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
n
•
pV = 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
•
p1V = p 2 V = constant
n
1
n
2
Remarks
p1V = p 2 V = constant
n
1
n
2
• 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 nÆ∞, it is called an isometric process.
• When n=k, it is an called isentropic process.
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:
dQ = 0
dU = nC v dT, and dW = PdV.
From the first law :
dU = dQ – dW
nC v dT = – PdV
Taking the derivative of the ideal gas law :
PV = nRT
results in
PdV + VdP = nRdT
Eliminating dT between these two equations and using
Cp – Cv = R
results in :
dp C p dV
+
=0
p
CvV
Adiabatic-Polytropic (Isentropic) Process
Denote Cp/Cv = k, the ratio of specific heat capacities of the gas.
Then
Integration gives
dp
dV
+k
=0
p
V
ln(P) + k ln(V) = ln(constant)
So
pV k = constant
Remarks
For an adiabatic process
p1V1k = p2V2k
Work done during an adiabatic process:
W12 = (p1V1 – p2V2)/(k-1)
Alternate expression: W12 = nCv(T1-T2), for
constant Cv
Ideal Gas Polytropic Process
From
p2 ⎛ V1 ⎞
= ⎜⎜ ⎟⎟
p1 ⎝ V2 ⎠
n
p1V1=nRT1 and p2V2=nRT2
n
we get
p2 ⎛ nRT1 / p1 ⎞ ⎛ T1 p2 ⎞ ⎛ p2 ⎞
⎟⎟ = ⎜⎜
⎟⎟ = ⎜⎜ ⎟⎟
= ⎜⎜
p1 ⎝ nRT2 / p2 ⎠ ⎝ T2 p1 ⎠ ⎝ p1 ⎠
T1 ⎛ p2 ⎞
or = ⎜⎜ ⎟⎟
T2 ⎝ p1 ⎠
Similarly:
n
T2 ⎛ V1 ⎞
= ⎜⎜ ⎟⎟
T1 ⎝ V2 ⎠
1− n
n
n −1
T2 ⎛ p2 ⎞
= ⎜⎜ ⎟⎟
or
T1 ⎝ p1 ⎠
n −1
n
n
⎛ T1 ⎞
⎜⎜ ⎟⎟
⎝ T2 ⎠
n