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 = C_{v}dT
and that δQ = 0, since the process is adiabatic, we have

C_{v}dT + PdV = 0 .

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

dT = d(PV)/R = (PdV + VdP)/R =
(PdV + VdP)/ (C_{p} – C_{v}) .

Substituting this expression for dT into the previous equation we get

C_{p}PdV + C_{v}VdP
= 0 .

It is convenient to introduce a quantity γ defined as

γ = C_{p}/C_{v} .

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 C_{p} and C_{v} 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

PV^{γ} = const .

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

TV^{γ-1} = const ,

P^{γ-1}/T^{γ} = const .

Since γ > 1, it follows from
TV^{γ-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.