Polytropic Process of an Ideal Gas

Polytropic Process of an Ideal Gas

Polytropic Process of an Ideal Gas
• The relationship between the pressure and volume during
compression or expansion of an ideal gas can be described
analytically. One form of this relationship is given by the
equation
pV
n =
constant
• where n is a constant for the particular process.
• A thermodynamic process described by the above equation is
called a Polytropic process.
• For a Polytropic process between two states 1-2
n n
p1 V1
= p2V2
=
constant

Remarks
n
1
p V = p V =
1 2 2
constant
• When n=0, p = constant, and the process is a
constant pressure or an isobaric process.
• When n=1, pV = constant, and the process is
a constant temperature or an isothermal
process.
• When n∞, it is called an isometric process.
• When n=k, it is an called isentropic process.
n

Adiabatic Process
A thermodynamic process in which there is no heat
into or out of a system is called an adiabatic process.
To perform an ideal adiabatic process it is necessary,
that the system be surrounded by a perfect heat
insulator. If a compression or expansion of a gas
takes place in a short time, it would be nearly
adiabatic, such as the compression stroke of a
gasoline or a diesel engine.

Adiabatic-Polytropic (Isentropic) Process
Let an ideal gas undergo an infinitesimal adiabatic process:
results in
results in :
dQ = 0
dU
nC
PV
=
From the first law :
dU = dQ – dW
v
PdV
dT
Taking the derivative of
=
nC
+
Eliminating dT between these two equations and using
Cp – Cv
v
=
nRT
dT, and dW =
VdP
=
– PdV
R
=
nRdT
PdV.
the ideal gas law :
dp
p
+
C
p
dV
C V
v
= 0

Adiabatic-Polytropic (Isentropic) Process
Denote C p /C v = k, the ratio of specific heat capacities of the gas.
Then
Integration gives
dp
p
+
k
dV
V
= 0
So
ln(P) + k ln(V) =
ln(constant)
pV
k =
constant

Remarks
For an adiabatic process
p 1 V 1k = p 2 V
k 2
Work done during an adiabatic process:
W 12 = (p 1 V 1 –p 2 V 2 )/(k-1)
Alternate expression: W 12 = nC v (T 1 -T 2 ), for
constant C v

Ideal Gas Polytropic Process
From
p
p
2
1
=
⎛ V
⎜
⎝V
1
2
⎞
⎟
⎠
n
p 1 V 1 =nRT 1 and p 2 V 2 =nRT 2
we get
p
p
2
1
=
⎛
⎜
⎝
nRT
nRT
1
2
/
/
p
p
1
2
⎞
⎟
⎠
n
=
⎛ T
⎜
⎝ T
1
2
p
p
2
1
⎞
⎟
⎠
n
=
⎛
⎜
⎝
p
p
2
1
⎞
⎟
⎠
n
⎛ T
⎜
⎝ T
1
2
⎞
⎟
⎠
n
or
T
T
1
2
=
⎛
⎜
⎝
p
p
2
1
⎞
⎟
⎠
1−n
n
or
T
T
2
1
=
⎛
⎜
⎝
p
p
2
1
⎞
⎟
⎠
n−1
n
Similarly:
T
T
2
1
=
⎛ V
⎜
⎝V
1
2
⎞
⎟
⎠
n−1