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

In summary, a rigid, thermally insulated container with a volume of 22.4 liters is filled with one mole of helium gas at a temperature of 0 Celsius. The container is surrounded by air at standard temperature and pressure. The pressure inside the container is 102,716 N/m^2. The calculated root-mean-square speed of the helium atoms is 1.69 * 10^-9 m/s. After opening a tiny square hole in the container, with an area of 10^-8 m^2, 5.68*10^8 helium atoms will escape and some air molecules from the room will enter the container through the same hole. During this 5 second period, the pressure inside the container decreases.
  • #1
Username1234
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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 Boltzman'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.
 
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  • #2
You need to enter the mass in kilograms in the formula for the rms speed.
 
  • #3
Which I did... .004Kg...
 
  • #4
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
 
  • #5
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!
 
  • #6
Username1234 said:
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.
 
  • #7
Username1234 said:

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
 

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

1. What is the pressure of helium gas?

The pressure of helium gas can vary depending on the temperature and volume of the gas. At standard temperature and pressure (STP), the pressure of helium gas is 1 atmosphere or 101.325 kilopascals.

2. How does the root-mean-square velocity of helium gas compare to other gases?

The root-mean-square velocity of helium gas is higher than most other gases due to its smaller atomic mass. This means that helium gas molecules move faster and have a higher average kinetic energy compared to other gases at the same temperature.

3. What is the relationship between temperature and the root-mean-square velocity of helium gas?

The root-mean-square velocity of helium gas is directly proportional to the square root of the temperature. This means that as the temperature increases, the velocity of helium gas molecules also increases.

4. How is helium gas used in industry?

Helium gas is used in various industries, including cryogenics, welding, and leak detection. Its low boiling point and non-reactive nature make it useful for cooling and pressurizing systems. It is also used in gas chromatography and as a carrier gas in analytical instruments.

5. Can helium gas be dangerous?

While helium gas itself is non-toxic, it can pose a danger if inhaled in large quantities. Inhaling helium gas can lead to asphyxiation as it displaces oxygen in the lungs. It is important to handle helium gas with caution and in well-ventilated areas.

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