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Ideal Gas Processes Calculator: Isochoric, Isobaric, Isothermal, Adiabatic, Polytropic

Choose the gas and the process, enter the initial state and one parameter of the end state: pressure, temperature or a ratio of pressures or volumes. The calculator finds state 2, the heat q, the work l, the changes Δu, Δh, Δs and draws the process on p-v and T-s diagrams with the work and the heat shaded.

Heat capacities are constant (about 300 K, NIST WebBook, Shomate equations), pressure is absolute. All values are per 1 kg, the right column gives totals for the mass m. For an adiabatic process the exponent n = k is taken automatically; a polytropic process with n = 1 is an isotherm, with n = 0 an isobar.
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What the calculator does

The five basic processes of an ideal gas with constant heat capacities. All of them are special cases of the polytrope p·vⁿ = const: n = 0 isobaric, n = 1 isothermal, n = k adiabatic, n = ±∞ isochoric. For one kilogram of gas it finds the end state, the heat q, the expansion work l, the technical work l_t and the changes of internal energy Δu, enthalpy Δh and entropy Δs.

Formulas

  • Isochoric: l = 0, q = Δu = cv·ΔT, Δs = cv·ln(T₂/T₁).
  • Isobaric: l = R·ΔT, q = Δh = cp·ΔT, Δs = cp·ln(T₂/T₁).
  • Isothermal: Δu = Δh = 0, q = l = R·T·ln(v₂/v₁) = R·T·ln(p₁/p₂), Δs = q/T.
  • Adiabatic: q = 0, l = −Δu = R·(T₁ − T₂)/(k − 1), Δs = 0, l_t = k·l.
  • Polytropic: l = R·(T₁ − T₂)/(n − 1), q = cn·(T₂ − T₁), cn = cv·(n − k)/(n − 1).

The first law q = Δu + l holds in every case; the calculator shows this check in the solution.

How to read the diagrams

In p-v the area under the curve of the process is the expansion work l, in T-s the area under the curve is the heat q (the sign follows the direction of the process). The thin lines are the isotherm and the adiabat through point 1 in p-v and the isochore and the isobar in T-s: their position shows where a polytrope goes. An adiabat in T-s is a vertical line, an isotherm a horizontal one.

Gas data

Molar masses and cp at 300 K come from the NIST Chemistry WebBook (cp from the Shomate equations); dry air is a mixture of N₂, O₂, Ar and CO₂ of standard composition. The exponent k = cp/(cp − R) is about 1.40 for air and diatomic gases, about 1.29 for CO₂ and 1.67 for monatomic gases. An own gas can be set by its molar mass and k.

FAQ

How does a polytropic process differ from an adiabatic one?
An adiabatic process has no heat exchange (q = 0) and n = k. A polytropic one does: for n below k the expanding gas receives heat, for n above k it gives heat away. The heat capacity of a polytrope cn = cv·(n − k)/(n − 1) can even be negative.
What is the work in an isothermal process?
l = R·T·ln(v₂/v₁) = R·T·ln(p₁/p₂). All the heat supplied goes into work, q = l, because the internal energy of an ideal gas depends on temperature only.
What is the technical work l_t?
The flow work of a machine such as a compressor or a turbine: l_t = −∫v dp = l + p₁v₁ − p₂v₂. For an adiabat l_t = k·l = −Δh, for an isotherm l_t = l, for an isobar l_t = 0.
How is the entropy change found?
Δs = cv·ln(T₂/T₁) + R·ln(v₂/v₁) = cp·ln(T₂/T₁) − R·ln(p₂/p₁). For an adiabat Δs = 0, for an isotherm Δs = R·ln(v₂/v₁), for an isochore cv·ln(T₂/T₁).
How accurate are constant heat capacities?
Up to 400–500 K the error is small. At 1000–2000 K the real cp is 10–25 % higher and k is lower. Courses of technical thermodynamics usually take constant heat capacities or tables of mean heat capacities.