Ruchardt's method

Hints for deriving the gas flushing equation

The gas flushing equation (equation number 2 on your handout) is as follows:

\[F=\frac{\text{Volume of gas}}{\text{Volume of Jar}}=\frac{V_{gas}}{V_{jar}}\]

where the gas can be CO\(_{2}\), Argon, etc. We can then write a small change in \(F\) as follows.

\[\begin{align}F+dF = \frac{1}{V_{jar}}\left[V_{gas}+dV_{flush}-FdV_{flush}\right]\end{align}\]

where \(V_{gas}\) is the original amount of gas in the jar, and \(dV_{flush}\) is a small amount of gas pumped in to flush the jar. The third term on the right hand side, \(-FdV_{flush}\) is motivated as follows. After \(dV_{flush}\) has instantaneously mixed with the whole volume of the mixture, a small amount of the mixture, \(dV_{mix}\) will escape.

\[dV_{mix}=dV_{flush}\]

and the amount of gas in the expelled \(dV_{mix}\) will be given by

\[(F+dF)dV_{mix}\approx FdV_{mix}=FdV_{flush}.\]

We can write the equation (1) as an ODE and solve it to get the final equation,

\[F=1-e^{-\frac{V_{flush}}{V_{jar}}}.\]

Good luck!