I have a container that has CO2 in it. The root-mean-square speed of the gas is equal to 329 m/s. I need to calculate: 1) the molar mass of the gas 2) the temperature of it (therefore I must...

I have a container that has CO2 in it. The root-mean-square speed of the gas is equal to 329 m/s. I need to calculate:

1) the molar mass of the gas

2) the temperature of it (therefore I must treat the gas as an ideal gas)

3) I have to demonstrate the dependence of the root-mean-square speed of the pressure exerted by the molecules of an ideal gas on the walls of a cubic recipient.

With regard to the 1) problem, I need to consider the formula of the root-mean-square speed. `rms = sqrt((3RT)/(M)) or rms = sqrt((3P)/(d))`

The first formula has got the molar mass in it. That's clearly a clue. The problem is that I don't know the temperature of the gas. So, there's no way to use the first formula.

With regard to the 2) problem, I clearly have to use the ideal gas law, `PV = RnT`

A lot of variables are missing, such as the volume (though I could calculate it with the second rms formula I posted above since I have the density but not the mass).

With regard to the 3) problem, I don't know what the problem is asking.

I appreciate your help.

Expert Answers
t-nez eNotes educator| Certified Educator

1) The molar mass of CO2  = 12.1 g/mol + 2(16.0 g/mol) = 44.1 g/mol

2) Knowing the molar mass allows you to use the rms velocity equation to calculate temperature:

`rms =sqrt((3RT)/M)`

Where R = 8.3145 (kg-m^2/sec^2)/K-mol,

and M = 44.1 g/mol = 0.0441 kg/mol

`(rms)^2 = (3RT)/M`

`T = (rms)^2(M)/(3(8.3145))`

 

`T = [(108241 m^2/s^2)(.0441)( kg)/(mol)]/(3((8.3145 kg (m)^2)/((sec)^2 mol K))`

` `

`T = 191K`

 

The equation `rms =sqrt((3P)/d)`  shows the dependence of of rms velocity on pressure: The rms velocity is proportional to the square root of pressure or pressure is proportional to the square of the rms velocity when other factors in the equation remain constant. 

This equation is derived from `rms=sqrt((3RT)/M)`  by:

1. susbstituting (mass/molar mass) for n in the ideal gas equation, and

2. substituting density for mass/volume:

PV=nRT

P=nRT/V

P=mRT/MV

P=dRT/M

P/d=RT/M

 

t-nez eNotes educator| Certified Educator

Yes, you are correct. P/d is substituted for RT/M. Thank you!

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