Theory — Macroscopic Ideal Gas Description
At the macroscopic level, many dilute gases obey a simple relationship among pressure \(P\), volume \(V\),
and absolute temperature \(T\). This chapter explains the classic gas laws as special cases and shows how to
compute changes between states.
1) Ideal gas law
- \(n\): amount of gas (moles)
- \(R\): ideal-gas constant (depends on chosen unit system)
- \(T\): absolute temperature in kelvins
If temperature is given in Celsius, convert first: \(\;T(\text{K})=T(^\circ\text{C})+273.15\).
2) Proportionality form and “state-to-state” relation
If \(n\) is constant, then \(nR\) is constant, so:
This is the most useful form for quickly solving two-state problems when one variable is unknown.
3) The classic gas laws (special paths)
In each case, the “constant” comes from \(nR\) being constant (fixed amount of gas) and the imposed path constraint.
4) Worked example (Charles’ law at constant pressure)
Start with \(V_1=1\,\text{L}\) at \(T_1=273\,\text{K}\). Heat to \(T_2=546\,\text{K}\) at constant pressure.
5) Multi-step (composite) processes
Real thermodynamic changes often occur in stages (e.g., first isochoric heating, then isothermal expansion).
If the gas stays ideal and \(n\) stays constant, you can apply the ideal gas law at each state and use the
appropriate gas-law relation along each segment.
The calculator’s composite mode implements exactly this: solve step 1 for a chosen unknown in state A, then step 2.
6) University tease: van der Waals correction
At higher densities, real gases deviate from ideal behavior due to finite molecular volume and intermolecular forces.
A common correction is the van der Waals model:
Your calculator can optionally compare \(P_\text{ideal}\) and \(P_\text{vdW}\) at the entered states as a preview.
