Hunter B. answered • 25d
M.S In Nuclear Engineering w/ B.S in Physics and 3+ years tutoring
Question 1 combines topics from thermodynamics with the ideal gas law. The ideal gas law tells us that for any gas which we assume is ideal, we can relate pressure, volume, the number of gas particles, and the temperature through the equation,
PV=nRT
The problem however asks us to find the mass of the gas which is not readily available in the ideal gas law. We could however look at the expression for the average velocity of a gas given by the Maxwell-Boltzmann. Assuming in this case that the average velocity is referring to the root mean squared or rms velocity, the equation is then,
vrms=√[(3RT)/(M)] ,where M is the molar mass or the mass of 1 mole of a given gas.
We can see that the velocity equation and the ideal gas law both contain "RT" so we can insert one into the other by rearranging to solve for "RT". If we do this to the velocity equation we obtain (after a little algebra),
RT =(v2rmsM)/3
Plugging this into are ideal gas law then gives us,
PV=(nv2rmsM)/3
We now have a way to connect a mass to the values given to us but we still have the issue that the number of gas particles is not given and that the molar mass is not the same as the total mass. We can remedy both of these by converting the molar mass into the mass per particle which will then cancel out our "n" and leave us with the total mass. This is done by using some chemistry ratios involving Avogadro's number (NA),
M = mass/mole * 1 mole/NA = mass/particle = m/n
If we perform this conversion and plug it into our equation the "n's" cancel out and we are left with,
PV=(v2rmsm)/3
From here we simple rearrange once more to solve for m and we are then given the equation,
m = 3PV/(v2rms)
Now we can plug in our numbers, being careful to convert liters to m^3, which gives us
m = 3*0.0015L*10^5Pa/(750m/s^2) = 0.0008Kg or 0.8g
