Ideal Gases — Question 6

Question 6

A flexible balloon changes volume under two competing influences

Treat the gas as ideal, $pV=nRT$, and use $R=8.314\,\mathrm{J/(mol\,K)}$. A closed flexible balloon initially has volume 3.00 m$^3$ at 100 kPa and 300 K. At a higher altitude its gas is at 250 K and 60.0 kPa. Neglect membrane overpressure and leakage.

1. Find the final volume and the separate pressure and temperature factors.

2. If the balloon can hold at most 5.00 m$^3$, find the lowest permissible absolute pressure at 250 K. Does cooling alone make the balloon expand?

Question 6 - Solution

A flexible balloon changes volume under two competing influences

(a) Method and calculation

$$\frac{V_f}{V_i}=\frac{T_f}{T_i}\frac{p_i}{p_f}=\frac{250}{300}\frac{100}{60}=\frac{25}{18}.$$$$V_f=4.16667\,\mathrm{m^3}.$$Cooling gives a factor $5/6$; the pressure fall gives $5/3$.

(b) Interpretation and further calculation

$$\frac{p_iV_i}{T_i}=\frac{p_{\min}V_{\max}}{T_f}\ \Rightarrow\ p_{\min}=\frac{100(3)(250)}{300(5)}=50.0\,\mathrm{kPa}.$$$$p\ge50.0\,\mathrm{kPa}\quad\text{at 250 K}.$$At fixed pressure cooling contracts an ideal gas; here the pressure reduction dominates.

Physical check

The maximum-volume condition constrains pressure only after temperature and gas amount are specified.