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Step-by-Step Solution
Step 1: Understand the relationship between $V_{\text{rms}}$ and temperature
For an ideal gas, the root-mean-square (rms) speed of molecules
$V_{\text{rms}}$ is proportional to the square root of the absolute temperature $T$.
Mathematically:
$$
V_{\text{rms}} \propto \sqrt{T}.
$$
Therefore, if $V_{\text{rms}}$ is doubled, the corresponding temperature must become four times its initial value.
Step 2: Convert the initial temperature to Kelvin and find the final temperature
The initial temperature is given as $27^\circ\text{C}$, which is
$27 + 273 = 300\,\text{K}$.
If $V_{\text{rms}}$ is doubled, then
$T_{\text{final}} = 4 \times T_{\text{initial}} = 4 \times 300 = 1200\,\text{K}$.
Step 3: Note that the process is isochoric (constant volume)
Because the volume does not change, we use the heat transfer formula for a constant-volume (isochoric) process:
$$
Q = n \, C_{v} \, \Delta T,
$$
where
n is the number of moles of the gas,
$C_{v}$ is the molar heat capacity at constant volume,
and $\Delta T$ is the change in temperature.
Step 4: Calculate the number of moles, $n$
Given the mass of nitrogen ($N_2$) is $15\,\text{g}$ and the molar mass of $N_2$ is $28\,\text{g mol}^{-1}$:
$$
n = \frac{\text{mass}}{\text{molar mass}}
= \frac{15}{28} \,\text{mol}.
$$
Step 5: Use the molar heat capacity at constant volume for a diatomic gas
For a diatomic gas like $N_2$,
$$
C_v = \frac{5}{2} \, R,
$$
where $R$ is the universal gas constant ($8.3\,\text{J mol}^{-1}\text{K}^{-1}$).
Step 6: Compute the heat transferred, $Q$
The change in temperature is $\Delta T = (1200 - 300) = 900\,\text{K}$.
Hence:
$$
Q
= n \, C_{v} \, \Delta T
= \left(\frac{15}{28}\right)
\times \left(\frac{5}{2} \times 8.3\right)
\times 900.
$$
Evaluating this step by step,
$$
\left(\frac{15}{28}\right) \approx 0.536, \quad
\frac{5}{2} \times 8.3 = 20.75,
$$
so
$$
Q
\approx 0.536 \times 20.75 \times 900
\approx 10000\,\text{J}
= 10\,\text{kJ}.
$$
Step 7: State the final answer
The heat that must be transferred to the $N_2$ gas to double its rms velocity is
10 kJ.
