Lecture 14 – Review for Exam 1
Wednesday September 26th
¾Chapter 1 – definitions (see lecture 2)
¾Chapter 2 – Equations of state
•Ideal gas law and van der Waals’ equation
•P-v-T surfaces
•Expansivity and compressibility
¾Chapter 3 – The first law
•Exact, inexact differentials, and state variables
•Work (configuration, dissipative and adiabatic)
•Heat
¾Chapter 4 – Applications of the first law
•Heat capacity and specific heat
•Enthalpy and heats of transformation
•Work done in adiabatic processes
Exam 1 – Friday (50-55 mins in class), Chs. 1-4.
There will be no homework due next week
Equation of State of an Ideal Gas
•In chapter 1, we used the zeroth law to show that a
relationship always exists between P, V and T.
General form: f (P,V,T) = 0
Example:
PV = nRT
(ideal gas law)
n quantifies the amount of the substance. The units
of R and n are linked such that their product nR
has the dimensions of joules/kelvin.
•If n measured in kilomoles, then R = 8.314 × 103 J/kilomole⋅K
•If n measured in moles, then R = 8.314 J/mole⋅K
•Ideal gas law may also be written in intensive form
Pv = RT
v is the specific volume in either m3/mole or m3/kilomole
Equation of State of a Real Gas
Van der Walls’ equation in intensive form:
a⎞
⎛
⎜ P + 2 ⎟ ( v − b ) = RT
v ⎠
⎝
100
90
Pressure (arb. units)
80
o
70
60
50
o
12
o
10
o
9
o
8
o
6
o
4
CP
40
11
o
9.5
o
8.5
o
7
o
5
o
3
30
20
10
0
0.1
0.2
0.3
0.4
0.5
0.6
Specific volume (arb. units)
PVT surface for a real substance
Regions where Van der Walls’ PVT surface is multi-valued
is where one finds co-existence of phases.
Expansivity and Compressibility
Two important measurable quantities:
Expansivity:
1 ⎛ ∂v ⎞
β≡ ⎜ ⎟
v ⎝ ∂T ⎠ P
Compressibility:
1 ⎛ ∂v ⎞
κ ≡− ⎜ ⎟
v ⎝ ∂P ⎠T
Some sophisticated multivariable calculus
In general, the differential dz = Mdx + Ndy is ‘exact’ if:
1.
2.
∂M ∂N
=
∂y
∂x
Read Also Appendix A
v∫ dz = 0
∫ dz is independent of path
b
3.
a
•In thermodynamics, all state variables are by definition
exact. However, there exist several thermodynamic
quantities whose differentials are inexact: work and heat
are the best examples.
•Mathematical techniques exist to handle the integration of
inexact differentials – see Appendix A for further details.
Configuration Work
This is the work done in a reversible process given by the
product of some intensive variable (y) and the change in
some extensive variable (X). The most general case would
be:
đW =
∑ y dX ,
i
i
i
i = 1, 2,....n.
Work and Internal Energy
•Differential work đW is inexact (work not a state variable)
•Configuration work is the work done in a reversible process
given by the product of some intensive variable (y) and the
change in some extensive variable (X).
•đW is the work done by ‘the system’, e.g. đW is positive
when a gas expands.
•Dissipative work is done in an irreversible process and is
always done ‘on the system’, i.e. đWirr < 0 always.
•Total work (configuration and dissipative) done in adiabatic
process between two states is independent of path. This
leads to the definition of internal energy (state variable).
∫
b
a
dU = U b − U a = −
∫
b
a
đWad = −Wad
and
dU = − đWad
The First Law of Thermodynamics
“The heat supplied to a system is equal to the
increase in internal energy of the system plus
the work done by the system. Energy is
conserved if the heat is taken into account.”
đQ = dU + đW
đQ is not a state variable. However, correct combination of
the inexact differentials đQ and đW leads to the exact
differential dU:
dU = đQ − đW
The increase in internal energy is equal to the heat flow
into the system, minus any work done by the system, and
integration of dU is independent of path.
A summary of the first law is given at
the bottom of page 46 in the text book
Properties of Heat
Though we have yet to prove this, it is the temperature of
a body alone that determines whether heat will flow to or
from a body,
“Heat energy is transferred across the boundary of a
system as a result of a temperature difference only.”
•However, this does not necessarily imply that the transfer
of heat to a body will increase its temperature. It may
also undergo a change of state (phase) from e.g. a liquid
to a gas, without a change in temperature.
•Also, if the temperature of a system increases, it does
not necessarily imply that heat was supplied. Work may
have been done on the system. Therefore,
“Heat is the change in internal energy of a system
when no work is done on or by the system.”
Heat Capacity and specific heat
The heat capacity C of a system is defined as:
⎛ Q
C ≡ lim ⎜
ΔT → 0 ΔT
⎝
⎞ đQ
⎟=
⎠ dT
•Heat capacity is an extensive quantity.
The specific heat capacity c of a system is:
1 ⎛ đQ ⎞ đq
c≡ ⎜
⎟=
n ⎝ dT ⎠ dT
•Specific heat is obviously an intensive quantity.
•SI units are J.kilomole-1.K-1.
Heat Capacity and specific heat
Because the differential đq is inexact, we have to specify
under what conditions heat is added.
•
the specific heat cv; heat supplied at constant volume
•
the specific heat cP; heat supplied at constant pressure
⎛ đq ⎞
cv ≡ ⎜
⎟
⎝ dT ⎠v
and
⎛ đq ⎞
cP ≡ ⎜
⎟
⎝ dT ⎠ P
Using the first law, it is easily shown that:
⎛ đq ⎞ ⎛ ∂u ⎞
cv ≡ ⎜
⎟ =⎜
⎟
⎝ dT ⎠v ⎝ ∂T ⎠v
•For an idea gas, the internal energy depends only on the
temperature of the gas T. Therefore,
du
cv ≡
dT
and
T
u − u0 = ∫ cv dT
T0
Enthalpy
When considering phase transitions, it is useful to define a
quantity h called ‘enthalpy’
h ≡ u + Pv
•Because h depends only on state variables, it too must be
a state variable – hence its usefulness.
•When a material changes phase (e.g. from a solid to a
liquid) at constant temperature and pressure, latent heat l
must be added. This heat of transformation is related
simply to the enthalpy difference between the liquid and
solid
l = ( u2 + Pv2 ) − ( u1 + Pv1 ) = h2 − h1
Enthalpy and specific heat
•The specific heat is not defined at any phase transition
which is accompanied by a latent heat, because heat is
transferred with no change in the temperature of the
system, i.e. c = ∞.
•However, enthalpy turns out to be a useful quantity for
calculating the specific heat at constant pressure
⎛ đq ⎞
⎛ ∂h ⎞
cP ≡ ⎜
⎟ =⎜
⎟
∂
dT
T
⎝
⎠P ⎝
⎠P
•For an idea gas, it will be shown in chapter 5 that the
enthalpy depends only on the temperature of the gas T.
Therefore,
dh
cP ≡
dT
and
T
h − h0 = ∫ cP dT
T0
Configuration Work and ideal gases
W = ∫ PdV = 0
Isochoric
W = P ∫ dV = P (V f − Vi )
Isobaric
⎛Vf ⎞
dV
W = ∫ PdV = nRT ∫
= nRT ln ⎜ ⎟
Isothermal
Vi V
⎝ Vi ⎠
Note: for an ideal gas, U = U(T), so W = Q for isothermal processes.
It is also always true that, for an ideal gas,
Vf
ΔU = ncV (T f − Ti )
and
ΔH = ncP (T f − Ti )
Adiabatic processes: đQ = 0, so W = −ΔU, also Pvγ = constant.
⇒
1
W = ncV (Ti − T f ) =
PV
(
i i − Pf V f )
γ −1
⎡
3R
5R
cP 5
=
=
=
=
γ
; cP
;
⎢ Monatomic: cV
2
2
cV 3
⎣
Diatomic: cV =
5R
7R
cP 7 ⎤
=
=
= ⎥
γ
; cP
;
2
2
cV 5 ⎦