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# Kinetic Theory and The Ideal Gas : Work Done in Thermal Isolation

Physics

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Kinetic Theory and
The Ideal Gas
Work Done in Thermal Isolation
A process which is carried out in thermal isolation is called an Adiabatic
process. This process is carried out by placing the cylinder containing the
ideal gas on a slab of insulating material. During this process no heat enters
or leaves the system. If the gas is allowed to expand, the work is done at the
cost of internal energy.
If we plot a graph between P and V, we get a hyperbolic type curve.
This curve is obtained according to the relation given by
PV
γ
= Constant
Where γ = C
p
/C
v
, Its value varies b/w 1.1 and 1.8
Since value of γ is greater than 1, so the curve obtained by the relation
PV
γ
= Constant is a bit steeper than the curve obtained by the relation
PV = Constant
Hence the work done in this process will be smaller than the work done in
Isothermal process. The PV diagram for Adiabatic process is given below. The
curve obtained in this process is known as Adiabat.

P
i
PV = Constant
PV
γ
= Constant
P
f
P V
i
V
f
V
Let us choose the initial point, where pressure is P
i
and volume V
i,
then we
can write
PV
γ
= P
i
V
i
γ
P = P
i
V
i
γ
/V
γ
……………1
Now the work done on an ideal gas during the adiabatic process can be
calculated as
W = -
vi
vf
Pdv
W = -
vi
vf
P iV i
γ
/V
γ
dv
W = - PiVi^ γ
vi
vf
dv /V
γ

W = - PiVi
vi
vf
V
γ
dv
W = - PiVi
V
γ +1
γ +1
vf
vi
W = - PiVi^γ /1-γ[
V f
γ +1
V i
γ +1
]
W = PiVi^γ /γ-1[
]
W = 1/γ-1[
PiV i
γ
Vf
V f
γ
PiV i
γ
Vi
V i
γ
]
W = 1/γ-1[
PiV i
γ
Vf
V f
γ
PiVi
]
As
PiV i
γ
=PfV f
γ
= Constant so
W = 1/γ-1[
Pf V f
γ
Vf
V f
γ
PiVi
]
Hence
W = 1/γ-1(P
f
V
f
- P
i
V
i
)
The above expression gives the work done in case of Thermal Isolation

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