Question

What is the mass of He in a uniform cylindrical tube, 5mm radius, 0.1m length. Pressure at z=0 is 0.06atm, pressure at z=0.1 is 0.02atm . Temperature of the system is 298K. There is another gas in the tube that causes the pressure gradient but I don't believe that information is necessary. I know to use the idea gas law, however, I can't figure out how to account for the pressure difference when computing mass of the He in the system.

Answer 1

first do we need to assume the shape of the gradient or is that a given? i will assume linear here so we can say \(p(z) = 0.06 - 0.4 \ z\)

the for the equation itself, written so that mass is in there, with M = Molar mass, R,T = constant
\[p \ V = m\frac{R \ T}{M_m} \]

we consider a small element of the cylinder within which we can apply the ideal gaw law, because the pressure gradient in small element \(\to 0\)



we can say
\[p(z) \ dV = dm\frac{R \ T}{M_m} \]
\[dV = \pi R^2 dz\]
\[\pi \ R^2 \ (0.06 - 0.4 \ z) \ dz = dm\frac{R \ T}{M_m}\]
\[\pi \ R^2 \int\limits_{0}^{0.1} \ (0.06 - 0.4 \ z) \ dz = \frac{R \ T}{M_m}\int\limits_{0}^{m_0} dm\]

\[m_o = 0.004 \ \frac{M_m}{R \ T} \pi \ R^2 = 0.04 \ V \frac{M_m}{R \ T} \]

IOW you can use the average of the 2 pressures if the gradient is linear.