Chemical thermodynamics

 1.1. The equation of state
The equation of state is the function that relates the pressure to the density,
molecular weight, and temperature at any place in the universe.
It is the relation between the thermodynamic properties of a substance at
equilibrium.
It is a solely an internal property of the gas.
Because pressure is caused by the collision of molecules, we might expect
the pressure P to be greater where there are more molecules (i.e., greater
density ρ), and where they are moving faster (i.e., greater temperature T).
P =f(ρ, 𝝁,T)
For dry air the gases in the atmosphere have a simple equation of state
known as the ideal gas law.
 1.2. Ideal Gases and Ideal Gases Law
What makes an ideal gas, ideal?
1. The volume of the molecules is negligible << the empty space
2. The molecules don’t interact with each other – there is no attraction or
repulsion between them.12/17/2018 2
Chapter One- Introduction

An ideal gas is a hypothetical sample of gas whose
pressure-volume-temperature behavior is predicted
accurately by the ideal gas equation.
A gas is composed of small hard particles.
The particles have an insignificant volume and are
relatively far apart from one another.
There is empty space between particles.
No attractive or repulsive forces between particles.
The particles in a gas move in constant random motion.
Particles move in straight paths and are completely
independent of each of other
Particles path is only changed by colliding with another
particle or the sides of its container.12/17/2018 3
Ideal Gases cont…

All collisions a gas particle undergoes are
perfectly elastic.
No energy is lost from one particle to another,
and the total kinetic energy remains constant.
• Gases seem to be weightless, but they are
classified as matter, which means they have mass.
• The density of a gas – the mass per unit of volume
– is much less than the density of a liquid or solid,
however.
It’s this very low density that allows us to be able
to walk through the room without concerning
ourselves with air resistance.
Since it is so easy to “swim” across the room we
don’t put much thought into the mass of a gas.
Really it is only noticeable if we have a large
collection of gas in a container.
12/17/2018 4
Ideal Gases cont…

Gas particles have a high velocity,
relative to their masses.
This gives them a lot of energy and
movement.
The movement causes the gases to
spread out, which leaves a lot of
space between molecules.
That empty space can be compressed
by pressure allowing gas particles less
room to move around thus
decreasing the volume.
12/17/2018 5
Ideal Gases con’t…

The ideal gas law is a combination of the combined gas
law and Avogadro’s Law.
 V∝
𝑻
𝑷
and V ∝ n
 ∴ V = constant x
𝒏𝑻
𝑷
The absolute pressure P of an ideal gas is directly
proportional to the Kelvin temperature T and the
number of moles n of the gas and is inversely
proportional to the volume V of the gas: P = R(nT/V). In
other words,
 PV = nRT 1.1
where R is the universal gas constant and has the value of
8.31 J/(mol·K).12/17/2018 6
Ideal Gases Law

Real molecules do take up space and do interact
with each other (especially polar molecules).
Need to add correction factors to the ideal gas
law to account for these.
Differences Between Ideal and Real Gases
Ideal Gases Real Gases
12/17/2018 7
1.3. Real Gas Laws
Obey PV=nRT Always Only at very
high P and low
T
Molecular volume Zero Small but
nonzero
Molecular attractions Zero Small
Molecular repulsions Zero Small

Volume Correction
 The actual volume free to move in is less because of particle size.
 More molecules will have more effect.
 Corrected volume V’ = V – nb
 “b” is a constant that differs for each gas.
 Pressure Correction
 Because the molecules are attracted to each other, the pressure on
the container will be less than ideal.
 Pressure depends on the number of molecules per liter.
 Since two molecules interact, the effect must be squared.
1.2
12/17/2018 8
Real Gases Law cont…

Van der Waal’s equation
 𝑃 − 𝑎
𝑛
𝑉
2
𝑉 − 𝑛𝑏 = 𝑛𝑅𝑇 1.3

 Corrected Pressure Corrected Volume
“a” and “b” are determined by experiment
 “a” and “b” are different for each gas bigger molecules have
larger “b”
 “a” depends on both size and polarity
12/17/2018 9
Real Gases Law

2.1 Basic SI Units
 The base units of the International System are listed in Table 1,
which relates the base quantity to the unit name and unit symbol
for each of the seven base units.
12/17/2018 10
Chapter Two. Units and Mathematics

Derived units are products or quotients of base
units.
Table 2. Examples of derived units in terms of
base units.
12/17/2018 11
2.2. Derived Units

The expression 𝒂 𝟐
means 𝒂 × 𝒂, 𝒂−𝟐
means
𝟏
𝒂 𝟐, 𝒂 𝟑
means 𝒂 𝒙 𝒂 𝒙 𝒂 , and
so on.
In addition, you can have exponents that are not integers. If we write
 y = 𝒂 𝒙 (1.4)
the exponent x is called the logarithm of y to the base 𝒂 and is denoted by
 𝒙 = 𝒍𝒐𝒈𝒂 𝒚 (1.5)
If 𝒂 is positive, only positive numbers possess real logarithms.
12/17/2018 12
2.3. Logarithms And Exponents

If the base of logarithms equals 10, the logarithms are called common
logarithms:
If 𝟏𝟎 𝒙
= y, then x is the common logarithm of y, denoted by log10(y). The
subscript 10 is sometimes omitted, but this can cause confusion.
For integral values of x, it is easy to generate the following short table of
common logarithms:
Tabe 1.3. Common logarithms
12/17/2018 13
Logarithms And Exponents con’t…

Natural Logarithms
Besides 10, there is another commonly used base of logarithms. This
is a transcendental irrational number called e and equal to 2.7182818
. . .
Logarithms to this base are called natural logarithms. The definition
of e is
 Logarithm Identities
There are a number of identities involving logarithms, some of
which come from the exponent identities. Table 1.4 lists some
identities involving exponents and logarithms. These identities
hold for common logarithms and natural logarithms as well for
logarithms to any other base.
12/17/2018 14

 Table 1.4. Logarithm Identities

12/17/2018 15

Thermodynamics: deals with the interconversion of various
kinds of energy and the changes in physical properties
involved.
It is also concerned with equilibrium states of matter and has
nothing to do with time.
3.1 Thermodynamic Terms
Terminology of Thermodynamics:
System is the part of the world, in which we have special
interest.
Surroundings is the region of the universe that we make our
observations or the part of the universe outside the boundary
of the system is referred.
12/17/2018 16
Chapter –Three. Thermodynamics
Universe
System
Open
Closed
IsolatedSurroundings

The type of system depends on the characteristics of the boundary which divides
it from the surroundings:
 (a) An open system can exchange matter and energy with its surroundings.
 Example: An ocean.
 (b) A closed system can exchange energy with its surroundings, but it cannot
exchange matter.
 A greenhouse is an example of a closed system exchanging heat but not work
with its environment. Whether a system exchanges heat, work or both is usually
thought of as a property of its boundary.
 (c) An isolated system can exchange neither energy nor matter with its
surroundings. Example: a completely insulated gas cylinder.
Except for the open system, which has no walls at all, the walls in the two other
have certain characteristics, and are given special names: 17
3.1. Thermodynamic Terms…

 A diathermic (closed) system is one that allows energy to escape as heat
through its boundary if there is a difference in temperature between the
system and its surroundings. It has diathermic walls.
 An adiabatic (isolated) system is one that does not permit the passage of
energy as heat through its boundary even if there is a temperature
difference between the system and its surroundings. It has adiabatic
walls.
12/17/2018 18

Homogeneous system:
The macroscopic properties are identical in all parts of the system. It consists a
single phase.
Example: liquid water
Heterogeneous System:
It contains more than one phase.
Example: Liquid water in equilibrium with water.
3.1.2. State of a system.
It is defined as physical variables (T, P, V, n) which are used to define the system
completely.
Only 3 of them need to be specified, together with the equation of state in order to
describe the system completely.
3.1.3 Properties of a System
Extensive and intensive properties
An extensive property is a property that depends on the amount of substance in
the sample.
Examples: mass, volume…12/17/2018 19
Thermodynamic Terms…

 An intensive property is a property that is
independent on the amount of substance in
the sample.
Examples: temperature, pressure, mass
density…
 A molar property Xm is the value of an extensive property X
divided by the amount of substance, n:Xm=X/n. A molar
property is intensive. It is usually denoted by the index m, or
by the use of small letters.
 The one exemption of this notation is the molar mass, which is
denoted simply M.
 • A specific property Xs is the value of an extensive property X
divided by the mass m of the substance: Xs=X/m. A specific
property is intensive, and usually denoted by the index s.
12/17/2018 20
3.1.3. Properties of A System

Thermodynamical Equilibrium: A system is said to have
attained a state of thermodynamical eqm, When it shows no
further tendency to change its property with time. For
example, if the pressure was not uniform, turbulence would
occur until the system reaches equilibrium.
1-Thermal equilibrium: ~ no flow of heat occurs from one
system to the other.
If a system and all its internal parts A,B,C satisfy that TA = TB
= TC = ... it is said that it is in thermal equilibrium.
2-Mechanical equilibrium: ~ uniformity of pressure through
out the system.
When all the forces in the system balance we say that the
system is in mechanical equilibrium.
3. Chemical Equilibrium: ~ no change of composition of any
part of the system with time.12/17/2018 21
3.1.4 Thermodynamic Equilibrium, Zeroth's Law
of Thermodynamics

12/17/2018 22
Zeroth’s Law
Thermodynamics If A is in thermal
equilibrium with B, and B is
in thermal equilibrium with
C, then C is also in thermal
equilibrium with A. All
these systems have a
common property: the same
temperature. It forms the
basis of concept of
temperatures. This
experimental finding is
extremely simple, and as far
as we can tell, it is always
true. We call this general
law the “Zeroth law of
thermodynamics.”
 Thermal Equilibrium

Energy flows as heat from a region at a higher temperature to
one at a lower temperature if the two are in contact through a
diathermic wall, as in (a) and (c). However, if the two regions
have identical temperatures, there is no net transfer of energy
as heat even though the two regions are separated by a
diathermic wall (b).
3.1.5 Thermodynamical Process
It deals with changes that occur in the properties of system
when the system goes from one equilibrium state to another.
It is the path or operation by w/c a system changes from one
state to another.
A) Adiabatic process: no exchange of heat takes place b/n the
system and its surroundings. i.e., thermally insulated from the
surroundings. (no constant variable).
12/17/2018 23
Zeroth’s Law Thermodynamics cont…

B) Isothermal Process (T remains constant)
~𝑩𝒐𝒚𝒍𝒆′
𝒔 𝑳𝒂𝒘~𝑷𝑽= Constant
C) Isobaric process (P remains constant) ~𝑪𝒉𝒂𝒓𝒍𝒆′
𝒔 𝑳𝒂𝒘~
𝑽
𝑻
=
Constant
Example, 2H2(g) + O2(g) → 𝟐𝐇 𝟐 𝐎(𝓵) in a cylinder w/c is fitted
with a weightless, frictionless, and airtight piston.
D) Isochoric process ( V remains
constant)~𝑮𝒂𝒚 − 𝑳𝒖𝒔𝒔𝒂𝒄′ 𝒔 𝑳𝒂𝒘~
𝑷
𝑻
= Constant
Example, N 𝟐 𝐎 𝟒(𝐠) → 𝟐𝐍𝐎 𝟐(𝒈) in a cylinder w/c is fitted with a
weightless, frictionless, and airtight piston.
E) Cycle process: It is the process that occurs when a system
undergoes a series of a state changes in a such way that the
final state becomes identical with the initial state.
12/17/2018 24
3.1.5 Thermodynamical Process cont…

 Use ideal gas assumption (closed system):
12/17/2018 25
3.1.5. Thermodynamical Process cont…
2
1
2 1
1 2
Isothermal process: T=constant
Energy balance U=Q-W, for ideal gas U= H=0
since both are functions of temperature only
Q=W, W= P
ln ln
Isobaric process:
mRT dV
dV dV mRT
V V
V P
mRT mRT
V P
  
 
   
    
   
  


2
2 1
1
2 1 2 1 2 1
2 2 1 1 2 1
P=constant
U=Q-W, W= PdV=P dV=P(V )
( ) ( ) ( )
( ) ( )
V
Q U P V V U U P V V
U PV U PV H H H
 
       
       
 

v
v
Constant volume process: V=constant
Q-W= U, W= 0, no work done
Q= U=m u=m c
Adiabatic process: Q=0
Q-W= U, -W= U
- W=dU (infinitesimal increment of work and energy)
dU+PdV=0, mc 0
v
PdV
dT
mRT
dT dV
V
c

 
 
 
 
  
 




v
2 2 2 1 1
1 1 1 2 2
1
0, , integrate and assume
c =constant
ln ln ,
v
v
v
R kc
cRT dT dV
dT dV
V R T V
c T V T V V
R T V T V V

 
    
 
       
          
       
12/17/2018 26
3.1.5. Thermodynamical Process cont...

12/17/2018 27
3.1.5. Thermodynamical Process cont...

State functions are properties that are determined by the state of
the system, regardless of how that condition was achieved.
A State Function is a function in which the value only depends on the
initial and final state….NOT on the pathway taken.
The changes in the values of state functions quantities do not depend
on how the change is carried out
Thermodynamic State Functions: Thermodynamic properties
that are dependent on the state of the system only. (Example:
E, H, S, G and so on. )
Path Functions.
Other variables will be dependent on pathway
(Example: q and w). These are NOT state functions.
The pathway from one state to the other must be
defined.
12/17/2018 28
3.1.6. State Functions

State functions values are completely determined by any two of
the thermodynamic variables and can be written as a function of
these variables.
For example, if we take the enthalpy (H), it is possible to write it
as a function of any two variables as:
H = f1(P,T) = f2(V,T) = f3(P,V) = f4(P,n) 3.1
If we take any one of the expressions H = f1(P,T), any change in
H resulting from changes in the values of pressure and
temperature is given by
∆H = Hf-Hi 3.2
For an infinitesimal change, dH with a change in both variables
can be written as
dH =
𝜕𝐻
𝜕𝑃 𝑇
dP +
𝜕𝐻
𝜕𝑇 𝑃
dT 3.312/17/2018 29
Mathematical Techniques
Interconnecting the State Functions

Thermodynamic properties are continuous point functions and
have exact differentials. A property of a single component
system may be written as general mathematical function z =
z(x,y). For instance, this function may be the pressure P = f(T,V).
The total differential of z is written as
3.4
Taking the partial derivative of M with respect to y and of N
with respect to x yields
3.5
12/17/2018 30

3.6
Since properties are continuous point functions
and have exact differentials, the following is
true
3.7
Eq. 3.7 is called the Euler reciprocal relation.
This is applicable to state functions only.
If dz is an exact deferential, the cyclic integral
of dz is equal to zero.
𝑑𝑧 = 0 3.8
Example 1. Verify whether dz = (51𝒙 𝟐 𝒚 + 47 𝒚 𝟒) + (17 𝒙 𝟑 +
𝟏𝟖𝟖𝒙𝒚 𝟑
) is an exact differential or not.
12/17/2018 31

The differential dH of a function H as defined by Eq.1.8 is called exact
differential w/c satisfy the Euler’s theorem of exactness as given below
in Eq.(3.9).
𝜕2 𝐻
𝜕𝑃𝜕𝑇
=
𝜕2 𝐻
𝜕𝑇𝜕𝑃
3.9
Example 2. Writing V as a function of T and P, show that for an ideal
gas, dv is an exact differential.
Solution: V = f(T, P)
dV =
𝜕𝑉
𝜕𝑇 𝑝
𝑑𝑇+
𝜕𝑉
𝜕𝑃 𝑇
𝑑𝑃 i
For an ideal gas, PV = RT so that V =
𝑅𝑇
𝑃
𝜕𝑉
𝜕𝑇 𝑝
=
𝑅
𝑃
and
𝜕𝑉
𝜕𝑃 𝑇
= -
𝑅𝑇
𝑃2 ii
Subustiting Eq. (ii) in Eq. (i), we get
dV =
𝑅
𝑃
dT -
𝑅𝑇
𝑃2 dP
According to the Euler’s rule, dV would be an exact differential if
𝜕
𝑅
𝑃
𝜕𝑃 𝑇
= -
𝜕
𝑅𝑇
𝑃2
𝜕𝑇 𝑃
-
𝑅
𝑃2 = -
𝑅
𝑃2
12/17/2018 32

Exercise 1. Show that for an ideal gas, the work differential 𝛿w is not an
exact function.
The Cyclic Rule. From Eq 3.4 we see that if the change occurs at constant z,
then dz = 0 so that
0 =
𝜕𝑧
𝜕𝑥 𝑦
𝑑𝑥+
𝜕𝑧
𝜕𝑦 𝑥
𝑑𝑦
Hence,
𝜕𝑥
𝜕𝑦 𝑧
= -
𝜕𝑧
𝜕𝑦 𝑥
𝜕𝑧
𝜕𝑥 𝑦
or
𝜕𝑧
𝜕𝑥 𝑦
𝜕𝑥
𝜕𝑦 𝑧
𝜕𝑦
𝜕𝑧 𝑥
= -1 3.10
Eq.3.10 is called the cyclic rule. This is applicable only in the case of state
functions.
Example 3. Verify the cyclic rule for one mole of an ideal gas.
PV = RT
Differentiating this equation, we have
PdV + VdP = RdT i
Again, at constant T, dT = 0 so that
𝜕𝑃
𝜕𝑉 𝑇
= -
𝑃
𝑉
ii
12/17/2018 33

At constant P, dp = 0 so that
𝝏𝑽
𝝏𝑻 𝑷
=
𝑹
𝑷
At constant V, dV = 0 so that
𝝏𝑻
𝝏𝑷 𝑽
=
𝑽
𝑹
Hence, from Eqs.(iii), (iv) and (v)
𝝏𝑷
𝝏𝑽 𝑻
𝝏𝑽
𝝏𝑻 𝑷
𝝏𝑻
𝝏𝑷 𝑽
= -
𝑷
𝑽
𝑹
𝑷
𝑽
𝑹
= -1 which obeys the cyclic
rule.
The Volume Expansivity () and The Isothermal
Compressibility ( )
𝜷 =
𝟏
𝑽
𝝏𝑽
𝝏𝑻 𝑷
and 𝜶 = −
𝟏
𝑽
𝝏𝑽
𝝏𝑷 𝑻
3.11
Exercise 2. Using the cyclic rule, show that the thermal coefficients 𝜶
and 𝜷 are related as 𝜶 = 𝜷𝜸𝑷 where 𝜸 is the pressure coefficient.
Exercise 3. Show that
𝝏𝑯
𝝏𝑷 𝑻
= 0 for ideal gas12/17/2018 34

The fundamental physical property in thermodynamics is work:
work is done when an object is moved against an opposing force.
Or which is the energy that is transferred from one body to another
due to forces that act between them.
(Examples: change of the height of a weight, expansion of a gas that
pushes a piston and raises the weight, or a chemical reaction which
e.g. drives an electrical current).
Heat refers to the energy that is transferred from one body or
location due to a difference in temperature. For example, a heater is
immersed in a beaker with water (the system), the capacity of the
water to do work increases because hot water can be used to do
more work than cold water.
Heat: This mode of transference of energy occurs if there is a
difference of temperature b/n system and surroundings.
Work: This mode of transference of energy is possible if the system
involves gaseous substances and there a difference of pressure b/n
system and surroundings.
12/17/2018 35
3.1.8 Heat and Work

Heat is exchanged if only internal parameters are changed
during the process.
Walls that permit the passage of energy as heat are called
diathermic walls.
Walls that do not permit the passage of energy as heat even
though there is a temperature difference are called adiabatic
walls.
A process that releases energy as heat is called exothermic, a
process that absorbs energy as heat is called endothermic.
Examples: All combustions are exothermic reactions.
C6H12O6(s)+ 6O2(g) ⟶ 6 CO2(g) + 6 H2O(l)
Dissociation of ammonium nitrate in water is the
endothermic process.
12/17/2018 36
3.1.8 Heat and Work cont…

Convention
Work, w, is positive (w > 0) if work is done on the system.
Work is negative (w < 0) if work is done by the system.
Heat, q, is positive (q > 0) if heat is absorbed by the system.
Heat is negative (q < 0) if heat is released from the system.
Examples: Consider a chemical reaction that is a net producer
gas, such as the reaction b/n urea, 𝑵𝑯 𝟐 𝟐 𝑪𝑶, and oxygen to
yield carbon dioxide, water and nitrogen:
𝑵𝑯 𝟐 𝟐 𝑪𝑶(𝒔) +
𝟑
𝟐
O2 (g) ⟶ CO2(g) +2H2O(l) + N2(g)
Suppose first that the reaction takes place inside a cylinder
fitted with a piston, then the gas produced drives out the
piston and raises a weight in the surroundings.
12/17/2018 37
3.1.8 Heat and Work cont…

It is commonly known as the Law of Conservation of Energy:
The total energy of the universe (Internal energy) is constant.
The internal energy is the total energy contained in a
thermodynamic system. It is the energy necessary to create the
system, but excludes the energy associated with a move as a whole,
or due to external force fields. Internal energy has two major
components, kinetic energy and potential energy.
Energy cannot be created or can not be destroyed.
Energy can be converted from one form to another or transferred
from a system to the surroundings or vice versa
Energy can be in two forms
a) Energy in transit form like heat and work transfer w/c are
observed at the boundaries of the system.
b) Energy in Storages
i) macroscopically – by virtue of motion, position, or configuration
of a system.
12/17/2018 38
3.2 First Law of
Thermodynamics

Chemical thermodynamics

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