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Chemical Thermodynamics
2013/2014
3rd Lecture: Work, Heat and the First Law of Thermodynamics
Valentim M B Nunes, UD de Engenharia
Introduction
As we saw before, thermodynamics it’s a science that studies energy
transformations but, as we will see, thermodynamics describes
macroscopic properties of equilibrium systems.
Although everybody as the feeling of knowing what
is energy, it is very difficult to give a precise
definition. For our purposes Energy can be defined
as the ability to cause changes or realize Work.
One of the fundamental laws of nature is the law
of conservation of energy. Energy in a system may
take on various forms (e.g. kinetic, potential, heat,
light) but energy may neither be created nor
destroyed.
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Some basic concepts
Thermodynamic System – it’s a part of the Universe that is being
studied.
Exterior or Surroundings of the system – all the rest of the Universe
Boundary of the system – its what divides the system from the rest
of the universe.
Surroundings
System
Universe = System + Surroundings
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Types of systems
Isolated System – An isolated system is that in which no transfer of
mass & energy takes place across the boundaries of system
Closed System - A closed system in which no transfer of mass takes
place across the boundaries of system but energy transfer is possible
Open System - An open system is one in which both mass & energy
transfer takes place across the boundaries
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Describing Systems
To describe a given system we need to indicate the components of
the system, their physical state (gas, liquid, solid, mixtures) and the
state properties, like pressure, p, volume, V, number of moles, n,
mass, m, and temperature, T.
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Properties of a system
When one system suffers a transformation it goes from an initial state
to a final state. The properties of the system that are univocally
determined by the sate of the system are state functions (or state
variables or state properties).
These properties may be either intensive or extensive. Extensive
properties depends on the size or extension of the system, like the
volume, V. Intensive properties are independent of the size of the
system, like temperature or pressure.
If we divide an extensive property by the number of moles we obtain
an intensive property, like the molar volume:
V
Vm 
n
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State of a System at Equilibrium
The state of equilibrium is defined by the macroscopic properties and
is independent of the history of the system.
Cooling
Heating
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Change of state
A process or transformation is a change in the state of the system over
time, starting with a definite initial state and ending with a definite
final state.
The process is defined by a path, which is the continuous sequence of
consecutive states through which the system passes, including the
initial state, the intermediate states, and the final state
There are many types of processes to change the state of a system at constant volume (isochoric), at constant pressure (isobaric), at
constant temperature (isotherm) and so one..
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Infinitesimal Changes
An infinitesimal change of the state function X is written dX. The
mathematical operation of summing an infinite number of infinitesimal
changes is integration, and the sum is an integral. The sum of the
infinitesimal changes of X along a path is a definite integral equal to •X:
X2
 dX  X
2
 X 1  X
X1
If dX obeys this relation—that is, if its integral for given limits has the
same value regardless of the path—it is called an exact differential. The
differential of a state function is always an exact differential
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Cycles
A cyclic process is a process in which the state of the system changes
and then returns to the initial state. In this case the integral of dX is
written with a cyclic integral.
Since a state function X has the same initial and final values in a cyclic
process, X2 is equal to X1 and the cyclic integral of dX is zero:
 dX  0
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Internal energy
The total energy of a system is the Internal Energy, U. The internal
energy is a state function. If a system as an initial energy Ui and
after a transformation as a n energy Uf then the variation of internal
energy, U is:
U  U f  U i
The internal energy is an extensive property, that is, it depends on
the size of the system. It can only be changed by two different
modes: Work, W, and Heat, Q, trough the boundary of the system.
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Work and Heat
Heat can be viewed as a disordered way of transferring energy
(caused by temperature gradient across the boundary) while work is an
order way of transferring energy (lifting a weight for instance)
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The 1st Law of Thermodynamics
The internal energy of an isolated system is constant. If the system is
closed it can only be transferred by heat flow or work done.
In differential form
dU  dQ  dW
In integrated form:
U  Q  W
dQ and dW are not exact differentials what means that they will
depend on the path! So heat and work are path functions, they are
associated with a process, not a state.
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The 1st Law of Thermodynamics
An equivalent formulation of the first law is the following: the work
necessary to change an adiabatic system from one state to another is
always the same, no matter the type of work done.
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The 1st Law of Thermodynamics
But… AU is the same
for all the processes!
f
 dU  U
f
 U i  U
i
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Signal convention
Process
System does work on the surroundings
Surroundings do work to the system
Heat absorbed by the system (endothermic process)
Heat absorbed by the surroundings (exothermic process)
Signal
+
+
-
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Work
p
Force
 Force  pext  A
Area
Work  Force  Dist ance
dW   pext  A  dx   pext dV
dW   pext dV
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Expansion work
Let us assume the work done by the expansion of a gas against
constant external pressure:
W    pext dV   pext  dV   pext V f  Vi 
Vf
Vf
Vi
Vi
W   pext V
In a free expansion, against the vacuum, the external pressure is
null (pext = 0), so W=0.
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Isothermal Perfect gas expansion (1 step)
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Isothermal Perfect gas expansion (two steps)
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Isothermal Perfect gas expansion (Infinite steps)
If at each step we have p = pext, we have infinite expansions, and
maximum work is delivered to the surroundings! This is obtained using a
reversible path.
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Reversible Process
Considering the reversible expansion of a perfect gas, he have:
Vf
Wmax, rev    pdV
Vi
Vf
Wmax, rev   
Vi
nRT
dV
V
Wmax, rev  nRT ln
Vf
Vi
R = 8.314 J.K-1.mol-1
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Summary
Process
If Pext = 0
If Pext = constant
If the expansion is reversible
W 0
W   pext V
W  nRT ln
Vf
Vi
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Reversible vs Irreversible
(T, p1, V1)
expansion
(T,p2,V2)
compression
(T,p1,V1)
Process I: expansion against pext = p2 and compression with pext = p1
Wcycle  Wexp  Wcomp   p2 V2  V1   p1 V2  V1 
Wcycle   p1  p2 V2  V1  0
Work done to System!
 dW  0
Process II: infinite expansions and compressions with pext = p along
the path
Wcycle  nRT ln
V2
V
 nRT ln 1  0
V1
V2
Reversible Process!
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Heat
It’s the quantity flowing between the system and the surroundings that
can cause a change in temperature of the system and/or the
surroundings. Like work, heat its not a state function!
What connects Heat with temperature it’s the Heat capacity, C.
Units SI are J.K-1.mol-1.
dq  CdT
At constant volume:
At constant pressure:
 q 
CV  

 T V
 q 
Cp  

 T  p
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Heat Capacity at constant volume
At constant volume, dw = 0 so, from the 1st Law, we can easily
obtain:
 U 
CV  

 T V
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Enthalpy, H
Chemical reactions and many other processes, including biological,
take place under constant pressure and reversible pV work. Let us
define a new function of state, the enthalpy, defined as:
H  U  pV
Differentiating:
dH  dU  pdV  Vdp
dH  dq  dw  pdV  Vdp
dH  dq  pdV  pdV  Vdp
dH  dq  Vdp
At constant pressure:
dH  dq
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Heat Capacity at constant pressure
Previous result shows that the enthalpy its equal to the heat in a
constant pressure process, and we can finally obtain:
 H 
Cp  

 T  p
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Relation between Heat Capacity’s for an ideal gas
We can now derive a relation between Cp and Cv for an ideal gas:
 H   U 
C p  CV  
 

 T  p  T V
But, H = U + pV =U + RT (per mole)
Assume this
is equal!
 U 
 U 
C p  CV  

R





T

T

p

p
C p  CV  R
For instance, for an ideal monatomic gas CV =3/2 R , so Cp = 5/2 R
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The Joule Experiment
 U 

 V T
Let us consider the free expansion of a gas, to get  T  
Adiabatic, q = 0
Expansion into the vacuum, w = 0
P~0
U  0
The experiment proofs that, for an ideal gas, U = U(T)!
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Total differential of U
If we regard the internal energy function as U =U(V,T) , then the
total differential of U comes:
 U 
 U 
dU  
dV



 dT
 V T
 T V
For an ideal gas
 U 
T  
  0 so dU = CvdT
 V T
This means that the internal energy of an ideal gas depends only on
temperature. As a consequence: ΔU = 0 for all isothermal expansions
or compressions of an ideal gas, and
U   CV dT
For any ideal gas change of state.
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