Specific Heat
August 2, 2010
Chapter 12, Section 6
Why do we Need it?

It is rather difficult to measure the total
energy of a system
–

It is very simple to measure the change in
energy of a system
–

Nearly impossible
Control how much you put in
Entropy isn’t exceptionally physically useful
–
Fairly abstract
Specific Heat Capacity

Defined as C  Eatom
T
Eatom 


Esystem
N atoms
Measures how temperature changes with the addition
of energy
Varies Depending on the material
–
Higher C  takes more energy to change the temperature

Good coolants
Heat Capacity vs. Specific Heat

Heat Capacity depends on the size of the
object
–

Specific heat is a constant of the material
–

Units J/K
Units J/K/atom (or J/K/gram)
Heat capacity = mass * (Specific Heat)
Clicker Question #1

Water has a specific heat capacity of 4.2
J/K/gram. Iron’s is 0.84 J/K/gram. How
many grams of iron would it take to match
the heat capacity of 1 gram of water?
–
–
–
–
A)
B)
C)
D)
1
8
6
5
Application: Quenching

Water is often used as a coolant due to its high specific
heat
–
–

450 grams of iron ≈ 1 lb. at 900°C
3790 grams of Water ≈ 1 gallon at 20°C
Find the final Temperature
Eobj  mCT
Esys  Ewater  Eiron  0
mwC w T f  Ti , w   miron Ciron T f  Ti ,iron   0
T f  40.4C
Heat Capacity and Temperature

Specific Heat varies
greatly with temperature
–

More variation at low
temperatures
Approaches some limit at
high temperatures
Classical Limit

At High temps the
specific heat approaches
a classical limit
C  Nk BT
–
N = number of 1-D
oscillators/atom
Another View of Heat Capacity

The number of ways one can store energy in
a molecule
–
–
–

Translation
Vibration
Rotation
Used for gasses
–
Can also be used for solids
Classical Specific Heat of a Solid

For a given 3-D Block
–
Write the energy equation
Etot 

1
1
m v x2  v 2y  v z2  k s s x2  s 2y  s z2
2
2



There is
1
kB
2
For every x2 term in the
energy equation
C
–

There are 6 of them so
1 
C  6 k B   3k B
2 

A Lead Nanoparticle (3 Atoms)
q  N  1!
S  k B ln 
q!  N  1!
Quanta

4
495
Eint
T
S
Eatom
C
T
6.2*kB
60.1 K
5
1287
7.16*kB
3.44*1023
67.7 K
6
3003
8.01*kB
J
Katom
Clicker Question #2

In a 1200 Watt Microwave how long would it
take to bring 240 g (~8 oz) of 20°C water up
to 100°C?

–
–
–
–
A)
B)
C)
D)
Specific Heat of Water is 4.2 J/K/gram
2 minutes
2 minutes, 30 Seconds
1 minute
1 minute, 30 Seconds
Specific Heat of a diatomic Gas

Atoms in gasses are free to
–
–
–

Rotate
Vibrate
Translate
Atoms in solids can really only vibrate
–
Semi – fixed
Diatomic Gas

Rotation
–

Vibration
–

Around which axis is
there a moment of inertia
1 axis of vibration
Translation
–
3 dimensions of motion
Diatomic Gas

Now write an equation for total energy
2
2
2
 1 2 1 2 1 2   m1  p1 1 2   Lrot , x Lrot , y 

E   mv x  mv y  mv z   1 
 ks s   


2
2
2
I
2 I y 
2
  m2  2m1 2

  x

Total Heat Capacity depends on the number of x2
terms in the energy equation
–
There are 7 of them
–
Comes from Virial/Equipartition Theorem
1
 Each squared term gets 2 k B
N
CV  k B
2
Temperature Dependence

All 7 degrees of freedom are not available at
all times
–
–
–
At low temperature: Translation only
Hotter Temps: Translation and Rotation
Very hot: Translation, vibration and rotation