Gas Pressure Gradients in Body-Force Fields
Mechanical equilibrium of a fluid element gives grad P = rho g_eff; with an isothermal ideal gas, density depends on pressure and the relation integrates exponentially.
Why this shows up in the exam
Rotating gas tubes · Accelerating sealed compartments · Atmospheric or chimney pressure differences
Learn the idea
A gas at rest in gravity, rotation, or an accelerating frame develops a pressure gradient that balances effective body force. Different gas layers must support or accelerate neighboring layers, so pressure need not be spatially uniform even at one temperature.
🧠 Memory hook: Pressure rises in the direction opposite the gas's effective weight.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- grad P = rho g_eff — local hydrostatic balance in an effective acceleration field
- dP/P = (M/RT) g_eff dot dr — isothermal ideal-gas form for integration
- dP/dr = rho omega² r — radial balance in uniform rotation
How to approach it
- 1Draw the effective acceleration
- 2Write force balance on a thin element
- 3Integrate with the stated temperature model
Common slip-ups that cost marks
- •Assuming uniform pressure in an accelerated gas
- •Using constant density across a compressible gas without justification
- •Choosing the wrong sign for effective acceleration
🌟 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
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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.