Adiabatic Relation Between P V And T in Physics

In an adiabatic process a gas changes without exchanging heat with its surroundings, usually because it is insulated or the change is very fast. For a reversible adiabatic change of an ideal gas, pressure, volume, and temperature are linked by special relations involving gamma.

These relations explain why air heats when compressed in a diesel engine and cools when expanding in a spray can or rising in the atmosphere. They are essential for engine cycles, sound speed, and atmospheric physics.

Key Concepts

Gamma, symbol gamma, equals Cp divided by Cv, ratio of molar heat capacities at constant pressure and constant volume. For monatomic ideal gas gamma = 5/3 = 1.67; for diatomic gas near room temperature gamma = 7/5 = 1.40. Adiabatic condition Q = 0 does not mean temperature constant. Reversible adiabatic is also isentropic, entropy constant. Relationships assume ideal gas and quasi-static change.

Formulas

P-V relation: P V^gamma = constant. T-V relation: T V^(gamma - 1) = constant. T-P relation: T^gamma / P^(gamma - 1) = constant, equivalently T2/T1 = (P2/P1)^((gamma - 1)/gamma). Variables: P pascal, V cubic metre, T kelvin. Derived from first law with Q=0 and PV=nRT. Not valid for irreversible free expansion.

Worked Example

A diatomic gas gamma=1.40 initially at 300 kelvin and volume 0.020 cubic metre is compressed adiabatically to 0.010 cubic metre. Find final temperature.

Step 1: Use T V^(gamma-1) constant, so T2 = T1 (V1/V2)^(gamma-1). Step 2: V1/V2 = 0.020/0.010 = 2. Step 3: exponent gamma-1 = 0.40. 2^0.40 = about 1.32. Step 4: T2 = 300 x 1.32 = 396 kelvin. Temperature rises by 96 kelvin with no heat added; work done on gas became internal energy.

Common Misconceptions

• Adiabatic expansion always cools to same temperature formula as free expansion: Wrong. Free expansion of ideal gas has no work and no temperature change, but reversible adiabatic expansion does work and cools.
• Gamma is same for all gases: Wrong. It depends on molecular degrees of freedom.
• Using Celsius in these ratios: Wrong. Temperature must be in kelvin for multiplicative relations.

Real-World Applications

Diesel engines use high adiabatic compression to ignite fuel without spark plugs. Rising air parcels cool adiabatically, forming clouds when reaching dew point. Refrigerators and gas liquefiers use adiabatic expansion cooling. Speed of sound in gas depends on gamma through relation c = square root of gamma RT/M.

For reversible adiabatic ideal gas, PV^gamma, TV^(gamma-1), and the T-P power law are constant. Compression heats and expansion cools, with gamma controlling how strong the temperature change is.