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During an adiabatic process a thermodynamical system does not exchange any heat with the surroundings.

### Equations

By definition, a thermodynamical adiabatic process does not exchange any heat with the surroundings. According to the First Law of thermodynamics we have for an adiabatic process:

0 = Q = dU + dW = cv dT + P dV
(1)

We consider 1 mol of ideal gas:

R T = P V
(2)

and differentiate both sides:

R dT = P dV + V dP
(3)

We compute dT from Eq. (3) and substitute it into Eq. (1) to get:

0 = (cv + R)
dV

V
+ cv
dP

P
(4)

Using cp = cv + R and γ = cp / cv we obtain:

0 = γ
dV

V
+
dP

P
(5)

This can be written as:

0 = d(γ ln V) + d(ln P) = d(ln Vγ + ln P)
(6)

By integrating we get:

const = ln V γ + ln P = ln(P V γ )
(7)

Hence, finally:

P V γ = const
(8)

Processes with the property (8) are called polytropic processes. When we insert the ideal gas law Eq. (2) we get a related property (with different constants const):

T V γ-1 = const
(9a)
V T 1-γ = const
(9b)