Short Notes: Thermodynamic Relations | Short Notes for Mechanical Engineering PDF Download

 Page 1


Thermodynamic Relations
Thermodynamic relations are based on the condition of exact differential. Let 
relation exist among the variables x, y and z.
z = f <*, v)
dz —
dz
d x -
dz
dx
r
Or
dy
dz
= M and b
dz
Let
dx
¥
Oy
dM ON
dy
•
dx .
dx dz d\
dv dx. dz
• For thermodynamic variables p ,V & T
dP d v dT
o r .
T
OT
f
dp
Maxwell's Equation
• Using thermodynamics relations and applying to each of those relations 
results in four equations that are known as Maxwell's relation. For a pure 
substance undergoing an infinitesimal reversible process,
Page 2


Thermodynamic Relations
Thermodynamic relations are based on the condition of exact differential. Let 
relation exist among the variables x, y and z.
z = f <*, v)
dz —
dz
d x -
dz
dx
r
Or
dy
dz
= M and b
dz
Let
dx
¥
Oy
dM ON
dy
•
dx .
dx dz d\
dv dx. dz
• For thermodynamic variables p ,V & T
dP d v dT
o r .
T
OT
f
dp
Maxwell's Equation
• Using thermodynamics relations and applying to each of those relations 
results in four equations that are known as Maxwell's relation. For a pure 
substance undergoing an infinitesimal reversible process,
dA = -SdT - PdV
dfi - TdS + VdP 
dG = — SdT + VdP
f d T \ _ /<?V\
^ W ) s ~ { d s ) F
of the four Maxwell's relations, the last two Eqs. (iii)p and (iv) are more valuable 
since they relate entropy derivatives
j 8.5 I
I 8V l 
and
as
to derivatives of pressure and volume
dp
OT
and
. r
\d v 
I 6T
TDS Equation
• First TdS equation,
TdS = C.-dT - T dV
• Second TdS equation,
Difference in Heat Capacity
I o v
Tds = c dr - r —
' \6T
dp
CP ~ Cr = — T
av * i dp
dT
? [ev\
• As
T - 0 =* C, — Cr 
or absolute zero expansivity
Page 3


Thermodynamic Relations
Thermodynamic relations are based on the condition of exact differential. Let 
relation exist among the variables x, y and z.
z = f <*, v)
dz —
dz
d x -
dz
dx
r
Or
dy
dz
= M and b
dz
Let
dx
¥
Oy
dM ON
dy
•
dx .
dx dz d\
dv dx. dz
• For thermodynamic variables p ,V & T
dP d v dT
o r .
T
OT
f
dp
Maxwell's Equation
• Using thermodynamics relations and applying to each of those relations 
results in four equations that are known as Maxwell's relation. For a pure 
substance undergoing an infinitesimal reversible process,
dA = -SdT - PdV
dfi - TdS + VdP 
dG = — SdT + VdP
f d T \ _ /<?V\
^ W ) s ~ { d s ) F
of the four Maxwell's relations, the last two Eqs. (iii)p and (iv) are more valuable 
since they relate entropy derivatives
j 8.5 I
I 8V l 
and
as
to derivatives of pressure and volume
dp
OT
and
. r
\d v 
I 6T
TDS Equation
• First TdS equation,
TdS = C.-dT - T dV
• Second TdS equation,
Difference in Heat Capacity
I o v
Tds = c dr - r —
' \6T
dp
CP ~ Cr = — T
av * i dp
dT
? [ev\
• As
T - 0 =* C, — Cr 
or absolute zero expansivity
J = —
V
QV
QT
• Isothermal compressibility kT is defined as
kj - V
8 V
dp
• Adiabatic compressibility is defined as
k, = ~ r
8 V
dp .
c -c,. =
fi I
TV 3'
Joule Thomson Coefficient
• If a real gas undergoes a throttling process then its temperature changes.
• Slope of a curve drawn between pressure and temperature is known as Joule 
Thomson coefficient generally denoted as pj.
• S entropic curve and inversion curve
Here, jU j < 0:
flf< 0 =
temperature increases
• jUj= 0 ^temperature constant
• jUj> 0 = > temperature decreases
Mj =
e r
A typical phase diagram 
1
f c Cp
5V
r j — ¦ - v
or
Mj=
• For an ideal gas, pj = 0 
Clausius-Clapeyron Equation
Clausius-Clapeyron equation is a way of characterizing a discontinuous phase 
transition between two phase of matter of a single constituent.
Page 4


Thermodynamic Relations
Thermodynamic relations are based on the condition of exact differential. Let 
relation exist among the variables x, y and z.
z = f <*, v)
dz —
dz
d x -
dz
dx
r
Or
dy
dz
= M and b
dz
Let
dx
¥
Oy
dM ON
dy
•
dx .
dx dz d\
dv dx. dz
• For thermodynamic variables p ,V & T
dP d v dT
o r .
T
OT
f
dp
Maxwell's Equation
• Using thermodynamics relations and applying to each of those relations 
results in four equations that are known as Maxwell's relation. For a pure 
substance undergoing an infinitesimal reversible process,
dA = -SdT - PdV
dfi - TdS + VdP 
dG = — SdT + VdP
f d T \ _ /<?V\
^ W ) s ~ { d s ) F
of the four Maxwell's relations, the last two Eqs. (iii)p and (iv) are more valuable 
since they relate entropy derivatives
j 8.5 I
I 8V l 
and
as
to derivatives of pressure and volume
dp
OT
and
. r
\d v 
I 6T
TDS Equation
• First TdS equation,
TdS = C.-dT - T dV
• Second TdS equation,
Difference in Heat Capacity
I o v
Tds = c dr - r —
' \6T
dp
CP ~ Cr = — T
av * i dp
dT
? [ev\
• As
T - 0 =* C, — Cr 
or absolute zero expansivity
J = —
V
QV
QT
• Isothermal compressibility kT is defined as
kj - V
8 V
dp
• Adiabatic compressibility is defined as
k, = ~ r
8 V
dp .
c -c,. =
fi I
TV 3'
Joule Thomson Coefficient
• If a real gas undergoes a throttling process then its temperature changes.
• Slope of a curve drawn between pressure and temperature is known as Joule 
Thomson coefficient generally denoted as pj.
• S entropic curve and inversion curve
Here, jU j < 0:
flf< 0 =
temperature increases
• jUj= 0 ^temperature constant
• jUj> 0 = > temperature decreases
Mj =
e r
A typical phase diagram 
1
f c Cp
5V
r j — ¦ - v
or
Mj=
• For an ideal gas, pj = 0 
Clausius-Clapeyron Equation
Clausius-Clapeyron equation is a way of characterizing a discontinuous phase 
transition between two phase of matter of a single constituent.
• On a p-T diagram, the line separating two phases is known as the coexistence 
curve.
dp Sj — Sj l l
dr Vf-l] T(Vf -V t) T-V
where dp / dT is the slope of the tangent to the co-existence curve at any point, / is 
the specific latent heat. T is the temperature and V is the specific volume 
change and S stands for specific entropy, 
where,
Sf = entropy of the final phase 
S 7 = entropy of the initial phase 
Vf = volume of the final phase 
Vi = volume of the initial phase.