### 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 = c

_{v}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 = (c

_{v}+ R)dV

V

V

+ c

_{v}dP

P

P

(4)

Using *c _{p} = c_{v} + R* and

*γ = c*we obtain:

_{p}/ c_{v}0 = γ

dV

V

V

+

dP

P

P

(5)

This can be written as:

0 = d(γ ln V) + d(ln P) = d(ln V

^{γ}+ 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)