In Section 25-4, we showed that the collision frequency per...

In Section $25-4$, we showed that the collision frequency per unit area is (Equation $25.48$ ) $$ z_{\mathrm{coll}}=\frac{\rho\langle u\rangle}{4} $$ Use Equation 1 and the ideal-gas law to show that $J_{N}$, the number of molecules striking a surface of unit area $\left(1 \mathrm{~m}^{2}\right)$ in one second, is $$ J_{N}=\frac{P N_{\mathrm{A}}}{(2 \pi M R T)^{1 / 2}} $$ where $M$ is the molar mass of the molecule, $P$ is the pressure of the gas, and $T$ is the temperature. How many nitrogen molecules will strike a $1.00-\mathrm{cm}^{2}$ surface in $1.00 \mathrm{~s}$ at $298.1 \mathrm{~K}$ and a gas pressure of $1.05 \times 10^{-6} \mathrm{~Pa}$ ?

In Section $25-4$, we showed that the collision frequency per unit area is (Equation $25.48$ )
$$
z_{\mathrm{coll}}=\frac{\rho\langle u\rangle}{4}
$$
Use Equation 1 and the ideal-gas law to show that $J_{N}$, the number of molecules striking a surface of unit area $\left(1 \mathrm{~m}^{2}\right)$ in one second, is
$$
J_{N}=\frac{P N_{\mathrm{A}}}{(2 \pi M R T)^{1 / 2}}
$$
where $M$ is the molar mass of the molecule, $P$ is the pressure of the gas, and $T$ is the temperature. How many nitrogen molecules will strike a $1.00-\mathrm{cm}^{2}$ surface in $1.00 \mathrm{~s}$ at $298.1 \mathrm{~K}$ and a gas pressure of $1.05 \times 10^{-6} \mathrm{~Pa}$ ?

Key Concepts

Maxwell-Boltzmann Speed Distribution

The Maxwell-Boltzmann speed distribution describes the statistical distribution of speeds among molecules in a gas at a given temperature. This distribution is used to derive average speeds and other statistical properties that are integral in computing collision rates and fluxes, influencing factors such as the numerical coefficients in the derived formulas.

Ideal Gas Law

The ideal gas law (PV = nRT) relates pressure, volume, and temperature of a gas to the number of moles present. By connecting macroscopic measurements with molecular properties, it enables the substitution of parameters like density and number density into equations derived from kinetic theory, facilitating the calculation of molecular flux.

Kinetic Theory of Gases

The kinetic theory of gases explains that gases consist of many molecules in constant, random motion, and their collisions with surfaces and among themselves account for properties like pressure and temperature. This theory provides the foundation for deriving expressions for collision frequency and molecular speeds, linking microscopic behavior to macroscopic thermodynamic variables.

Molecular Flux (Collision Frequency)

Molecular flux refers to the rate at which gas molecules collide with a surface per unit area per unit time. In kinetic theory, this flux is calculated by relating the number density of molecules and their average speed. It is central to understanding how microscopic collisions give rise to macroscopic observables such as pressure.

Transcript

00:01 So we're doing chapter 18, problem 49 here.

00:04 So part a says show that the number of collisions a molecule makes per second called the collision frequency, which we call f, is given by b average over this for the mean free path.

00:19 And if we reevaluate that, that is the same as 4, root 2, pi r squared, v average in all.

00:32 Over v.

00:35 Okay.

00:36 So this is what we want to show.

00:40 And to start this, we can think of the average speed as the molecule as v bar.

00:45 Then that means the average time between collisions is the mean -free path divided by that.

00:53 Right.

00:54 So this gives us our distance over velocity gives us a time.

01:00 And if we want the frequency of that, that's just one over the time.

01:06 So this is the same as 1 over delta t average or v bar over lm.

01:15 And as we can see here, we already know that v bar over lm is the same as 4 root 2 pi r squared, v bar n over v.

01:27 It's as simple as that.

01:29 Cool.

01:31 Let's move on to part b.

01:35 So part b asks us to figure out what is the collision frequency for into molecules in air at a temperature 20 degrees celsius and pressure of 1 times 10 in the negative 2 atmospheres.

01:54 Okay.

01:56 So first let's figure out what the mean speed is and that's given as 8 kbt over the pi times the molar mass.

02:04 So this is 8 times 1 .38, 10 to negative 23 times 293 kelvin all over pi times what is the molar mass...

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