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Jotun.uu
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Homework Statement
A sphere is filled with a gas which has a constant temperature T. The pressure in the sphere starts at p0 and the spheres radius is r. Assume that a small area (A) of the sphere is held at a very low temperature, while the rest of the surface holds the temperature T. Also assume that all the molecules that hit the surface A condense and do not return to their gas state. Calculate the time it takes for the pressure to fall from p0 to p.
Given information
r = 10 cm
A = 1 cm^2
T = 300 K (kelvin)
m = 2.98*10^-26 kg
p0 = 10 torr
p = 1*10^-4 torr
Homework Equations
(1) Amount of collisions ns = p / sqrt(2*PI*m*Kb*T)
(2) pV = NKbT, where Kb is Boltzmanns constant
(3) Collisions per area ns*A
The Attempt at a Solution
If you multiply the amount of collisions with the area being held at a low temperature i reckon you should get the amount of molecules hitting it, which then integrated should give a time.
Thus, dN/dp = (dN/dp)*(dp/dt). From (2) i know that dN/dp = V/(KbT).
and dNdp/dpdt is the amount of molecules which change the pressure, so dNdp/dpdt = -ns*A (i make it negative because they decrease go down as they condense).
So far i think i have it mostly right, from looking at the correct answer written by my teacher. But if you see a problem with the way i think please tell me, as my teachers answer is very non descriptive in its thought process.
So now i have (from (1) and(3)) dNdp/dpdt = -A*p / sqrt(2*PI*m*Kb*T).
With some math-e-magic [dN/dp = V / (Kb*T)];
V*dp/(Kb*T*dt) = -A*p / sqrt(2*PI*m*Kb*T).
I isolate dp/p = (-A*Kb*T / (V * sqrt(2*PI*m*Kb*T))) dt
And here is where things go awfully wrong for me! I must be making some fundamental mistake when integrating or something! I feel like i should integrate with regard to the time so i should get something like
ln(p0/p) = -t*V *sqrt(2*PI*m*Kb*T) / (A*Kb*T) and then isolate t but my teachers answer tells me this is wrong...
please help!
t should become 3.24 seconds. The equation my teacher gets for t is -(V/A)ln(p/p0)(2*PI*M/(R*T)) where M = m * Avogadros constant and R = Kb * Avogadros constant. I don't understand where the square root has gone for 1 thing!