Collision Frequency of a Gas

[rhochipi]

Oxygen Molecules collide with a 19.0 cm^2 wall. Assume that all molecules travel with a speed of 490 m/s and strike head on. How many collisions are there per second if the oxygen pressure is 1.00atm?


Pressure = Force/Area
Force = 2*Mass*Velocity
Collision Frequency = PI*Molecular_Diameter^2*Average_Velocity*Number_of_Molecules_per_Unit_Volume
Collision Frequency also = Velocity_Average/Mean_Free_Path
Pressure*Volume = Number_of_Molecules*Boltzmann's_Constant*Temperature
Average Velocity = sqrt(8*Boltzmann's Constant*Temperature/(PI*Mass))

A previous problem calculated the number of molecules per unit volume from the ideal gas law: N/V = P/(KbT). However, I need temperature, which I thought of getting from the average velocity equation: v_Ave = sqrt(8KbT/(PI*m)). However, I don't know the mass, so I've become stumped.

A friend worked with momentum. By Kinetic Theory, p = 2mv = Force*Time_Duration, for all molecules, N: p = 2mvN = Force*Time. By equating that force with that from P = FA, N/T = PA/(2mv), where T = 1 second, P = 1.013e5 Pa, v = 490m/s, A = .0019 m^2. But, he assumed m to be the molar mass of Oxygen, 32g. When I tried it, I received an answer of 6.1 collisions per second, which didn't make sense and was obviously wrong.

Any Suggestions?

 

[denverdoc]

Maybe its asking for a real simple minded approach that treats the molecules as they were billiard balls, and not spinning, vibrating, etc.

I would just use mass here as 32gm/N, convert pressure in atm to pascals, and that in turn to total force over the 19 sq cm plate, and use your elastic collsion formula to see what the number is/second I'd not solve for mass explicitly but express it in terms of N/s (I got about 6.14 N/s where
N=6.02E23)

 

[m2003]

Thank you for your question. It seems like you and your friend are on the right track in using the ideal gas law and kinetic theory to calculate the collision frequency of oxygen molecules with a wall. However, there are a few things that need to be clarified in your calculations.

Firstly, the mass that should be used in the equations is the mass of a single oxygen molecule, not the molar mass of oxygen. The molar mass is the mass of one mole of oxygen molecules, which contains Avogadro's number (6.022 x 10^23) of molecules. In this problem, we are interested in the collision frequency of a single molecule, so we need to use its individual mass.

Secondly, the average velocity equation you used (v_Ave = sqrt(8KbT/(PI*m))) is actually the root mean square velocity, which is the average velocity of all the molecules in a gas. In this problem, we are only interested in the velocity of the molecules that are colliding with the wall, which is equal to the average velocity (v_Ave) divided by 2. This is because only half of the molecules in a gas are moving towards the wall at any given time.

With these corrections, the calculation should look like this:

1. Calculate the number of oxygen molecules per unit volume:
N/V = P/(KbT)
= (1.013e5 Pa)/(1.38e-23 J/K * 273 K)
= 2.3 x 10^25 molecules/m^3

2. Calculate the average velocity of the oxygen molecules:
v_Ave = sqrt(8KbT/(PI*m))
= sqrt((8 * 1.38e-23 J/K * 273 K)/(PI * 5.31e-26 kg))
= 493 m/s

3. Calculate the velocity of the molecules colliding with the wall:
v_wall = v_Ave/2 = 493 m/s / 2 = 246.5 m/s

4. Calculate the collision frequency:
Collision Frequency = Velocity_Average/Mean_Free_Path
= v_wall/2r
= (246.5 m/s)/(2 * (0.5 * 19.0 cm))
= 6.5 x 10^6 collisions/second

So, the collision frequency of oxygen molecules with a 19.0 cm^2 wall at a pressure of