Question

You are given two flasks of equal volume. Flask A contains \(\mathrm{H}_{2}\) at \(0{ }^{\circ} \mathrm{C}\) and 1 atm pressure. Flask \(\mathrm{B}\) contains \(\mathrm{CO}_{2}\) gas at \(0{ }^{\circ} \mathrm{C}\) and 2 atm pressure. Compare these two samples with respect to each of these properties. (a) Average kinetic energy per molecule (b) Average molecular velocity (c) Number of molecules

Short answer

(a) Same for both, (b) Higher in Flask A, (c) More in Flask B

Step-by-step solution

Average Kinetic Energy Per Molecule

According to the kinetic molecular theory of gases, the average kinetic energy per molecule of a gas is dependent only on the temperature, not on the type of gas or its pressure. The formula for the average kinetic energy per molecule is \( KE_{\text{average}} = \frac{3}{2} k T \), where \( k \) is the Boltzmann constant and \( T \) is the temperature in Kelvin. Since both gases are at the same temperature (0°C or 273 K), the average kinetic energy per molecule for \( \mathrm{H}_2 \) in Flask A and \( \mathrm{CO}_2 \) in Flask B is the same.

Average Molecular Velocity

The average molecular velocity is determined using the root-mean-square speed formula: \( v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \), where \( m \) is the mass of a molecule. Since \( \mathrm{H}_2 \) has a smaller molar mass compared to \( \mathrm{CO}_2 \), the molecules of \( \mathrm{H}_2 \) will have a higher average molecular velocity at the same temperature. Thus, the average molecular velocity of molecules in Flask A is higher than in Flask B.

Number of Molecules

To compare the number of molecules in the flasks, use the ideal gas law \( PV = nRT \) and also consider \( n = \frac{N}{N_A} \), where \( N \) is the number of molecules, and \( N_A \) is Avogadro's number. Rearranging this gives \( N = \frac{PVN_A}{RT} \). Since the temperature \( T \) and volume are constant for both flasks, the number of molecules is proportional to the pressure \( P \). Flask A is at 1 atm and Flask B is at 2 atm, so Flask B contains twice the number of molecules as Flask A.

Key concepts

Average Kinetic Energy

Understanding the average kinetic energy in gases begins with the Kinetic Molecular Theory. This theory tells us that the average kinetic energy of gas molecules primarily depends on the temperature. Regardless of the gas type or its pressure, the molecules at the same temperature will have the same average kinetic energy.
The formula used is \( KE_{\text{average}} = \frac{3}{2} k T \), where

• \(k\) is the Boltzmann constant
• \(T\) is the temperature in Kelvin

At 0°C (or 273 K), both \( \mathrm{H}_2 \) in Flask A and \( \mathrm{CO}_2 \) in Flask B have identical average kinetic energies because they are at the same temperature. It's a great reminder that for gas molecules, temperature is the key factor in determining energy, not the identity or pressure of the gas.

Root-Mean-Square Speed

The root-mean-square (RMS) speed is another crucial concept in gas behavior. This concept helps us understand the average velocity of gas molecules and differs from the kinetic energy because molecular mass plays a role. The RMS speed formula is \( v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \), where

• \(m\) is the mass of a single molecule
• The rest of the variables are as previously defined

Notably, lighter molecules, like \( \mathrm{H}_2 \), move faster on average than heavier molecules, such as \( \mathrm{CO}_2 \), when at the same temperature. That’s why in Flask A, the \( \mathrm{H}_2 \) molecules have a higher average velocity compared to \( \mathrm{CO}_2 \) in Flask B, illustrating the fun and intricate dance of molecules in motion.

Ideal Gas Law

The ideal gas law is an essential tool for comparing the number of gas molecules under different conditions. It is expressed as \( PV = nRT \), where

• \(P\) is the pressure
• \(V\) is the volume
• \(n\) is the number of moles
• \(R\) is the gas constant
• \(T\) is the temperature in Kelvin

This formula helps us relate these properties to the number of molecules (\(N\)) using \( n = \frac{N}{N_A} \), where

• \(N_A\) is Avogadro's number

Rearranging gives \( N = \frac{PVN_A}{RT} \). This demonstrates that if the volume and temperature are constant, the number of molecules in a gas is directly proportional to its pressure. Thus, with Flask A at 1 atm and Flask B at 2 atm, \( \mathrm{CO}_2 \) in Flask B contains twice as many molecules as \( \mathrm{H}_2 \) in Flask A. This neatly shows how pressure helps determine molecule count in gases.