Connect an ideal-gas state to a molecular energy distribution

Declare the state, system and observation

The ideal gas law pV=nRT connects equilibrium pressure, volume, amount and absolute temperature. Pressure must be absolute, volume must use compatible units and T must be in kelvins. The model neglects molecular volume and interactions except during idealized collisions; it is not a universal equation for dense gas or a liquid.

Build the algebraic model

This packet contains 0.5 mol of a hypothetical monatomic gas, R=8.314 J/(mol K), with molar mass 0.020 kg/mol. At 300 K and 10 L=0.010 m³, pressure is 124710 Pa. Doubling volume at fixed amount and temperature halves pressure, while doubling kelvin temperature at fixed volume doubles it. Those are different controlled comparisons, not descriptions of every unconstrained gas change.

Check the original evidence packet

For the declared monatomic ideal model, internal energy is 3nRT/2=1870.65 J at 300 K. Average translational energy per molecule is 3kBT/2, while root-mean-square speed is √(3RT/M)≈611.662 m/s. Individual molecules have a distribution of speeds; the mean vector velocity may be zero despite nonzero mean squared speed. Equal temperatures imply equal average translational kinetic energy per molecule, not equal speeds for different molecular masses.

Challenge the domain and transfer

Pressure arises from momentum transfer in wall collisions; it is not caused by molecules all moving in one direction. Real phase changes and saturation require additional material evidence. At a specified temperature, relative humidity compares water-vapor partial pressure with the supplied saturation pressure: if these are 1 and 2 kPa, relative humidity is 50%; at unchanged partial pressure and a new supplied saturation value 4 kPa it is 25%. Boiling requires vapor pressure matching surrounding pressure, not a universal temperature independent of pressure. These separate conceptual packets do not turn the monatomic model into a real-water phase model.

Worked original case

At 400 K and 5 L, p=332560 Pa and U=2494.2 J. Compared with 300 K, rms speed scales by √(4/3), whereas average translational energy scales by 4/3.

Bounded investigation

Hold amount fixed and compare temperature changes with volume changes. Retain absolute pressure, internal energy and rms speed, then compare a hypothetical species with four times the molar mass at the same temperature.

Changed-case numerical practice

Try the changed case before feedback

An ideal gas stays at fixed amount and volume. Its absolute temperature increases from 250 K to 400 K. Find the pressure ratio final/initial.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. Packets are supplied or generated teaching evidence, not measured apparatus. No heating, pressure vessels, physical oscillators, sound exposure or construction is requested. Public checks and saved reports do not establish supervised practical skill, equipment safety or independent mastery.