© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Express thermal energy of an ideal monoatomic gas
The internal (thermal) energy $U$ of a monoatomic ideal gas of $n$ moles is given by:
$U = \frac{3}{2} n R T$
Using the ideal gas law $PV = n R T$, we can substitute $n R T$ with $PV$. This gives:
$U = \frac{3}{2} P V
$
Step 2: Determine volume $V$ from the given data
We know:
Pressure $P = 4 \times 10^{4}\,\text{N/m}^2$
Density $\rho = 8\,\text{kg/m}^3$
Mass $m = 2\,\text{kg}$ of the gas
The density $\rho$ is defined as mass per unit volume:
$\rho = \frac{m}{V} \quad \Longrightarrow \quad V = \frac{m}{\rho}
$
Substitute the given values to find $V$:
$V = \frac{2\,\text{kg}}{8\,\text{kg/m}^3} = \frac{2}{8} \,\text{m}^3 = 0.25\,\text{m}^3
$
Step 3: Calculate the thermal energy
Using $U = \frac{3}{2} P V$, substitute $P$ and $V$:
$U = \frac{3}{2} \times \left(4 \times 10^{4}\,\text{N/m}^2\right) \times 0.25\,\text{m}^3
$
Perform the multiplication:
$U = \frac{3}{2} \times 4 \times 10^{4} \times 0.25
= \frac{3}{2} \times 10^{4}
= 1.5 \times 10^{4} \,\text{J}
$
Step 4: Conclude the order of magnitude
The calculated thermal energy is on the order of $10^{4}\,\text{J}$. Hence, the correct answer is:
$\boxed{10^{4}\,\text{J}}
