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Chapter 9: Problem 1283

If the molecular weight of two gases are \(\mathrm{M}_{1}\) and \(\mathrm{M}_{2}\), then at a given temperature the ratio of root mean square velocity \(\mathrm{v}_{1}\) and \(v_{2}\) will be (A) \(\sqrt{\left(\mathrm{M}_{1} / \mathrm{M}_{2}\right)}\) (B) \(\sqrt{\left(M_{2} / M_{1}\right)}\) (C) $\left.\sqrt{[}\left(\mathrm{M}_{1}-\mathrm{M}_{2}\right) /\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)\right]$ (D) $\left.\sqrt{[}\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right) /\left(\mathrm{M}_{1}-\mathrm{M}_{2}\right)\right]$

Short Answer

Expert verified
The short answer based on the provided step-by-step solution is: The ratio of root mean square velocities for the two gases is given by \(\frac{v_{1}}{v_{2}} = \sqrt{\frac{M_{2}}{M_{1}}}\), which matches option (B).

Step by step solution

01

Recall the formula for root mean square velocity

The root mean square velocity of a gas, v_rms, can be calculated using the formula: \(v_{rms} = \sqrt{\frac{3RT}{M}}\) where R is the gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas.
02

Calculate the root mean square velocities for the two gases

Using the formula from step 1, we can calculate the root mean square velocities for the two gases: \(v_{1} = \sqrt{\frac{3RT}{M_{1}}}\) \(v_{2} = \sqrt{\frac{3RT}{M_{2}}}\)
03

Find the ratio of the two root mean square velocities

Divide the root mean square velocity of gas 1 by that of gas 2: \(\frac{v_{1}}{v_{2}}=\frac{\sqrt{\frac{3RT}{M_{1}}}}{\sqrt{\frac{3RT}{M_{2}}}}\) Simplify the expression: \(\frac{v_{1}}{v_{2}}=\sqrt{\frac{M_{2}}{M_{1}}}\)
04

Compare the result with the given options

Now, we compare our result from step 3, \(\frac{v_{1}}{v_{2}} = \sqrt{\frac{M_{2}}{M_{1}}}\) , with the given options: (A) \(\sqrt{\left(M_{1} / M_{2}\right)}\) (B) \(\sqrt{\left(M_{2} / M_{1}\right)}\) (C) \(\sqrt{\left(\frac{M_{1}-M_{2}}{M_{1}+M_{2}}\right)}\) (D) \(\sqrt{\left(\frac{M_{1}+M_{2}}{M_{1}-M_{2}}\right)}\) Our result matches option (B), so the correct answer is: (B) \(\sqrt{\left(M_{2} / M_{1}\right)}\)

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Most popular questions from this chapter

A monoatomic gas molecule has (A) Three degrees of freedom (B) Five degrees of freedom (C) Six degrees of freedom (D) Four degrees of freedom

At what temperature pressure remaining constant will the \(\mathrm{rms}\) speed of a gas molecules increase by \(10 \%\) of the \(\mathrm{rms}\) speed at NTP? (A) \(57.3 \mathrm{~K}\) (B) \(57.3^{\circ} \mathrm{C}\) (C) \(557.3 \mathrm{~K}\) (D) \(-57.3^{\circ} \mathrm{C}\)

At what temperature is the kinetic energy of a gas molecule double that of its value at \(27^{\circ} \mathrm{C}\) (A) \(54^{\circ} \mathrm{C}\) (B) \(108^{\circ} \mathrm{C}\) (C) \(327^{\circ} \mathrm{C}\) (D) \(300^{\circ} \mathrm{C}\)

The root mean square speed of hydrogen molecules of an ideal hydrogen kept in a gas chamber at \(0^{\circ} \mathrm{C}\) is \(3180 \mathrm{~ms}^{-1}\). The pressure on the hydrogen gas is (Density of hydrogen gas is $8.99 \times 10^{-2} \mathrm{~kg} / \mathrm{m}^{3}, 1 \mathrm{~atm}=1.01 \times 10^{5} \mathrm{Nm}^{-2}$ ) (A) \(1.0 \mathrm{~atm}\) (B) \(3.0 \mathrm{~atm}\) (C) \(2.0 \mathrm{~atm}\) (D) \(1.5 \mathrm{~atm}\)

The product of the pressure and volume of an ideal gas \(1 \mathrm{~s}\) (A) A constant (B) Directly proportional to its temperature. (C) Inversely proportional to its temperature. (D) Approx. equal to the universal gas constant.
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