Adiabatic Work Calculator

For a reversible adiabatic process of an ideal gas:

• P·V^γ = constant
• T·V^γ−1 = constant

Work done by the system (common thermodynamics sign):

If temperatures are known, you can also use:

For adiabatic processes, heat transfer Q = 0. For an ideal gas, ΔU = nC_v(T₂ − T₁) with C_v = R/(γ−1).

1. Pick an input set: PV with V₂, PV with P₂, or temperatures with n.
2. Choose units for pressure, volume, and temperature.
3. Enter γ. Typical values: air ≈ 1.4, monatomic gases ≈ 1.67.
4. Enter the required fields. Optional T₁ enables a T₂ estimate in PV modes.
5. Click Calculate. Results appear above the form.
6. Use Download CSV or Download PDF to save results.

Values are rounded and depend on unit choices and γ.

1. Purpose and scope

This calculator estimates work for a reversible adiabatic process in an ideal gas. It is designed for fast checks in thermodynamics problems, lab reports, and engineering sizing where friction, heat leak, and real‑gas effects are small. Results are reported in joules, with optional final state values (P₂, V₂, and T₂) when the chosen inputs allow them.

2. Key inputs and typical data

The most important parameter is the heat capacity ratio γ = C_p/C_v. Common reference values are γ≈1.40 for dry air near room temperature, γ≈1.67 for monatomic gases such as helium and argon, and γ≈1.30–1.35 for many polyatomic vapors. Small γ changes can shift work by several percent in strong compressions.

3. Adiabatic state relations

For a reversible adiabatic path, the calculator uses P·V^γ = constant and T·V^γ−1 = constant. These relations connect the initial and final states, enabling calculation of P₂ from V₂ (or V₂ from P₂) and optional prediction of T₂ when T₁ is entered.

4. Work model and sign

Work by the system is computed from W_by = (P₂V₂ − P₁V₁)/(1 − γ). Expansion usually yields positive W_by, while compression yields negative W_by. If you prefer the engineering convention, the tool also reports W_on = −W_by and lets you highlight either sign convention.

5. Temperature form and energy balance

When n, T₁, and T₂ are provided, the calculator uses W_by = nR(T₂ − T₁)/(1 − γ). For adiabatic behavior, Q = 0, so the first law implies ΔU = −W_by. With an ideal gas, ΔU = nC_v(T₂ − T₁) and C_v = R/(γ − 1).

6. Units and numerical stability

Internally, pressure is converted to pascals and volume to cubic meters to keep P·V in joules. The display can use Pa, kPa, MPa, bar, atm, or psi, and volumes in m³, liters, or imperial units. Very small or large values are displayed using scientific notation to reduce rounding error.

7. Practical interpretation

In rapid piston–cylinder motion or well‑insulated equipment, adiabatic modeling is often appropriate. For example, doubling volume at γ=1.40 reduces pressure by about 2.64×, while halving volume increases pressure by about 2.64×. Use the example table as a sanity check against your own inputs.

8. Limitations and verification

The model assumes a reversible path and ideal‑gas behavior. If there is significant heat transfer, strong friction, phase change, or high pressure where real‑gas effects matter, the predicted work can deviate. Validate results by checking that P₁V₁ and P₂V₂ are physically consistent and that the sign matches expansion or compression.

1) What does “reversible adiabatic” mean here?
It assumes no heat transfer (Q=0) and no dissipative losses, so the path can be described by P·V^γ=constant. Real systems approximate this when insulation is good and processes are fast.

2) Which γ value should I use for air?
For dry air near room temperature, γ≈1.40 is a widely used reference. At higher temperatures or with humidity, γ can shift slightly, which changes computed work and T₂.

3) Why is work negative during compression?
With the “work by system” convention, compression means the surroundings do work on the gas, so W_by becomes negative. The calculator also reports W_on=−W_by if you prefer that sign.

4) Can I compute T₂ without entering T₁?
Not uniquely. The adiabatic relations can give pressure–volume changes without temperature, but T₂ requires at least one temperature reference (such as T₁) or additional thermodynamic information.

5) Why does the calculator convert to SI units internally?
Using pascals and cubic meters keeps P·V in joules, preventing unit confusion and improving consistency checks. You can still enter and view values in the units you select.

6) Is this valid for real gases or steam?
It is best for ideal-gas regions. Steam near saturation, high-pressure gases, or mixtures can deviate from ideal behavior; use real‑gas models or property tables if accuracy is critical.

7) How can I sanity-check my result quickly?
Confirm expansion gives higher V and lower P, while compression does the opposite. Then verify P₂≈P₁(V₁/V₂)^γ (or V₂≈V₁(P₁/P₂)^1/γ) and that the work magnitude seems reasonable.