Thermodynamics: Adiabatic Process
- Mole
- Gas Volume
- States of Matter
- Heat
- Enthalpy
- Thermodynamics
- Adiabatic Process 🢀
- Mass Energy Conservation
- Carnot Engine
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
V
+ cv
dP
P
P
(4)
Using cp = cv + R and γ = cp / cv we obtain:
0 = γ
dV
V
V
+
dP
P
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)