Dr Roger Bennett
R.A.Bennett@Reading.ac.uk
Rm. 23 Xtn. 8559
Lecture 6
Partial Derivatives
• For functions of several variables
• z=z(x,y), z is continuous function of
independent variables x and y. Here z is
called an explicit function.
• This function can be represented as a surface
in Cartesian coordinates.
Partial Derivatives
• Imagine the surface intersects a planar
surface parallel to the z-x plane which cuts
the y-axis at y.
• The gradient of the line of intersection is
given by the partial derivative of z with
respect to x, and is defined as:-
z  z 
 z ( x  x, y )  z ( x, y ) 
    lim 


x

0
x  x  y
x


Partial Derivatives
• This is the same expression as for normal
differentials but with y considered as a
constant.
• It effectively gives the gradient in the x
direction. A similar expression can be written
for the y direction.
z  z 
 z ( x  x, y )  z ( x, y ) 
    lim 

x  x  y x0 
x

Partial Derivatives - example
• Let z= x2+3xy +y3
z ( x  x, y)  ( x  x)  3 y( x  x)  y
2
3
z( x  x, y)  x  2 xx  x  3 y( x  x)  y
2
2
z  z 
 z ( x  x, y )  z ( x, y ) 
    lim 

x  x  y x0 
x

3
Partial Derivatives - example
• Ignoring terms second order in the limit x0
z( x  x, y)  x  2 xx  3 y( x  x)  y
2
3
z ( x, y)  x  3xy  y
2
3
z  z 
 2 xx  3 yx 
    lim 
 2x  3 y

x  x  y x0 
x

Partial Derivatives
• Differential form take the implicit function
f(x, y, z) = 0, means x, y,and z are related and
only two variables are independent so x = x(y,z)
 x 
 x 
dx    dy    dz
 z  y
 y  z
• Similarly we can write y = y (x,z) and
 y 
 y 
dy    dx    dz
 x  z
 z  x
Partial Derivatives
 x 
 x 
dx    dy    dz
 z  y
 y  z
 y 
 y 
dy    dx    dz
 x  z
 z  x
• Substituting for dy
  x   y   x  
 x   y 
dx      dx          dz



y

x

y

z

z






 z
z
x
y
 z
Partial Derivatives
  x   y   x  
 x   y 
dx      dx          dz
  y   z   z  
 y  z  x  z
x
y
z

• We can choose x and z to be independent
variables so choosing dz = 0 is valid, dx can
then be non-zero.
 x   y 
dx      dx
 y  z  x  z
 x   y 
1     
 y  z  x  z
1
 x 
 y 
     Reciprocal relation
 x  z  y  z
Partial Derivatives
  x   y   x  
 x   y 
dx      dx          dz
  y   z   z  
 y  z  x  z
x
y
z

• We can choose x and z to be independent
variables so choosing dx = 0 is valid, dz can
then be non-zero.
 x   y   x 
0        
 y  z  z  x  z  y
 x   z   y 
1        Cyclical relation
 y  z  x  y  z  x
Partial Derivatives – Chain rule
• Supposing variables x, y, z, are not independent (so any
two variables define the third) we can define the
function  = (x, y) and re-arrange to x = x(, y).
 x 
 x 
dx    d    dy
   y
 y 
• Dividing by dz and hold  constant.
 x   x   y 
      
 z   y   z 
Chain Rule
Higher Partial Derivatives
• Repeated application allows the definition of
higher derivatives.
 z   z 
  
2
x
x  x  y
2
 z
  z 
  
yx
y  x  y
2
 z
  z 
  
xy
x  y  x
2
Higher Partial Derivatives
• The order of differentiation does not matter as
long as the derivatives are continuous.
 z
  z 
  z 
 z
      
yx
y  x  y x  y  x xy
2
2
Pfaffian forms
• The differential equation
df  Xdx  Ydy  Zdz
2

• In general the integral df depends on the
path of integration.
In this case this is
1
called an inexact differential.
• But the integral is path independent if it can be
expressed as a single valued function f(x,y,z).
• df is then an exact differential.
Pfaffian forms – exact differentials
• If we write
 f 
 f 
X    Y   
 x  y , z
 y  x , z
 f 
Z  
 z  x , y
• It follows from the double differentials that
 X

 y

 Y 
   
 x , z  x  y , z
 Z 
 Y 
    
 z  y , x  y  z , x
 Z 
 X 
  

 x  z , y  z  x , y
• These are necessary and sufficient for the Pfaffian form
to be an exact differential.
Dr Roger Bennett
R.A.Bennett@Reading.ac.uk
Rm. 23 Xtn. 8559
Lecture 7
Processes – functions of state
• The internal energy U of a system is a
function of state. For a fluid (ideal gas) we
could write U=U(P,T) or U=U(V,T).
Processes – functions of state
• The internal energy U of a system is a
function of state. For a fluid (ideal gas) we
could write U=U(P,T) or U=U(V,T).
• It is straightforward to imagine that in
changing from state 1 to state 2 we can
define U1 and U2 such that the difference in
energies U=U2-U1 is the change of energy of
the system.
Processes – functions of state
• The internal energy U of a system is a function of
state. For a fluid (ideal gas) we could write U=U(P,T)
or U=U(V,T).
• It is straightforward to imagine that in changing from
state 1 to state 2 we can define U1 and U2 such that
the difference in energies U=U2-U1 is the change of
energy of the system.
• We cannot make the same argument for Work or
Heat as you cannot define a quantity of either to be
associated with a state. They are only defined during
changes.
Processes – functions of state
• The internal energy U of a system is a function of
state. For a fluid (ideal gas) we could write U=U(P,T)
or U=U(V,T).
• It is straightforward to imagine that in changing from
state 1 to state 2 we can define U1 and U2 such that
the difference in energies U=U2-U1 is the change of
energy of the system.
• We cannot make the same argument for Work or
Heat as you cannot define a quantity of either to be
associated with a state. They are only defined during
changes.
• Work and heat flow are different forms of energy
transfer.
Heat and Work
• The physical distinction between these two
modes of energy transfer is:
– Work deals with macroscopically
observable degrees of freedom.
– Heat is energy transfer on the microscopic
scale via internal degrees of freedom of the
system.
Heat and Work
• The physical distinction between these two
modes of energy transfer is:
– Work deals with macroscopically
observable degrees of freedom.
– Heat is energy transfer on the microscopic
scale via internal degrees of freedom of the
system.
• First Law of Thermodynamics dU = đW + đQ
is always true as it is just conservation of
energy – you just have to account for all
possible heat and work processes.
Work
• First Law of Thermodynamics dU = đW + đQ
is always true as it is just conservation of
energy – you just have to account for all
possible heat and work processes.
– Stretching a wire (spring) đW = Fdx
– Surface tension – bubble đW = dA
– Charging a capacitor
đW = EdZ where
E is the emf and Z stored charge.
Analysis of processes
• Quasistatic work
–
đW = -PdV
P
a
b
b
(Wab ) quasistatic    PdV
a
V
Analysis of processes
P
a
• Isothermal Work (ideal gas)
b
b
b
V
dV
(Wab ) Isothermal ,quasistatic    PdV   nkT 
a
a V
 Vb 
(Wab ) isothermal,quasistatic   nkT ln  
 Va 
Heat Transfer
• Constant pressure heating
– đQ = dU - đW
– đQ = dU + PdV
P
a
 U 
 U 
dU  
 dT  
 dV
 T V
 V T
U 

 U  
– đQ  
 dT   P  
 dV
 T V
 V T 

 U  
 U   V 
Cp  
 P 
 

 T V 
 V T  T  P
b
• (đQ/T)P = CP specific heat capacity at constant
pressure.
V
Heat Transfer Ideal Gas Only
• Cv = (đQ/T)v = (dU/T)v
• Cp = (đQ/T)p
 U  
 U   V 
Cp  
 P 
 

 T V 
 V T  T  P
• For ideal gas PV = (-1)U = NRT
NR
 V 
 U 

  0  T   P

P
 V T
C p  CV  R
C p  CV  R
Heat Transfer Ideal Gas Only
• Cv = (đQ/T)v = (dU/T)v = 3/2 R
• Cp = (đQ/T)p
C p  CV  R
Cp  5 R
2