Calculate the root mean square velocities of \(\mathrm{CH}_{4}(g)\) and \(\mathrm{N}_{2}(g)\) molecules at 273 \(\mathrm{K}\) and 546 \(\mathrm{K} .\)

To find the root mean square velocity of a gas, we will use the following formula: \[v_{rms} = \sqrt{\frac{3RT}{M}}\] Where \(v_{rms}\) is the root mean square velocity, \(R\) is the ideal gas constant (8.314 J/mol K), \(T\) is the temperature in Kelvin, and \(M\) is the molecular mass in kg/mol.

To find the molecular mass of CH4 and N2 in kg/mol, we'll need to multiply the atomic mass of each element by its frequency in the molecule and then convert the result to kg. For CH4: \(M_{CH4} = (12.01\,\mathrm{g/mol}\times 1) + (1.01\,\mathrm{g/mol}\times 4)\) \(M_{CH4} = 12.01\,\mathrm{g/mol} + 4.04\,\mathrm{g/mol}\) \(M_{CH4} = 16.05\,\mathrm{g/mol}\) Converting to kg/mol, we get: \(M_{CH4} = 0.01605\,\mathrm{kg/mol}\) For N2: \(M_{N2}= (14.01\,\mathrm{g/mol}\times 2)\) \(M_{N2} = 28.02\,\mathrm{g/mol}\) Converting to kg/mol, we get: \(M_{N2} = 0.02802\,\mathrm{kg/mol}\)

Now that we have the molecular masses and the temperature values, we can calculate the root mean square velocities for CH4 and N2 at 273 K and 546 K. For CH4 at 273 K: \(v_{rms} = \sqrt{\frac{3\times 8.314\,\mathrm{J/(mol\cdot K)}\times 273\,\mathrm{K}}{0.01605\,\mathrm{kg/mol}}}\) \(v_{rms} \approx 922.65\,\mathrm{m/s}\) For CH4 at 546 K: \(v_{rms} = \sqrt{\frac{3\times 8.314\,\mathrm{J/(mol\cdot K)}\times 546\,\mathrm{K}}{0.01605\,\mathrm{kg/mol}}}\) \(v_{rms} \approx 1304.97\,\mathrm{m/s}\) For N2 at 273 K: \(v_{rms} = \sqrt{\frac{3\times 8.314\,\mathrm{J/(mol\cdot K)}\times 273\,\mathrm{K}}{0.02802\,\mathrm{kg/mol}}}\) \(v_{rms} \approx 661.54\,\mathrm{m/s}\) For N2 at 546 K: \(v_{rms} = \sqrt{\frac{3\times 8.314\,\mathrm{J/(mol\cdot K)}\times 546\,\mathrm{K}}{0.02802\,\mathrm{kg/mol}}}\) \(v_{rms} \approx 935.68\,\mathrm{m/s}\)

The root mean square velocities for each gas at the given temperatures are: For CH4 at 273 K: \(v_{rms} \approx 922.65\,\mathrm{m/s}\) For CH4 at 546 K: \(v_{rms} \approx 1304.97\,\mathrm{m/s}\) For N2 at 273 K: \(v_{rms} \approx 661.54\,\mathrm{m/s}\) For N2 at 546 K: \(v_{rms} \approx 935.68\,\mathrm{m/s}\)