Helium Gas: Pressure, root-mean-square velocity, and more

[Username1234]

Homework Statement

A rigid, thermally insulated container with a volume of 22.4 liters is filled with one mole of helium gas (4 grams per mole( at a temperature of 0 Cesius (273K). The container is sitting in a room, surrounded by air at standard temperature and pressure (STP:1atm, 0 Celsius).
a) Calculate the pressure inside the container in N/m^2
b) Calculate the root-mean-square speed of the helium atoms.
c) Now open a tiny square hole in the container, with area 10^-8 m^2. After 5 seconds, how many helium atoms will have left the container?
d) During the same 5 sec. some air molecules from the room will enter the container _ through the same hole. How many air molecules will enter the container?
e) Does the pressure inside the container increase or decrease during this 5 second _ period?

Homework Equations

a) P=nkT where n=N/V, N is # of molecules (using avagadro's number) and V is in m^3. k is Boltzmann's constant, T is temp. in Kelvin.
b) Vrms = sqrt((3kT)/m) where k and T are as above and m is mass.
c) .25nAv[avg] = number of molecules crossing area A per second. n is as above and v[avg] is average velocity.

The Attempt at a Solution

For part one I calculated 102,716 N/m^2, and for part two I calculated 1.69 * 10^-9 m/s. For part c) I got 5.68*10^8 He atoms escape, and part e, that pressure inside decreases. I am not sure if these answers are right so far, especially the Vrms, as it is so small. The value I got for the pressure 102,716 N/m^2 came directly from the pressure equation above, which I thought should have resulted in atm rather than Pa (N/m^2), but seeing that 102,716 was of the order of magnitude of Pa in this case, I assumed that it was, and left it at that. I wanted to check and make sure I haven't made some grave error. I am also not quite sure how to approach d). Could anyone help? Thanks in advance.

 

[Count Iblis]

You need to enter the mass in kilograms in the formula for the rms speed.

 

[Username1234]

Which I did... .004Kg...

 

[Count Iblis]

You need to mass of an helium atom, which is approximately 4 u, where u is the atomic mass unit:

u = 1.660539*10^(-27) kg

 

[Username1234]

Oh, so you're saying the mass in the equation is of one Helium atom, not the total mass in the container. OH; I used the mass in the container, and seeing that there was one mole of gas, the mass would have been 4g. But I see now. Thank you!

 

[ideasrule]

For part one I calculated 102,716 N/m^2, and for part two I calculated 1.69 * 10^-9 m/s. For part c) I got 5.68*10^8 He atoms escape, and part e, that pressure inside decreases. I am not sure if these answers are right so far, especially the Vrms, as it is so small. The value I got for the pressure 102,716 N/m^2 came directly from the pressure equation above, which I thought should have resulted in atm rather than Pa (N/m^2), but seeing that 102,716 was of the order of magnitude of Pa in this case, I assumed that it was, and left it at that. I wanted to check and make sure I haven't made some grave error. I am also not quite sure how to approach d). Could anyone help? Thanks in advance.

Pressure is indeed approximately 102.7 N/m^2. It isn't it atmospheres because if you do dimensional analysis on PV=nkT with the units that you used, you'll get P in N/m^2.

For part b, you used v=sqrt(3kT/m), where m is the mass of one helium atom. So you need to find the mass of a single helium atom.

For d), can you calculate the average (not rms) speed of the air molecules? If so, you can apply the equation N=0.25nAv again. Remember that in the ideal gas approximation, gas molecules don't interact, so the gas molecules exiting the hole has no effect on the gas molecules entering it.

 

[ehild]

Homework Statement

A rigid, thermally insulated container with a volume of 22.4 liters is filled with one mole of helium gas (4 grams per mole( at a temperature of 0 Cesius (273K). The container is sitting in a room, surrounded by air at standard temperature and pressure (STP:1atm, 0 Celsius).
a) Calculate the pressure inside the container in N/m^2
...
For part one I calculated 102,716 N/m^2,

This is definitely wrong.

ehild