The Adiabatic Expansion of an Ideal Gas — Question 9

Question 9

A confined gas acts as an adiabatic spring for a small piston motion

An ideal gas at equilibrium pressure $p_0=100$ kPa and volume $V_0=.00500$ m$^3$ supports a frictionless piston of area $A=.0100$ m$^2$ and moving mass $M=2.00$ kg. External force is constant and balances the equilibrium pressure force. Small displacements are slow enough for uniform gas pressure but fast enough for negligible heat exchange. Take $\gamma=1.40$.

1. Linearize the pressure response to an outward displacement $x$ and derive the effective spring constant and oscillation frequency.

2. Find the corresponding frequency if heat exchange were fast enough to keep the gas isothermal. Compare the two frequencies.

Question 9 - Solution

A confined gas acts as an adiabatic spring for a small piston motion

(a) Method and calculation

$$V=V_0+Ax,\quad pV^\gamma=p_0V_0^\gamma\ \Rightarrow\ \delta p\simeq-\gamma\frac{p_0A}{V_0}x.$$$$F_{\mathrm{net}}=A\delta p=-k_{\mathrm{gas}}x,\quad k_{\mathrm{gas}}=\frac{\gamma p_0A^2}{V_0}=2800\,\mathrm{N/m}.$$$$f_{\mathrm{ad}}=\frac1{2\pi}\sqrt{k_{\mathrm{gas}}/M}\approx5.95503\,\mathrm{Hz}.$$

(b) Interpretation and further calculation

$$k_T=\frac{p_0A^2}{V_0}=2000\,\mathrm{N/m},\quad f_T=\frac1{2\pi}\sqrt{k_T/M}\approx5.03292\,\mathrm{Hz}.$$$$f_{\mathrm{ad}}/f_T=\sqrt\gamma\approx1.18322.$$Adiabatic compression raises temperature and produces a stronger restoring pressure than isothermal compression.

Physical check

The linear approximation requires $|Ax|\ll V_0$; thermal and acoustic time scales determine whether this model applies.