Question

A bottle contains \(1.0 \mathrm{~mol} \mathrm{He}(\mathrm{g})\) and a second bottle contains \(1.0 \mathrm{~mol} \mathrm{} \operatorname{Ar}(\mathrm{g})\) at the same temperature. At that temperature, the root mean square speed of He is \(1477 \mathrm{~m} \cdot \mathrm{s}^{-1}\) and that of Ar is \(467 \mathrm{~m} \cdot \mathrm{s}^{-1}\). What is the ratio of the number of He atoms in the first bottle to the number of Ar atoms in the second bottle having these speeds? Assume that both gases behave ideally.

Short answer

The ratio of the number of He atoms to Ar atoms is 1:1.

Step-by-step solution

Understand the Relationship Between Molar Amount and Number of Atoms

Since both bottles contain 1.0 mol of their respective gases (He and Ar), they contain the same number of atoms. This is due to Avogadro's law which states that equal volumes of all gases, at the same temperature and pressure, have the same number of molecules. The number of atoms is given by Avogadro's number, which is approximately \(6.022 \times 10^{23}\) atoms/mol.

Calculate the Ratio of Speeds

Given the root mean square speeds of He and Ar, we can calculate the ratio of the speeds. The ratio of the speeds of He to Ar is \(\frac{v_{He}}{v_{Ar}} = \frac{1477 \, \mathrm{m/s}}{467 \, \mathrm{m/s}}\).

Simplify the Ratio

To find the simplified ratio of speeds, divide the speed of He by the speed of Ar: \(\frac{1477}{467}\).

State the Ratio of the Number of He Atoms to Ar Atoms

Since the number of molecules of each gas is the same (1.0 mol each), the ratio of the number of He atoms to the number of Ar atoms is 1:1, regardless of their individual speeds at the same temperature.

Key concepts

Root Mean Square Speed

On our journey to understand gas behavior, we come across a concept known as the root mean square speed (RMS speed). This is a way to express the average speed of particles in a gas. Think of it like how you would calculate the average speed of a group of runners in a race, but a bit more math-heavy.

RMS speed is pertinent when evaluating the kinetic energy of a gas because it ties into the energy each gas molecule has. The formula for calculating the RMS speed, \( v_{rms} \), for the particles of an ideal gas is given by: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \] where \( k \) is the Boltzmann constant, \( T \) is the absolute temperature in Kelvin, and \( m \) is the molar mass of the gas molecule. Because \( m \) is the molar mass of the individual molecules, which is typically expressed in kilograms in this equation, we typically convert the molar mass from grams per mole to kilograms per mole.

This equation shows that at a given temperature, lighter molecules, like helium (He), will typically have higher RMS speeds than heavier molecules like argon (Ar), given what we observed in our exercise.

Molar Mass

On to another cornerstone of chemical understanding: Molar mass. This value is essentially the weight of one mole of a substance, often expressed in grams per mole (g/mol). Picture molar mass as a way to bridge the microscopic world of atoms and molecules with scales we can measure on.

For example, helium (He) has a molar mass of about 4.00 g/mol, while argon (Ar) has a molar mass of roughly 39.95 g/mol. This substantial difference in molar mass explains why, at the same temperature, helium atoms zip around much faster than argon atoms, as seen through the vast difference in their RMS speeds from our exercise.

Molar mass is an invaluable tool for chemists. It allows us to measure out amounts of a substance in the lab in grams that will correspond to exact mole ratios for chemical reactions. This correlation is principal in stoichiometry calculations and in understanding the behavior of gases.

Ideal Gas Behavior

The ideal gas law is a beautiful simplification in chemistry. It's like assuming all cars on the highway are driving at the speed limit—unlikely, but it gives us an easy way to predict traffic flow. An ideal gas is a hypothetical gas that perfectly follows the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature.

For ideal gases, we assume that the particles don't attract or repel each other and take up no space. Of course, real gases aren't quite so obedient, but this assumption works surprisingly well under many conditions, especially at higher temperatures and lower pressures where gases behave more 'ideally'.

In the context of our exercise, this ideal behavior allows us to say that one mole of helium behaves the same as one mole of argon under the same conditions of temperature and volume, leading us to Avogadro's law, which ties into another concept - Avogadro's number.

Avogadro's Number

Wrapping our heads around Avogadro's number sometimes feels like trying to visualize the vastness of the universe. This number, approximately \(6.022 \times 10^{23}\), represents the number of particles, whether they are atoms or molecules, in one mole of a substance.

We don't often encounter numbers this large, but in chemistry, they're indispensable. Every time you measure out a mole of a substance, you're handling Avogadro's number of things! It's like knowing the exact number of jellybeans in a really huge jar, without having to count each one.

In the exercise, understanding Avogadro's number was critical because it tells us that each bottle, one containing helium and the other argon, had the same number of gas particles. Consequently, despite their differing RMS speeds, the ratio of helium atoms to argon atoms in the two bottles remains a straightforward 1:1.