas can be expressed as:
\[ KE = \frac{5}{2} nRT \]
where:
- \( n \) is the number of moles of the gas,
- \( R \) is the universal gas constant (approximately \( 8.314 \, \text{J/(mol K)} \)),
- \( T \) is the temperature in Kelvin.

### Step 2: Use the ideal gas law
From the ideal gas law, we know:
\[ PV = nRT \]
This allows us to express the kinetic energy in terms of pressure (P) and volume (V):
\[ KE = \frac{5}{2} PV \]

### Step 3: Calculate the volume of the gas
We can find the volume (V) of the gas using the mass and density:
\[ V = \frac{m}{\rho} \]
where:
- \( m = 1 \, \text{kg} \) (mass of the gas),
- \( \rho = 4 \, \text{kg/m}^3 \) (density of the gas).

Calculating the volume:
\[ V = \frac{1 \, \text{kg}}{4 \, \text{kg/m}^3} = \frac{1}{4} \, \text{m}^3 \]

### Step 4: Substitute values into the kinetic energy formula
Now we can substitute the values of pressure and volume into the kinetic energy formula:
\[ KE = \frac{5}{2} PV \]
Given \( P = 8 \times 10^4 \, \text{N/m}^2 \) and \( V = \frac{1}{4} \, \text{m}^3 \):
\[ KE = \frac{5}{2} \times (8 \times 10^4) \times \left(\frac{1}{4}\right) \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ KE = \frac{5}{2} \times 8 \times 10^4 \times \frac{1}{4} \]
\[ = \frac{5 \times 8 \times 10^4}{2 \times 4} \]
\[ = \frac{40 \times 10^4}{8} \]
\[ = 5 \times 10^4 \, \text{J} \]

### Final Answer
The energy of the gas due to its thermal motion is:
\[ KE = 5 \times 10^4 \, \text{J} \]