Polytropic Process Calculator

Calculator

Fields adapt to your chosen known final value.

Choose what you know about the final state

If you know T2 and n=1, the change is isothermal.

Pressure Unit

Volume Unit

Temperature Unit

Initial Pressure (P1)

Initial Volume (V1)

Initial Temperature (T1)

Polytropic Exponent (n)

Common values: 0 (not allowed), 1 (isothermal), γ (isentropic).

Final Pressure (P2)

Final Volume (V2)

Optional for T2 mode with n = 1.

Final Temperature (T2)

Moles (optional)

Used for ΔU, ΔH, and Q estimates.

Heat Capacity Ratio γ (optional)

Must be greater than 1.

Formula Used

A polytropic path follows the relation P·Vⁿ = constant. With an ideal gas and fixed amount, temperature is consistent with T ∝ P·V.

P1·V1ⁿ = P2·V2ⁿ
If P2 is known: V2 = V1·(P1/P2)^1/n
If V2 is known: P2 = P1·(V1/V2)ⁿ
If T2 is known and n≠1: T·V^n-1 = constant so V2 = V1·(T1/T2)^1/(n-1)
Work (n≠1): W = (P2·V2 − P1·V1)/(1−n)
Work (n=1): W = P1·V1·ln(V2/V1)
Optional energy: ΔU = N·c_v·(T2−T1), Q = ΔU + W, using c_v = R/(γ−1)

How to Use This Calculator

Select what you know about the final state: P2, V2, or T2.
Pick units for pressure, volume, and temperature.
Enter the initial state values P1, V1, and T1.
Provide the polytropic exponent n for your process.
Enter the known final value (P2 or V2 or T2).
Optionally enter moles and γ to estimate heat transfer.
Press Calculate to view results above the form.
Use CSV/PDF buttons to export your computed report.

Example Data Table

Sample scenarios for verification and comparison.

Case	P1 (kPa)	V1 (m³)	T1 (K)	n	Known	Value	Computed (P2, V2, T2)
A	200	0.050	300	1.30	P2	500 kPa	V2 ≈ 0.0260 m³, T2 ≈ 390 K
B	150	0.030	320	1.00	V2	0.060 m³	P2 ≈ 75 kPa, T2 ≈ 320 K
C	400	0.010	350	1.20	T2	450 K	P2 and V2 from T·V^(n−1)

Values are rounded for display. Use the calculator for exact outputs.

Note: This tool assumes a fixed amount of ideal gas for temperature relations. If your system has real-gas effects or changing mass, results require corrections.

1) Meaning of a Polytropic Path

A polytropic process follows P·Vⁿ = C, where the exponent n models heat exchange and irreversibilities in a compact way. It is widely used to approximate real compression/expansion in cylinders, compressors, and turbines when the path is between isothermal and adiabatic behavior.

2) Choosing the Exponent n

Typical reference values are: n = 0 (constant pressure), n = 1 (isothermal ideal-gas limit), n = γ (reversible adiabatic for an ideal gas), and very large n trending toward constant volume. In practice, fitted values such as 1.2–1.4 are common for gas compression with cooling losses.

3) Linking States (P1, V1) to (P2, V2)

With a known n, the end state is obtained from P1·V1ⁿ = P2·V2ⁿ. If you know P2, the calculator solves V2; if you know V2, it solves P2. The tool also reports converted SI values to keep unit consistency.

4) Temperature Relation for an Ideal Gas

For a fixed amount of ideal gas, T ∝ P·V, so temperature changes along the polytrope can be computed after P2 and V2 are determined. When T2 is the known target and n ≠ 1, the calculator uses T·Vⁿ⁻¹ = const to solve the final volume and then pressure.

5) Work of Compression or Expansion

The boundary work is the most common engineering output. For n ≠ 1, W = (P2·V2 − P1·V1)/(1−n). For the isothermal limit n = 1, W = P1·V1·ln(V2/V1). The sign indicates whether the system does work (expansion) or receives work (compression).

6) Heat and Internal-Energy Estimates

If you provide moles and γ, the calculator estimates ΔU, ΔH, and heat transfer using ideal-gas relations. With c_v = R/(γ−1) and c_p = γR/(γ−1), it computes ΔU = N c_v(T2−T1), ΔH = N c_p(T2−T1), and Q = ΔU + W.

7) Data Quality Checks and Edge Cases

Reliable results require physically valid inputs: positive pressures/volumes, temperatures above absolute zero, and consistent targets. The calculator warns when n approaches 1 (logarithmic work form) or when optional parameters are missing. For real gases at high pressure or near condensation, treat the results as an approximation.

8) Practical Use in Design and Testing

Engineers often infer n from measured pressure–volume data by fitting a straight line to ln(P) versus ln(V), where the slope is −n. Once n is known, you can quickly estimate end states, work, and heat trends for performance comparisons, sizing, and sanity checks during experiments.

FAQs

1) What does n physically represent?

It is an empirical exponent describing how pressure changes with volume. It indirectly captures heat transfer and losses, placing the path between isothermal and adiabatic limits.

2) When should I use n = 1?

Use n = 1 for an ideal-gas isothermal process, where temperature is effectively constant. In real systems, strong cooling can make n close to 1.

3) Why is n = γ called adiabatic?

For an ideal gas undergoing a reversible adiabatic process, the relation is P·V^γ = const. It is a special polytropic case with no heat transfer.

4) Can the calculator handle unit mixing?

Yes. You may choose different units for each field. The tool converts to consistent SI units internally, then reports both computed SI and display-unit results.

5) What if my known target is T2?

If n ≠ 1, it uses T·Vⁿ⁻¹ = const to solve V2, then computes P2 from P·Vⁿ = const. For n = 1, isothermal implies T2 ≈ T1.

6) Why are moles and γ optional?

They are only needed for energy estimates (ΔU, ΔH, and Q). If omitted, the calculator still computes end-state values and boundary work.

7) Are results accurate for steam or refrigerants?

They can be rough at best. Near phase change or at high pressure, real-gas effects matter. Use property tables or real-gas models if accuracy is critical.