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Step-by-Step Solution
Step 1: Calculate the Volume of the Gas
The density of the gas is given as $4 \,\text{kg/m}^3$, and the mass of the gas is $1 \,\text{kg}$. The volume can be found using the relation between mass, density, and volume:
$ \text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{1 \,\text{kg}}{4 \,\text{kg/m}^3} = \frac{1}{4} \,\text{m}^3 $
Step 2: Identify the Formula for Thermal Energy of a Diatomic Gas
For a diatomic gas, the internal (translational + rotational) energy or the kinetic energy due to thermal motion is given by:
$ K.E. = \frac{5}{2} P \, V $
Here, $P$ is the pressure and $V$ is the volume of the gas.
Step 3: Substitute the Known Values into the Formula
Given pressure $P = 8 \times 10^4 \,\text{N/m}^2$ and volume $V = \tfrac{1}{4} \,\text{m}^3$, substitute these into the formula:
$ K.E. = \frac{5}{2} \times P \times V = \frac{5}{2} \times 8 \times 10^4 \,\text{N/m}^2 \times \frac{1}{4} \,\text{m}^3 $
Step 4: Perform the Calculations
First calculate the product of $P$ and $V$:
$ P \times V = 8 \times 10^4 \times \frac{1}{4} = 2 \times 10^4 $
Now multiply by $\frac{5}{2}$:
$ K.E. = \frac{5}{2} \times 2 \times 10^4 = 5 \times 10^4 \,\text{J} $
Step 5: State the Final Answer
Therefore, the energy of the gas due to its thermal motion is:
$ 5 \times 10^4 \,\text{J} $
