#$4.$ A gas microbalance is a device for accurately measuring the molecular weight of a gas. Briefly, this device consists of a closed container, into which we introduce the unknown gas. Inside the closed volume is the actual balance, which is simply an evacuated glass bulb $($with mass $m$ and volume $V$ at the end of a balance beam. As the unknown gas is introduced into the system, we measure the exact pressure at which the balance condition is met. At this point, the mass of the bulb equals the mass of the displaced unknown gas. If we do this for both the unknown gas and a known calibration gas, we can determine the molecular weight.

a$)$ To help visualize this device, draw a simple diagram of the apparatus.

b$)$ Suppose the balance point occurs at $427\mathrm{.}22$ Torr of $CH{F}_3,$ the calibration gas. Write a simple equation for the mass of the bulb in terms of the volume and density of $CH{F}_3$ at the balance point.

c$)$ Now, suppose the balance point occurs at $293\mathrm{.}22$ Torr for some unknown gas. Write another equation similar to that in $($b$),$ but for the unknown. Then calculate the molecular weight, assuming ideal gas behavior.