Estimate how many air molecules rebound from a wall in a typical room per second, assuming an ideal gas of N molecules contained in a cubic room with sides of length ℓ at temperature T and pressure P.
(a) Show that the frequency f with which gas molecules strike a wall is ƒ = ($\overline{{\upsilon }_x}$ /2)(P/kT) ℓ² where $\overline{{\upsilon }_x}$ is the average x component of the molecule’s velocity.
(b) Show that the equation can then be written as ƒ≈ Pℓ² /$\sqrt{4mkT}$ where m is the mass of a gas molecule.

Verified step by step guidance

1. Step 1: Start by understanding the problem. We are tasked with deriving two expressions for the frequency (ƒ) at which gas molecules strike a wall in a cubic room. The first expression involves the average x-component of velocity (υ¯ₓ), pressure (P), temperature (T), and Boltzmann constant (k). The second expression simplifies this further using the mass of a gas molecule (m).
2. Step 2: Recall the relationship between pressure and molecular motion in an ideal gas. Pressure (P) is related to the average kinetic energy of the molecules. For a single direction (x-direction), the pressure can be expressed as P = (1/3)Nmv²/ℓ³, where N is the number of molecules, m is the mass of a molecule, v² is the mean square velocity, and ℓ³ is the volume of the cubic room.
3. Step 3: The average x-component of velocity (υ¯ₓ) is related to the root mean square velocity (v_rms) by υ¯ₓ = v_rms/√3. The root mean square velocity is given by v_rms = √(3kT/m), where k is the Boltzmann constant, T is the temperature, and m is the mass of a molecule. Substitute this into the expression for υ¯ₓ.
4. Step 4: To derive the first expression for ƒ, consider the number of molecules striking a unit area of the wall per second. This depends on the number density of molecules (N/ℓ³), the average x-component of velocity (υ¯ₓ), and the area of the wall (ℓ²). Use the relationship between pressure, temperature, and molecular motion to express ƒ in terms of P, k, T, and ℓ.
5. Step 5: For the second expression, simplify the result from Step 4 by substituting υ¯ₓ = √(kT/m) and rearranging terms. This will yield ƒ ≈ Pℓ² / √(4mkT), which is the desired simplified form. Ensure all constants and variables are correctly accounted for in the derivation.

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas through the equation PV = nRT. This law assumes that gas molecules do not interact and occupy no volume, allowing for simplified calculations of gas behavior under various conditions. Understanding this law is crucial for estimating the behavior of gas molecules in a confined space, such as a room.

Average Velocity of Gas Molecules

The average velocity of gas molecules, denoted as υ¯ₓ, is a key factor in determining how frequently these molecules collide with the walls of a container. This average is derived from the kinetic theory of gases, which states that the temperature of a gas is proportional to the average kinetic energy of its molecules. Knowing this average velocity helps in calculating the frequency of collisions with the walls.

Kinetic Theory of Gases

The Kinetic Theory of Gases provides a microscopic explanation of gas behavior, positing that gases consist of a large number of small particles in constant random motion. This theory helps explain macroscopic properties such as pressure and temperature in terms of molecular motion and collisions. It is essential for deriving equations related to the frequency of molecular impacts on surfaces, as seen in the problem statement.