Connected Vessels and Final Gas Equilibrium
For connected ideal-gas compartments at final common pressure P_f, conservation of amount gives sum(P_iV_i/RT_i) = P_f sum(V_j/RT_j), unless matter enters or leaves the system.
Why this shows up in the exam
Bulbs maintained at different temperatures · Removing partitions between gas chambers · Steady pressure after joining vessels
Learn the idea
Connected chambers reach one pressure, while each chamber may retain its own temperature and volume. Count the molecules before opening the connection, then distribute the same total amount across the final chambers using their local temperatures.
🧠 Memory hook: One final pressure, but keep each chamber's own V over T.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- sum(P_i V_i/T_i) = P_f sum(V_j/T_j) — mole conservation for connected ideal-gas chambers with possibly unequal final temperatures
- P_f(V1+V2) = (n1+n2)RT_f — special case with a common final temperature
How to approach it
- 1Compute initial total moles
- 2Write final moles chamber by chamber
- 3Equate totals and check limiting cases
Common slip-ups that cost marks
- •Averaging pressures directly
- •Assuming equal final temperatures when only pressure equalizes
- •Losing total-mole conservation
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A gas has rms molecular speed 300 m/s at 300 K. What is its rms speed at 1200 K, assuming ideal behavior?
More from Kinetic Theory of Gases
Ideal gas law and gas laws
The ideal gas law and related gas laws describe the relationships between pressure, volume, temperature, and number of moles for ideal gases.
Degrees of freedom and thermal properties
Degrees of freedom determine the distribution of energy among molecules, affecting internal energy, specific heats, and the ratio of specific heats (γ).
Kinetic theory and molecular motion
The kinetic theory explains the behavior of gases in terms of the motion and collisions of their molecules, relating properties like pressure, temperature, and kinetic energy.
RMS speed and temperature dependence
The root mean square (rms) speed of gas molecules depends on temperature and molar mass, and is a key measure of molecular motion in gases.
Mean free path and collisions
Mean free path is the average distance a molecule travels between collisions, and depends on molecular size and number density.
Ideal-Gas Equation and Molecular Form
For a dilute ideal gas in thermal equilibrium, the state variables satisfy PV = nRT = Nk_B T, where intermolecular potential energy and molecular volume are neglected.