Teaching a piston experiment
Students compare 240, 300 and 360 K curves for 1.5 mol and see why the same 25 L volume supports different pressures.
Physics and thermodynamics
Ideal Gas Graph Calculator
Plot three ideal-gas pressure-volume isotherms and calculate reference pressure, local slope, endpoint ratio, and reversible isothermal work.
Ideal gas isotherm graph
Read the curve, slope, and area of an ideal-gas P-V map
This page builds three equilibrium P-V curves around a chosen temperature. It quantifies the local slope at one reference volume and the reversible work under the center isotherm.
Center-isotherm pressure-
Local dP/dV-
Center-isotherm work-
Endpoint pressure ratio-
Current model evidence
Pressure samples across the volume domain
The table and plot use the same nine volume samples and unrounded state model.
| Volume (L) | Low-T pressure (kPa) | Center-T pressure (kPa) | High-T pressure (kPa) | Amount (mol) |
|---|
DETAILED CALCULATION PROCESS
Formula, units, default substitution, and reconciliation
1. Governing relation
P(V,T) = nRT/V; dP/dV = -nRT/V²; Wrev = nRT ln(Vmax/Vmin)
Using kPa and liters is numerically compatible with R in J/(mol K), because 1 kPa L equals 1 joule.
2. Symbols and default basis
| Symbol | Meaning | Unit | Default basis |
|---|---|---|---|
| n | Fixed gas amount | mol | 1.5 |
| Tlow, Tc, Thigh | Three absolute temperatures | K | 240, 300, 360 |
| V | Gas volume | L | 10 to 60 |
| Vref | Slope reference volume | L | 25 |
| P | Equilibrium absolute pressure | kPa | calculated |
| Wrev | Reversible isothermal expansion work | J | calculated |
3. Unit and sign normalization
- kPa multiplied by liters equals joules, so no extra 1000 factor appears in the work expression.
- The logarithm uses the dimensionless ratio Vmax/Vmin.
- The displayed slope is local to Vref and becomes less negative as volume grows.
4. Current numerical substitution
5. Independent reconciliation
HOW TO USE THIS CALCULATOR
Five steps from measurements to a defensible result
- Enter a fixed amount of gas.
- Choose the center absolute temperature and a span that leaves the low curve above 0 K.
- Set an ordered positive volume range.
- Place the reference volume within that range.
- Compare all three curves, then use the slope and work only for the center isotherm.
IDEAL-GAS BASICS FOR THIS MODEL
Concepts that control this specific decision
- Isotherm
- A P-V relation evaluated at one constant temperature.
- Inverse shape
- At fixed n and T, doubling volume halves pressure.
- Curve separation
- At the same volume, pressure is proportional to absolute temperature.
- Local slope
- dP/dV is negative and varies with volume; the curve is not a straight line.
- Area as work
- For quasistatic expansion, the area under P(V) is boundary work.
DEEP ANALYSIS 1
A graph is not a trajectory
Points on an isotherm are equilibrium states. The graph does not say how quickly a piston moves or how heat is supplied.
DEEP ANALYSIS 2
Slope changes strongly at small volume
Because slope scales with 1/V², small-volume uncertainty can dominate a local stiffness interpretation.
DEEP ANALYSIS 3
Reversible work is a limiting path
The logarithmic work assumes quasistatic expansion with the center temperature maintained; irreversible work can differ.
RESULT INTERPRETATION
What the current output does—and does not—decide
Use vertical separation to compare thermal pressure at the same volume and horizontal movement to inspect compression or expansion at one temperature.
The endpoint ratio is a Boyle-law check; the local slope is not an average slope across the full range.
REAL USE CASES
Two decisions with different boundary conditions
Checking an expansion estimate
A thermodynamics exercise uses the center curve between 10 and 60 L; the logarithmic area provides reversible isothermal work, not compressor electricity.
EVIDENCE AND DATA QUALITY
What to retain with the exported result
Record the amount basis, whether pressure data are absolute, the temperature uncertainty, volume calibration, chosen graph bounds, and whether a real experiment was slow enough to approximate equilibrium.
LIMITS AND EXCLUSIONS
Where the model stops
- Assumes ideal-gas equilibrium states and fixed composition.
- The three curves are isotherms, not time histories.
- Work applies only to a reversible center-isotherm path.
- The local slope is evaluated at one volume and should not be treated as constant.
- Near condensation or at high density, use real-gas property data.
TERMS USED HERE
Six terms that keep the calculation unambiguous
- Isotherm
- Curve of states at constant temperature.
- P-V diagram
- Plot of pressure against volume.
- Local slope
- Instantaneous derivative at one point.
- Reversible work
- Maximum quasistatic boundary work for the stated path.
- Boyle relation
- Inverse pressure-volume relation at fixed n and T.
- Reference volume
- Point where card values and slope are evaluated.
RELIABLE SOURCES
References supporting the equation and units
FREQUENTLY ASKED QUESTIONS
Questions specific to this calculation
Why are the curves not straight?
P is proportional to 1/V, producing a hyperbola.
Why is the high-temperature curve above the others?
At the same n and V, pressure is proportional to kelvin temperature.
Can the span exceed the center temperature?
No. That would place the low isotherm at or below absolute zero.
Is the slope an effective spring constant?
It can describe local pressure sensitivity, but mechanical force also depends on piston area and system geometry.
Does the work include losses?
No. It is reversible boundary work for the center isotherm only.
Can I use gauge pressure on the graph?
No. The ideal-gas relation requires absolute pressure.
IMPORTANT BOUNDARY
Use the result as analysis, not certification
The graph is an educational equilibrium model, not a compressor map, piston design, or real-gas property chart.