How Many Air Molecules are Contained in a Smooth Cylinder at Equilibrium?

[pfunk22]

Homework Statement

A smooth, well-greased insulated cylindrical container with a metal base has a light (so you can neglect its weight), airtight piston that can frictionlessly move up and down the inside of the container. A mass M = 73 kg sits on top of the piston. The piston has a circular cross-section of area 0.032 m2, and is in equilibrium at a height of 4.7 cm above the base of the cylinder. The cylinder sits in equilibrium in a room that is at atmospheric pressure and a temperature of 22 °C.

How many air molecules are contained inside the cylinder beneath the piston?

Homework Equations

pV=nRT

p= (force)/(area) in N/m^2 Or Pa
Volume of cylinder= pi*r^2*h
A=area
Force = mass*gravity => mg
Avogadro's # = 6.02x10^23
R=8.31 J/(mol K)
T=temp in Kelvin

The Attempt at a Solution

pV=nRT

(m*g)/A)*(pi*r^2*h) = nRT

(mg/A)*(pi*(A/pi)*h) = nRT

(mg/A)*(A*h) = nRT

mgh = nRT

mgh/RT =n

n * avogadro's Number gives molecules.

im not sure what I am doing wrong...

 

[rock.freak667]

I think your method is correct. Did you convert all your numbers to the correct units?

 

[pfunk22]

I think your method is correct. Did you convert all your numbers to the correct units?

I think so, i converted height to meters and temp to kelvin.

 

[pfunk22]

i think my mistake is in the amount of pressure. i didn't take into account the whole system. how can i calculate the pressure of the entire system? specifically, the pressure of the air inside the cylinder pushing upward on the piston .Edit:
ok i figured it out.

 

[pfunk22]

Solution:

pV / RT=n

( ((Atmospheric pressure) + (Pressure of system) ) * (volume at eq) ) / (R*T)

[ (101,325 pa) + ( ((73 kg)*(9.8 m/s^2)) / (.032 m^2) ) * ( ( .032 m^2) * (.047 m) ) ] / ( (8.3 J/mol K)*(295 K) ) = .076 mols

.076 mols * (6.02*10^23 mol^-1) = 4.6*10^22 molecules