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Chapter 9: Problem 1297

The root mean square velocity of a gas molecule of mass \(\mathrm{m}\) at a given temperature is proportional to (A) \(\mathrm{m}^{0}\) (B) \(\mathrm{m}^{-1 / 2}\) (C) \(\mathrm{m}^{1 / 2}\) (D) \(\mathrm{m}\)

Short Answer

Expert verified
The root mean square velocity of a gas molecule with mass m at a given temperature is proportional to \(m^{-1/2}\), as described by the formula \(v_{rms} = \sqrt{\frac{3kT}{m}}\), where \(v_{rms}\) is the root mean square velocity, \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of the gas molecule. Comparing this expression to the given options, we find the correct answer is Option (B): \(v_{rms} \propto m^{-1/2}\).

Step by step solution

01

Root Mean Square Velocity Formula

The root mean square velocity \(v_{rms}\) of a gas molecule can be found using the kinetic theory of gases, which relates the molecular mass, temperature, and heat capacity: \[v_{rms} = \sqrt{\frac{3kT}{m}}\] Here, \(v_{rms}\) = root mean square velocity \(k\) = Boltzmann constant \(T\) = temperature \(m\) = mass of the gas molecule
02

Find Proportionality

Now, let's find the proportionality between the mass of the gas molecule and the root mean square velocity. From the formula, we have: \[v_{rms} \propto \sqrt{\frac{3kT}{m}}\] Since \(3kT\) is a constant term, we can write: \[v_{rms} \propto \frac{1}{\sqrt{m}}\] or equivalently \[v_{rms} \propto m^{-1/2}\] Comparing this expression to the given options, we can see that it matches Option (B): \[v_{rms} \propto m^{-1/2}\] Thus, the root mean square velocity of a gas molecule with mass m at a given temperature is proportional to \(m^{-1/2}\).

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Most popular questions from this chapter

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.

A gas at \(27^{\circ} \mathrm{C}\) has a volume \(\mathrm{V}\) and pressure \(\mathrm{P}\). On heating its pressure is doubled and volume becomes three times. The resulting temperature of the gas will be (A) \(1527^{\circ} \mathrm{C}\) (B) \(600^{\circ} \mathrm{C}\) (C) \(162^{\circ} \mathrm{C}\) (D) \(1800^{\circ} \mathrm{C}\)

If three molecules have velocities \(0.5,1\) and 2 the ratio of rms speed and average speed is (The velocities are in \(\mathrm{km} / \mathrm{s}\) ) (A) \(0.134\) (B) \(1.34\) (C) \(1.134\) (D) \(13.4\)

The volume of a gas at \(20 \mathrm{C}\) is \(200 \mathrm{ml}\). If the temperature is reduced to \(-20^{\circ} \mathrm{C}\) at constant pressure, its volume will be. (A) \(172.6 \mathrm{~m} 1\) (B) \(17.26 \mathrm{ml}\) (C) \(19.27 \mathrm{ml}\) (D) \(192.7 \mathrm{ml}\)

The rms velocity of gas molecules is \(300 \mathrm{~ms}^{-1}\). The rms velocity of molecules of gas with twice the molecular weight and half the absolute temperature is (A) \(300 \mathrm{~ms}^{-1}\) (B) \(150 \mathrm{~ms}^{-1}\) (C) \(600 \mathrm{~ms}^{-1}\) (D) \(75 \mathrm{~ms}^{-1}\)
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