The ratio of heat capacities at constant pressure and volume, describing how pressure changes with density in a gas.The adiabatic index, often denoted by the Greek letter gamma (γ), is a fundamental property of gases that describes how they respond to compression or expansion when no heat is exchanged with the surroundings — a process called adiabatic. It is defined as the ratio of the heat capacity at constant pressure (Cp) to the heat capacity at constant volume (Cv). For a simple ideal gas, this ratio is also the exponent in the relationship between pressure and density during an adiabatic process: pressure is proportional to density raised to the power γ. This means that a gas with a higher adiabatic index becomes 'stiffer' — its pressure rises more sharply when compressed.
The concept was developed in the 19th century, with key contributions from Sadi Carnot, Émile Clapeyron, and later Rudolf Clausius and William Rankine, who formalized the ratio of heat capacities in the context of thermodynamics.
The index arises from the different ways a gas stores energy. At constant volume, all added heat goes into raising the internal energy (and thus temperature). At constant pressure, the gas also does work by expanding, so more heat is needed for the same temperature rise. The ratio of these heat capacities reflects how much energy goes into work versus internal energy. For monatomic gases like helium, γ is about 5/3; for diatomic gases like nitrogen or oxygen, it is about 7/5; for more complex molecules, it approaches 1. During a rapid adiabatic compression, the temperature rises, and the pressure increases more than it would if heat could escape — the index quantifies exactly how much more.
The adiabatic index is crucial in meteorology for predicting temperature changes in rising air parcels, in engineering for designing internal combustion engines and gas compressors, and in astrophysics for modeling the structure of stars and the behavior of interstellar gas.
In the Sun's core, where temperatures are millions of degrees, the adiabatic index of the plasma is close to 5/3, and it plays a role in determining the star's stability against gravitational collapse.
| c_s | variable | adiabatic sound speed — sound speed metre per second |
| T | variable | temperature — temperature kelvin |
| m | variable | mean particle mass — mass kilogram |
| γ | constant | ratio of specific heats (Cp/Cv) of the gas — adiabatic index = varies (e.g., 5/3 for monatomic ideal ga |
| k_B | constant | relates temperature to kinetic energy of particles — boltzmann constant = 1.380649e-23 joule per kelvin |
| \Gamma_1 | variable | First adiabatic index — adiabatic index |
| M | variable | Mass of the star — mass kilogram |
| R | variable | Radius of the star — radius metre |
| c | variable | Speed of light — speed metre per second |
| G | constant | Newtonian gravitational constant — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| k | constant | Numerical constant of order unity depending on stellar structure — structure constant = 1 |
| p_1 | variable | Pre-shock pressure — pressure pascal |
| p_2 | variable | Post-shock pressure — pressure pascal |
| \gamma | variable | Adiabatic index (ratio of specific heats) — adiabatic index |
