relationships between partial derivatives
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Relationships between partial derivatives
Reminder to the chain rule
,...),...)y,x(v,...),y,x(u(F,...)y,x(F
composite function: ....),v,u(F
,...),y,x(u ,...,...)y,x(v
You have to introduce a new symbol for this function, also the physicalmeaning can be the same
Example: Internal energy of an ideal gas Tcnun)T(U V0
nR
PV)V,P(T
R
PVcun)V,P(U V0
,...),...)y,x(v,...),y,x(u(F,...)y,x(F
Let’s calculate x
,...)y,x(F
with the help of the chain rule
...x
,...)y,x(v
v
,...)v,u(F
x
,...)y,x(u
u
,...)v,u(F
x
,...)y,x(F
Example: xysinyx,...)y,x(F2/322
xycosyyxxysinx2yx2
3
x
,...)y,x(F 2/32222
explicit:
Now let us build a composite function with: 22 yx)y,x(u and xy)y,x(v
vsinu)v,u(F 2/3 vsinu2
3
u
)v,u(F 2/1
x2x
)y,x(u
vcosuv
)v,u(F 2/3
yx
)y,x(v
x
)y,x(v
v
)v,u(F
x
)y,x(u
u
)v,u(F
x
)y,x(F
u
)v,u(F
x
)y,x(u
v
)v,u(Fy
x
)y,x(v
vsinu
2
3 x2 vcosu 2/3 y xycosyyxxysinx2yx2
3
x
)y,x(F 2/32222
Composite functions are important in thermodynamics
-Advantage of thermodynamic notation:
Example: ))Z,X(Y,X(F)Z,X(F
If you don’t care about new Symbol for F(X,Y(X,Z))
wrong conclusion from X
Y
Y
F
X
F
X
F
0X
Y
Y
F
-Thermodynamic notation: ZXYZ X
Y
Y
F
X
F
X
F
can be well distinguished
Apart from phase transitions thermodynamic functions are analytic
y
)y,x(F
xx
)y,x(F
y yx
)y,x(F
xy
)y,x(F 22
See later consequences for physics
(Maxwell’s relations, e.g.)
Inverse functions and their derivatives
Reminder: )x(yfunction )y(xinverse function defined according to
y))y(x(y
Example:1x
1)x(y
function 1y)1x( 1yxy xyy1
y
y1)y(x
y
y)y1(
y
1y
y1
1
1)y(x
1))y(x(y
X
Y0 2 4 6 8 100
2
4
6
8
10
Y
X
y=y(x,z=const.)
What to do in case of functions of two independent variables y(x,z)
keep one variable fixed (z, for instance)
)z,x(y )z,y(xis inverse to if y)z),z,y(x(y
y)z),z,y(x(y Let’s apply the chain rule to
1dy
dx
dx
)z),y(x(dy
Result from intuitive relation: 1y
x
x
y
Thermodynamic notation:
1Y
X
X
Y
ZZ
0 1 2 30
1
2
3
dY/d
X*d
X/d
Y
X
0 2 4 6 8 100
2
4
6
8
10
Y
X
0 2 4 6 80
2
4
6
dY/d
X
X
0 2 4 6 8 100
2
4
6
8
10
x
Y0 2 4 6 8
0
2
4
6
dX
/dY
Y
Numerical example
Application of the new relation 1Y
X
X
Y
ZZ
Definition of isothermal compressibilityT
T P
V
V
1
Definition of the bulk modulus
Remember the
TT V
PVB
With 1V
P
P
V
TT
1
V
BV T
T
1BTT TT B/1or
Application of yx
)y,x(F
xy
)y,x(F 22
Isothermal compressibility:T
T P
V
V
1
TT
VP
V
Volume coefficient of thermal expansion:P
V T
V
V
1
VP
VT
V
TP
V
PT
V 22
P
T
T
)V(
T
V
P
)V(
P
TT
P TV
T
V
T
VV
T PV
P
V
=
P
TT TVV
V
=T
VVT P
VV
T
V
P
T
PT
PTP
V
TTPT
V
P
We learn: Useful results can be derived from general mathematical relations
Are there more such mathematical relations
Consider the equation of state: )T,V(PP or )T,P(VV
)T),T,P(V(PP
For .constP .const)T),T,P(V(P
Total derivative with respect to temperature
0T
P
T
V
V
P
VPT
VP
T
0P
T
T
P
P
T
T
V
V
P
VVVPT
1
1P
T
T
V
V
P
VPT
(before we calculated derivative with respect to P @ T=const.
now derivative with respect to T @constant P)
1P
T
T
V
V
P
VPT
Is a physical counterpart of the general mathematical relation:
1X
Z
Z
Y
Y
X
YXZ
Let’s verify this relation with the help of an example
X
Y
Z
2222 Rzyx Surface of a sphere
X=0 plane
z
y
y=0 plane
x
z
z=0 plane
x
y
2222 Rzyx
222 zyRx
222 zxRy
222 yxRz
x
y
zyR
y
y
x
222z
y
z
zxR
z
z
y
222x
z
x
yxR
x
x
z
222y
for x,y,z 1st quadrant
yxz x
z
z
y
y
x
z
x
y
z
x
y 1
Physical application: Change in pressure caused by a change in temperature
1P
T
T
V
V
P
VPT
V
PTP
T
1
T
V
V
P
VT
P
VT
P
PT T
V
V
1
V
PV
VTB
X
Y Z
cyclic permutation
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