Heat capacity and Specific Heat
Q 
Cp  

 dT  p
 H 
 

 T  p
Q 
 U 
CV  




 dT  V
  T V
ci 
Ci
n

1 Q
n dT
.
fixed P
fixed V
heat capacity per mole (or per gm…)
i can correspond to either P or V.
Heat capacity and Specific Heat
Q 
 H 
Cp  
  

dT

T

p

p
CV
dh  c p dT
du  cV dT
 h  h f  hi 

Tf
Ti
c p dT
Q 


 dT  V
u  u f  ui 
 U 
 

  T V

Tf
Ti
cV dT
Usually the for real substances specific heats are functions of temperature
and we need to know ci (T) before we can evaluate h or u.
Heat capacity and Specific Heat
large heat capacity
Q  nc  T
Q
small heat capacity
T
What physical processes control the heat capacity of a substance?
Ideal Gas:
Monatomic gas; 3 translational DOF. Each contributes kBT/2 to the internal
energy.
u 
3
2
N A k BT
Heat capacity and Specific Heat
Monatomic Ideal Gas cont:
3
3
 du 

  cV  N A k B  R
2
2
 dT  V
Diatomic Ideal Gas:
3 translational and two rotational DOF. Each contributes kBT/2 to the internal
energy.
cV 
5
R
2
Note that the specific heat of an ideal gas does not change with temperature.
Heat capacity and Specific Heat
What physical processes control the heat capacity of a substance?
Crystalline solids
3 translational DOF + 3 vibrational DOF  3R Law of Dulong and Petit.
Turns out that there is a temperature dependence to the specific heat .
Low Temperatures
Lattice vibrations ~ T 3
energy absorption by electrons ~ T
3
 T 
c v  c v  lattice   c v  electronic   K 
  T
 TD 
K = 1940 J/mol-K and  is material dependent
Heat capacity and Specific Heat
TD
Aluminium
428 K
Cadmium
209 K
Chromium
630 K
Copper
343 K
Gold
165 K
Iron
470 K
Lead
105 K
Also note that at very low T
cv  c p
Law of Dulong and Petit
Heat capacity and Specific Heat
High temperatures
Specific heats at temperatures (~above room temperature) are usually expressed
by an empirical equation of the form:
c p  a  bT  cT
2
 J / m ol  K 
Here, a, b and c are material dependent constants. Note that since the equation
is empirical the constants have no simple physical interpretation and that such
equations should only be used over the temperature range for which the constants
were experimentally determined.
Heat capacity and Specific Heat
Relation between cp and cV for an ideal gas
 q  du  Pdv  cV dT  Pdv Statement of the 1st Law
Pv  RT
Ideal gas law (n = 1)
P dv  vdP  R dT
Total derivative
 q  du  P dv  cV dT  R dT  vdP   c v  R  dT  vdP
q 
 dh 

 
  c p   c v  R  dT
 dT  p  dT  p
c p  cv  R
Heat capacity and Specific Heat
Recall the definitions of heat capacity
Q 
Cp  

dT

p
 H 
Q 
 U 
 
and
C


V




 .
 T  p
 dT  V
  T V
In terms of the specific internal energy u and enthalpy h the heat capacities
per unit mass are called specific heats and take on the following forms:
q 
cp  

 dT  p
 h 
q 
 u 
 
 and cV  
 
 .
 T  p
 dT  V
  T V
The specific heat ratio, k, is a quantity that later we will find useful and is
defined by,
k 
cp
cv
Heat capacity and Specific Heat
Approximations for Condensed Phases
It turns out that the specific volume and specific internal energy of condensed
phases vary little with pressure,
v T , p   v f T

u T , p   u f T

In the case of liquids the specific volume and internal energy may be evaluated
for engineering purposes at the condensed liquid state at the temperature of
interest. This is the incompressible substance model. It’s
an approximation.
Heat capacity and Specific Heat
A similar approximation can be made for the specific enthalpy.
Since, h = u + pv,
h  T , p   u  T , p   pv  T , p 
h  T , p   u f  T   pv f  T

T his equation for the enthalpy can be re w ritten as,
u f T

h  T , p   h f  T   p sat  T  v f  T   pv f  T
h  T , p   h f  T   v f  T   p  p sat  T  

Heat capacity and Specific Heat
In the incompressible substance approximation since the specific internal energy
and volume are taken as functions of temperature only,
cv  T

du
( incom pressible )
dT
Note that this is an ordinary derivative since u is only a function of T in this
approximation.
Also in this approximation, since h (T , p )  u (T )  pv taking derivatives while
holding p fixed yields,
du
 h 



dT
 T  p
( incom pressible ).
This says that in this model,
c p  cv
( incom pressible ).
Heat capacity and Specific Heat
Then the changes in specific internal energy and enthalpy can be calculated
according to:
T2
 u 12  u 2  u 1 
 c (T ) dT
T1
T2
 h12  h 2  h1  u 2  u 1  v ( p 2  p1 ) 
 c (T ) dT
T1
incompressible substance
 v ( p 2  p1 )
The Ideal Gas Model
The equation of state of an ideal gas is given by P V  nR T where P is the
pressure, V the volume, R the gas constant, n the number of moles
and T the absolute temperature (K).
The gas constant R = 8.314 J/mole-K. There are various forms of this equation
that you should become familiar with.
Some text books are a little confusing on this issue. For example they define a
symbol R different from the customary definition where the value of R is a
constant for a particular gas of molecular weight M. Then the units of this
R is kJ/kg-K. This is ugly since here R varies from material to material
since the molecular weight is different for different materials. That is,
R = R/M. This is ugly! It’s best just to know how to do the conversions.
For example,
The Ideal Gas Model
P V  nR T
n
m ass
PV 
RT 
m olecular w eight
P
V
m
pv 

m
RT
M
RT
M
RT
M
Also note that since R = 8.314 J/mole-K and 1 mole of a substance contains
6.02 x 1023 molecules (Avogadro’s number, NA), R/NA= 1.38 x 10-23J/molecule ºK. = kB.
kB is known as Boltzmann’s constant.
So, we can rewrite the ideal gas law as
pV  nR T  nN A
R
NA
T  N k BT
N is the no. of molecules
The Ideal Gas Model
Now, there’s a real interesting question to ask. How does the internal energy,
or enthalpy of an ideal gas depend on the state of the gas, i.e., the P, V, T values?
Do you remember how an ideal gas is defined?
The molecules do not interact with one another. The ideal gas equation connects
the 3 variables, so it’s not possible to hold two of them constant and alter the 3rd to
examine this problem.
It turns out that the internal energy of an ideal gas only depends on temperature.
u  u (T )
pv  RT
h  h (T )  u (T )  R T
c p  cv  R
The First Law and the Ideal Gas Model
du = q - w = q - pdv
For the ideal gas u = u(T) so,
du
du 
dT
dT  cV dT
 q  c v dT  pdv
and,
(a)
Also for 1 mole of an ideal gas, pv = RT, pdv  vdp  RdT and substitution
for pdv in Equation (a),
 q  c p dT  vdp
In a quasi-static adiabatic process since, q = 0,
vdp  c p dT
p d V   cv d T
(b)
The First Law and the Ideal Gas Model
Dividing the first equation by the second and rearranging,
dp
p
Integration yields,

c p dv
 
cv v
dv
; w here  
v
ln p    ln v  constant
or

pv  constant
cp
cv