Part V Thermal Properties of Materials
Chap. 18 Introduction
Chap. 19 Fundamentals of Thermal Properties Chap. 20 Heat Capacity
Chap. 21 Thermal Conduction
Chap. 22 Thermal Expansion
19.1 Heat, Work, And Energy
First law of thermodynamics
E W Q
∆ = +
: energy change of the system: the work on the system
: the heat received by the system E
W Q
∆
In this chapter, we limit our consideration to processes for which W can be considered to be zero.
E Q
∆ =
Energy, work, and heat have same unit.
1 cal = 4.184 J 1 J = 0.239 cal
Heat capacity : the amount of heat which needs to be transferred to a substance in order to raise its temperature by a certain temperature interval.
v
v
C E
T
∂
′ = ∂
2
v p
C C α TV
′ = ′ − κ
Generally, it is interested in two kinds : at constant volume & at constant pressure
at constant volume
p
p
C H
T
∂
′ = ∂ at constant pressure
These relationship is….
= volume expansion coefficient of a material = compressibility of a material
T = temperature V = volume α
κ
19.1 Heat Capacity C'
2
v p
C C α TV
′ = ′ − κ Proving
p v
P V P P V
( )
H U U PV U
C C
T T T T T
∂ ∂ ∂ ∂ ∂
− =∂ −∂ = ∂ + ∂ − ∂
and
T V T S T V
U U S U S P
T P T P
V S V V V T
∂ ∂ ∂ ∂ ∂ ∂
= + = − = −
∂ ∂ ∂ ∂ ∂ ∂
P P
(PV) ( )V
P PV
T T α
∂ ∂
= =
∂ ∂
P V
T
V
P T
T V
P
α κ
∂
∂ ∂
= =
∂ ∂
∂
P T P V T V
U U V U U U
T V T T αV V T
∂ ∂ ∂ ∂ ∂ ∂
= + = +
∂ ∂ ∂ ∂ ∂ ∂
T
U T
V P
α κ
∂
∴ ∂ = −
2
P T V V V
U U U T U TV U
V V P PV
T V T T T
α α
α α α
κ κ
∂ ∂ ∂ ∂ ∂
= + = ⋅ − + = − +
∂ ∂ ∂ ∂ ∂
2 2
p v
V V
2
v p
TV U U TV
C C PV PV
T T
C C TV
α α α α
κ κ
α κ
∂ ∂
− = − + ∂ + − ∂ =
∴ = −
19.1 Heat Capacity C'
Specific heat capacity is the heat capacity per unit mass
c C
m
= ′
v
v
c E E
C m
T T
∂ ∆
∴ ′ = = ∂ = ∆ ∆ = = ∆ E Q m T c
v19.3 Specific Heat Capacity, c
298K
At room temperature, molar heat capacity
at constant volume is approximately 25 J/mol·K for most solids. (discovered by Dulong and Petit)
Exception : carbon reach 25J/mol·K at high temperature
Molar heat capacity is the heat capacity per mole v
C
v vC c M
n
= ′ = ⋅
0
n N
= N
19.4 Molar Heat Capacity, C
vAll heat capacities are zero at T = 0K.
Near T= 0K, heat capacities are climb in proportion to T3
Debye Temperature θD: a Temperature at which heat capacities reach 96% of their final value.
19.4 Molar Heat Capacity, C
vunit ( J/m s K )
Q
J K dT
= − dx ⋅ ⋅
Heat flux is proportional to the temperature gradient.
The proportionality constant is called Thermal conductivity .
heat flux
temperature gradient
(or ) thermal conductivity J
QdT dx
K λ
=
=
=
Negative sign indicates that the heat flows from the hot to the cold end.
19.5 Thermal Conductivity, K
19.5 Thermal Conductivity, K
In ideal gas,
PV = nRT
0
8.314(J/mol K) 1.986(cal/mol K)
R = k N
B= ⋅ = ⋅
pressure of the gas volume of the gas
the amount of substance universal gas constant
the thermodynamics temperature P
V n R T
=
=
=
=
=
0
Boltzmann constant Avogadro constant kB
N
=
=
This equation is a combination of two experimentally obtained thermodynamics.
PV = constant ( discovered by Boyle and Mariotte ) V ~ T, at constant P ( discovered by Gay-Lussac )
19.6 The Ideal Gas Equation
In Figure 19.2
velocity dV = Adx = Avdt v =
v
1
z = 6 n v
v v v
1 1
number of particles
6 6
z ′ = n dV ⋅ = n Avdt n =
A The number of particles reaching the end face
The number of particles per unit time & unit area that hit the end face
v
N the number of particles per unit volume n = V
19.7 Kinetic Energy of Gases
2
0
1
3
B BPV = Nmv = nRT = nk N T = k NT
1 2 kin 2
E = mv
1 1
22
3 2 2 3
B kin
k NT = N mv = NE
3
kin
2
BE k T
∴ =
* 2
v
1 1
2 2
6 3
p z mv n v mv N mv
= = = V
* 2
( ) / / 1
3 F ma d mv dt dp dt N
P p mv
A A A A V
= = = = = =
The momentum per unit time & unit area
This yields, for the pressure,
Inserting ,
