Ideal Gases — Question 5

Question 5

A spring-loaded piston requires force balance and the equation of state

Treat the gas as ideal, $pV=nRT$, and use $R=8.314\,\mathrm{J/(mol\,K)}$. A vertical frictionless piston of area .0100 m$^2$ holds .100 mol gas. At displacement $x=0$, volume is .00200 m$^3$ and the total baseline resisting pressure is 100 kPa. A spring adds force $kx$, with $k=1000$ N/m. At final temperature 300 K, the piston is in mechanical equilibrium.

1. Set up the equation for $x$ and find the physically allowed nonnegative root.

2. Find final pressure and volume, and check both force balance and $pV=nRT$.

Question 5 - Solution

A spring-loaded piston requires force balance and the equation of state

(a) Method and calculation

$$p=100000+\frac{1000x}{.0100}=100000+100000x,\quad V=.00200+.0100x.$$$$(100000+100000x)(.00200+.0100x)=249.42.$$$$1000x^2+1200x-49.42=0\ \Rightarrow\ x\approx.0398594\,\mathrm m.$$

(b) Interpretation and further calculation

$$p\approx103985.94\,\mathrm{Pa},\quad V\approx.002398594\,\mathrm{m^3}.$$$$pV=249.42\,\mathrm J=nRT,\quad(p-100000)A=kx.$$The other root gives a negative volume and must be discarded.

Physical check

Neither pressure nor volume is independently fixed: the mechanical constraint couples them.