# 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!