Question

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 = ( $\overline{υ_{x}}$ /2)(P/kT) ℓ² where $\overline{υ_{x}}$ is the average x component of the molecule’s velocity.
(b) Show that the equation can then be written as ƒ≈ Pℓ² / $the square root of 4 m k T$ where m is the mass of a gas molecule.

Accepted Answer

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). 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. 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. 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. 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.

Risposta video verificata per un problema simile:

Concetti chiave

Ideal Gas Law

Average Velocity of Gas Molecules

Kinetic Theory of Gases