M A hydrogen molecule (diameter 1.0 ×10^-8 cm ), traveling at the rms speed, escapes from a 400

M A hydrogen molecule (diameter 1.0 ×10^-8 cm ), traveling...

M A hydrogen molecule (diameter $1.0 \times 10^{-8} \mathrm{~cm}$ ), traveling at the rms speed, escapes from a $4000 \mathrm{~K}$ furnace into a chamber containing cold argon atoms (diameter $3.0 \times 10^{-8} \mathrm{~cm}$ ) at a density of $4.0 \times 10^{19} \mathrm{atoms} / \mathrm{cm}^3$. (a) What is the speed of the hydrogen molecule? (b) If it collides with an argon atom, what is the closest their centers can be, considering each as spherical? (c) What is the initial number of collisions per second experienced by the hydrogen molecule? (Hint: Assume that the argon atoms are stationary. Then the mean free path of the hydrogen molecule is given by Eq. 19.5.2 and not Eq. 19.5.1.)

Key Concepts

Root Mean Square Speed

The root mean square (rms) speed is a statistical measure of the average speed of particles in a gas, derived from the kinetic theory of gases. It is given by the square root of the average of the squares of the speeds of all the particles, and is calculated using the temperature of the gas and the mass of its molecules. This concept is crucial for determining the kinetic energy distribution and for analyzing the motion of gas particles at a given temperature.

Hard Sphere Model

The hard sphere model is a simplified way of modeling gas molecules as non-deformable spheres that interact only on contact. In collisions, the molecules are considered as rigid spheres with a defined diameter. This model is instrumental in determining the effective collision cross section since it provides a geometric basis for computing the distance of closest approach between molecule centers during a collision.

Collision Cross Section

The collision cross section is a measure of the effective area that a molecule presents as a target for collisions. For spherical particles, it is derived from the sum of the radii of the colliding particles, typically resulting in a value proportional to the square of the sum of the individual diameters. This parameter directly influences the rate of collisions and is used in mean free path calculations to estimate how frequently collisions occur in a gas.

Mean Free Path and Collision Frequency

Mean free path is the average distance a molecule travels between successive collisions, and it is calculated using the collision cross section and the number density of target particles. The collision frequency indicates how many collisions a molecule undergoes per unit time and is inversely related to the mean free path. Together, these concepts are vital for understanding molecular dynamics and transport phenomena in gases, as they link microscopic interactions to macroscopic observables such as diffusion and viscosity.

Transcript

00:01 Now, we know that the temperature of the hydrogen atoms is 4 ,000 kelvin.

00:10 We also know that the diameter, we also know that the diameter of the hydrogen atoms is two times, one times 10 par minus 8 centimeters, so that's 1 .10 par minus 10 meters.

00:33 This is for hydrogen, and for argon, which is the gas, the hydrogen will colloquium.

00:40 With the diameter is 3 times 10 power minus 10 centimeters.

00:49 So for part a, let's calculate the rms velocity.

00:59 Now we know that vrms is square root of 3 rt by m, where m is the molar mass.

01:04 So that's square root of 3.

01:06 R is 8 .314 jule per mole kelvin, temperature is 4 ,000 kelvin, divided by the molar mass of hydrogen is 2 grams per mole so that's 2 into 10 bar minus 3 kilograms per mole the moles cancel the kelvin cancel and jule per kilogram is meter per second square and square root of that is meter per second which is exactly the units we need and the number turns out to be 7 into 10 part 3 so that's 7 ,000 meter per second now for part b the closest distance the distance they can, the two atoms can get to is when they're in contact with each other.

01:55 So the radius of the hydrogen atom plus the radius of the argon atom is the closest distance they can get.

02:05 So the closest distance d is radius of hydrogen atom, so that's 0 .5 into 10 power minus 10 meters.

02:14 And for argon it's 1 .5 into 10 power minus 10 meters.

02:18 So that's 2 into 10 power minus 10 meters or two angst...

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