Lesson 6 of 13 · 6 min
State variables and equation of state
NCERT §11.7
Vikram's nitrogen cylinder for tyre filling has a valve to an empty test chamber. When he opens it fast, the gas rushes in with a hiss. For that moment, what is 'the pressure' of the gas?
The lesson in notes
In short
An equilibrium state is fully described by the values of a few macroscopic state variables. For a gas: pressure, volume, temperature and mass (and composition, for a mixture).
A system is not always in equilibrium. A gas rushing into vacuum after a partition is suddenly removed (free expansion) has non-uniform pressure; a petrol-vapour and air mixture exploding after a spark has non-uniform temperature and pressure. State variables cannot describe these states.
Such a system eventually settles to uniform temperature and pressure and to thermal and mechanical equilibrium with its surroundings. State variables describe only equilibrium states.
The relation connecting the state variables is the equation of state. For an ideal gas it is PV = μRT, so for a fixed amount of gas only two variables are independent, e.g. P and V, or T and V. Real gases can have more complicated equations of state.
A pressure–volume curve drawn at fixed temperature is an isotherm.
Extensive variables tell the 'size' of the system; intensive ones do not. Split a system in equilibrium into two equal halves: variables that are halved in each part are extensive, those that stay the same are intensive.
Extensive: internal energy U, volume V, total mass M. Intensive: pressure P, temperature T, density ρ.
Checking an equation with this idea: in ΔQ = ΔU + PΔV every term is extensive. P (intensive) × ΔV (extensive) is extensive, and ΔQ is proportional to the mass even though Q is not a state variable.