eyeprolog

Ideal gas law

Is the pressure in this gas cell close to one atmosphere?

The ideal gas law P·V = n·R·T relates pressure, volume, amount of gas and temperature. This example computes the pressure of one gas cell, classifies it against a tolerance band, and explains the result down to each multiplication and division.

Run it in the playground · Source

The question

A cell holds 1 mole of gas at 300 kelvin in 0.024 cubic metres. Using a gas constant of 8 (rounded so the numbers come out cleanly), what is the pressure, and does it fall between 95 000 and 105 000 pascal, roughly one atmosphere?

The facts

gas_cell(cell1, 1.0, 8.0, 300.0, 0.024).
pressure_limit(cell1, low_Pa, 95000.0).
pressure_limit(cell1, high_Pa, 105000.0).

gas_cell/5 records the amount n, the constant R, the temperature T and the volume V. The two limits define the tolerance band.

The rules

pressure(Cell, Pressure) :-
  gas_cell(Cell, Moles, Gasconstant, Temperature, Volume),
  (Nr is Moles * Gasconstant),
  (Nrt is Nr * Temperature),
  (Pressure is Nrt / Volume).

near_atmospheric(Cell) :-
  pressure(Cell, Pressure),
  pressure_limit(Cell, low_Pa, Low),
  pressure_limit(Cell, high_Pa, High),
  (Pressure > Low),
  (Pressure < High).

The physics is P = n·R·T / V, computed in three explicit steps. The classification is a separate rule, so a reviewer can dispute the band without touching the physics.

The answer

eyeprolog examples/ideal-gas-law.pl

pressure_Pa(cell1, 100000.0).
status(cell1, near_atmospheric).
reason(cell1, "pressure is inside the one-atmosphere tolerance band").

1 × 8 × 300 / 0.024 = 100 000 pascal, comfortably inside the band.

The reason

eyeprolog --proof examples/ideal-gas-law.pl

step(8.0 is 1.0 * 8.0, builtin, [], []).
step(2400.0 is 8.0 * 300.0, builtin, [], []).
step(100000.0 is 2400.0 / 0.024, builtin, [], []).

The pressure rests on the cell’s recorded values and three arithmetic steps: n·R = 8, n·R·T = 2400, and 2400 / 0.024 = 100 000. The status then rests on two comparisons against the limits.

The check

condition('C1', resolution, ok, 8).
condition('C5', re_decision, ok, 5).
condition('C7', relevance, ok, 16).
verdict(checked).

Eight steps replay against the program’s clauses, and the five built-in steps (three arithmetic, two comparisons) are recomputed by the checker. The proof checks the derivation; whether the sensor readings were right is a question for the source of the facts.

Try it

Raise the temperature to 350.0. The pressure becomes about 116 667 pascal, and the status/2 and reason/2 answers disappear.
Add a second cell with a different volume and predict its status before running.
Add a reason/2 clause that explains why a cell is outside the band (too high or too low).

Learn more in The Art of EyeProlog, Chapter 14.

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