Question

A mixture of 2 moles of helium gas (atomic mass \(=4 \mathrm{amu}\) ) and 1 mole of argon gas (atomic mass \(=40 \mathrm{amu}\) ) is kept at \(300 \mathrm{~K}\) in a container. The ratio of the \(\mathrm{rms}\) speeds \(\left(\frac{v_{n n s}(\text { helium })}{v_{r m s}(\text { argon })}\right)\) is (A) \(0.32\) (B) \(0.45\) (C) \(2.24\) (D) \(3.16\)

Short answer

The ratio of the rms speeds of helium to argon is approximately 3.16.

Step-by-step solution

Understand Root Mean Square (rms) Velocity

The root mean square (rms) velocity of a gas particle can be calculated using the formula: \(v_{rms} = \[ \frac{3kT}{M} \]\), where \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas in kilograms per mole. In this formula, \(k = 1.38 \times 10^{-23} J/K\), and \(T = 300K\) in this exercise.

Calculate the molar mass of helium and argon in kg/mol

First, we need to convert the given atomic masses from amu to kg/mol. To do this, we use the conversion factor that 1 amu = \(1.66 \times 10^{-27}\) kg. For helium, \(M_{\text{He}} = 4 \text{ amu} = 4 \times 1.66 \times 10^{-27}\) kg. For argon, \(M_{\text{Ar}} = 40 \text{ amu} = 40 \times 1.66 \times 10^{-27}\) kg.

Calculate the rms speed of helium and argon

Now, we plug the molar masses of helium and argon into the rms velocity formula: \(v_{\text{rms}}(\text{helium}) = \[ \frac{3kT}{M_{\text{He}}} \]\) and \(v_{\text{rms}}(\text{argon}) = \[ \frac{3kT}{M_{\text{Ar}}} \]\). The Boltzmann constant and temperature are the same for both gases, so they cancel out when taking the ratio.

Calculate the ratio of rms speeds of helium to argon

To find the ratio of rms speeds, we divide the rms speed of helium by the rms speed of argon: \(\frac{v_{\text{rms}}(\text{helium})}{v_{\text{rms}}(\text{argon})} = \frac{\sqrt{\frac{3kT}{M_{\text{He}}}}}{\sqrt{\frac{3kT}{M_{\text{Ar}}}}} = \frac{(M_{\text{Ar}})^{1/2}}{(M_{\text{He}})^{1/2}}\). We just need the square root of the ratio of the molar masses because \(kT\) cancels out.

Complete the calculation

Plugging in the molar masses that were calculated in kg/mol, we get the following ratio: \(\left( \frac{(4 \times 1.66 \times 10^{-27})}{(40 \times 1.66 \times 10^{-27})} \right)^{1/2}\), which simplifies to: \(\left( \frac{1}{10} \right)^{1/2} = \frac{1}{\sqrt{10}} \approx 0.316\). Therefore, the correct answer is (D) \(3.16\).

Key concepts

Root Mean Square Velocity

Root Mean Square (rms) velocity represents the average speed of particles in a gas. It's a statistical measure that provides an idea of how fast the particles are moving on average in a given gas, under certain conditions. By using the formula
\(v_{rms} = \sqrt{\frac{3kT}{M}}\),
we can obtain this mean value where

• \(k\) corresponds to the Boltzmann constant,
• \(T\) is the absolute temperature in Kelvin,
• \(M\) is the molar mass of the gas in kilograms per mole.

This concept is crucial in the study of thermodynamics and kinetic theory of gases, giving insights into the relationship between temperature and the kinetic energy of particles.

Boltzmann Constant

The Boltzmann constant, denoted by \(k\) and approximately equal to \(1.38 \times 10^{-23} J/K\), is a fundamental physical constant that links the average kinetic energy of particles in a gas to the temperature of the gas.

It is named after the Austrian physicist Ludwig Boltzmann, and it provides a microscopic connection between macroscopic variables (like temperature) and the microscopic behavior of atoms and molecules. The constant is a key feature in the rms velocity formula and plays an indispensable role in various gas laws and universal principles of statistical mechanics.

Molar Mass Conversion

Molar mass conversion is an essential step when working with gas laws and physics problems. It involves converting the molecular weight from atomic mass units (amu) to kilograms per mole (kg/mol), which is necessary for calculations in SI units.

The process is straightforward yet vital: by using the conversion factor \(1 \text{ amu} = 1.66 \times 10^{-27} \text{ kg}\), molar masses of elements like helium and argon become analyzable in calculations involving kinetic energy, rms speed, and other physical properties.

Gas Particle Physics

Gas particle physics encompasses the study of the behavior, properties, and motion of particles that make up a gas. Understanding this field involves exploring concepts such as the rms speed, which reflects the kinetic energy resulting from the random, chaotic movement of gas particles.

The principles governing these particles are rooted in the kinetic theory, which is used to explain macroscopic properties of gases, such as pressure and temperature, in terms of microscopic particles' behavior and interactions.