Adiabatic Processes (Expansion or Compression)

In adiabatic process heat is neither allowed to enter nor leave the system. Therefore

q = 0

So, according to the firs law of thermodynamics, we have

U = q + w

or U = w

If there is an expansion, w will be negative and, therefore U will be negative i.e. internal energy will decrease and hence its temperature will fall. This indicates that the work is done by the system at the expense of internal energy.

Adiabatic Processes (Expansion or Compression)

For compression, w will be positive and, therefore, U will also be positive, i.e., internal energy will increase and hence temperature will rise. Here, the work is done by the surroundings on the system, which is stored as the internal energy. The equation shown above can be written as,

U =  w

For a finite change, we have

 w = C_v (T₂ T₁) For reversible expansion

dU =  PdV ; C_v dT =  PdV

For one mole of an ideal gas, P = RT/V; hence C_vdT = −RT ^dV

V

C_v ^dT

T = −R ^dV

V

For a finite volume change from V₁ at temperature T₁ to V₂ at temperature T₂, we have

C_T^T2 _v ^dT_T

1 = −R _V^{V2 dV}_V

1

Since C_v is constant, C_v ln ^T2

T₁ = −R ln ^V2

V₁

Again, C_p  C_v = R and for 1 mole of gas R/C_v =   1, where  = C_p/C_v, the ratio of molar heat capacities.

Inserting this value transforms the above equation to equation

ln ^𝑇2

𝑇₁ = − 𝛾 − 1 ln ^𝑉2

𝑉₁

This can be written in the form,

𝑇₂

𝑇₁ = ^𝑉1

𝑉₂

𝛾−1

or 𝑇₁𝑉₁^𝛾−1 = 𝑇₂𝑉₂^𝛾−1

For an ideal gas we know

T₂

T₁ = ^P2V2

P₁V₁ Therefore, ^P2V2

P₁V₁ = ^V1

V₂

γ−1

P₁V₁^γ = P₂V₂^γ ; PV^ = constant

Since C_p is always greater than C_v the ratio C_p/C_v is larger than unity. When pressure is plotted against volume it will be noted that the curve for an adiabatic process is steeper than that for an isothermal process.