Adiabatic process. Poissons equation

 

A process during which no heat is received or lost is called adiabatic. Let us find the relation between the parameters of an ideal gas during a quasi-static adiabatic process. 

Using the I Law of thermodynamics

 

δQ = dU +δW

or

δQ = dU +PdV

 

and noting that  dU = CvdT and that δQ = 0, since the process is adiabatic, we have

 

CvdT + PdV = 0 .

 

Using the equation of state for an ideal gas we can write

 

dT = d(PV)/R = (PdV + VdP)/R = (PdV + VdP)/ (Cp – Cv) .

 

Substituting this expression for dT into the previous equation we get

 

CpPdV + CvVdP = 0 .

 

It is convenient to introduce a quantity γ defined as

 

γ = Cp/Cv .

 

Then

 

γPdV + VdP = 0 .

 

This is a differential equation for a quasi-static adiabatic process for an ideal gas. The molar specific heats of ideal gas Cp and Cv may depend on temperature. However, in many cases they remain constant in a wide range of temperatures. If it is the case, γ will also be constant. Then the above differential equation can be easily integrated leading to

 

PVγ = const .

 

This equation is called the Poissons equation. Since for an ideal gas PV = RT this equation can be expressed in two other forms:

 

TVγ-1 = const ,

 

Pγ-1/Tγ = const .

 

Since γ > 1, it follows from TVγ-1 = const that an ideal gas heats up during the adiabatic compression, while it cools down upon adiabatic expansion. This effect is used in diesel engines where the ignition of the fuel/air mixture is achieved via the adiabatic compression. The reason for the heating up of the gas during the adiabatic compression is that during this process work is done on the gas, which results in the increase of its internal energy. Since the internal energy depends solely on temperature, the temperature of the gas increases. In the same fashion one can rationalise the cooling of the gas upon adiabatic expansion.