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Step-by-Step Solution
Step 1: Understand the Question and Given Information
We have a closed vessel containing two diatomic gases A and B. The molar mass of A is 16 times that of B. The mass of gas A present in the vessel is 2 times that of gas B.
From these facts, we analyze four statements regarding their kinetic energies, root mean square velocities, pressures, and relative numbers of molecules to determine which one is false.
Step 2: Relate Molar Masses and Masses
Let the molar mass of B be $M_B$. Then:
$M_A = 16 \, M_B$.
Similarly, let the mass of B be $m_B$. Then:
$m_A = 2 \, m_B$.
Step 3: RMS Velocity Comparison
The root mean square (rms) velocity of a gas at temperature $T$ is given by:
$$
v_{\mathrm{rms}} = \sqrt{\frac{3RT}{M}},
$$
where $M$ is the molar mass (in kg/mol if $R$ is in J/(mol·K)). For A and B:
$$
v_{\mathrm{rms},A} = \sqrt{\frac{3RT}{M_A}},
\quad
v_{\mathrm{rms},B} = \sqrt{\frac{3RT}{M_B}}.
$$
Taking the ratio:
$$
\frac{v_{\mathrm{rms},B}}{v_{\mathrm{rms},A}}
= \sqrt{\frac{M_A}{M_B}}
= \sqrt{\frac{16\,M_B}{M_B}} = 4.
$$
Hence, $v_{\mathrm{rms},B}$ is four times $v_{\mathrm{rms},A}$. Therefore, any statement claiming that B’s rms speed is four times that of A is correct.
Step 4: Determine Number of Moles and Molecules
The number of moles $n$ of a gas is given by:
$$
n = \frac{m}{M}.
$$
For A and B:
$$
n_A = \frac{m_A}{M_A},
\quad
n_B = \frac{m_B}{M_B}.
$$
Substituting $m_A = 2\,m_B$ and $M_A = 16\,M_B$:
$$
n_A
= \frac{2\,m_B}{16\,M_B}
= \frac{m_B}{8\,M_B},
\quad
n_B = \frac{m_B}{M_B}.
$$
Thus,
$$
n_B = 8 \, n_A.
$$
Since the number of molecules is directly proportional to the number of moles, the number of molecules of B is 8 times that of A. A statement indicating an 8:1 ratio of molecules/B to A is correct.
Step 5: Compare Partial Pressures
From the ideal gas law $P\,V = n\,R\,T$, under the same conditions of volume $V$ and temperature $T$, the partial pressure exerted by each gas is proportional to its number of moles. Hence,
$$
P_A \propto n_A,
\quad
P_B \propto n_B.
$$
Since $n_B = 8\,n_A$, we get:
$$
P_B = 8\,P_A.
$$
Therefore, gas B’s partial pressure is 8 times that of gas A, not 6 times.
Step 6: Identify the False Statement
The statement that “Pressure exerted by B is six times that of A” conflicts with our calculation, which shows an 8:1 ratio. Hence, this statement is false.
Final Conclusion
The false statement is: “Pressure exerted by B is six times of that exerted by A.” In reality, B’s partial pressure is eight times that of A.
