Download Engineering Chemistry Unit-5 and more Exams Engineering Chemistry in PDF only on Docsity!
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
UNIT 5
ENGINEERING MATERIALS
Phase Equilibria: Gibbs Phase Rule: Terms involved in Phase Rule – Phase Diagram of Water
System – Thermal Method of Analysis – Construction of Simple Eutectic System (Lead-Silver
System).
Fuels: Classification of Fuels – Determination of Calorific Values of Solid Fuels by Bomb
Calorimeter – Manufacture of Synthetic Petrol by Fischer-Tropsch Method – Knocking in IC
Engines: Chemical Structure – Octane and Cetane Rating of Fuels.
Nanomaterials: Size Dependent Properties of Nanomaterials – Synthesis of Gold and Silver
Nanoparticles by Chemical Reduction Method – Applications of Nanoparticles in Medicine.
5.1 INTRODUCTION TO PHASE EQUILIBRIA
1. Write mathematical statement of phase rule.
Gibb’s phase rule is mathematically stated as:
F = C ‒ P + 2
Where, P is the number of phases; C is the number of components; and F is the number
of degrees of freedom.
2. How many phases, components and degrees of freedom are available in the
following systems? (i) CaCO 3 (s) ↔ CaO (s) + CO 2 (g) (ii) (i) MgCO 3 (s) ↔ MgO (s) +
CO 2 (g) (iii) NH 4 Cl (s) ↔ NH 3 (g) + HCl (g).
(i) P = 3; C = 2; F = 2‒3 + 2 = 1
(ii) P = 3; C = 2; F = 2‒3 + 2 = 1
(iii) P = 2; C = 1; F = 1 – 2 + 2 = 1
3. Define the term ‘phase’.
A phase is a homogeneous, physically distinct, and mechanically separable part of a
system, which is separated from other parts of system by definite boundaries.
4. Define the term ‘components’.
It is defined as the minimum number of chemical constituents required to express the
composition of all the phases present in the system.
5. Define the term ‘variance or degrees of freedom’.
It is defined as the minimum number of independent variable factors such as temperature,
pressure and concentration (composition) required to describe the system completely.
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
6. What are the limitations of phase rule?
(i) The influence of factors such as electric, magnetic, gravitational, surface forces is
ignored in phase rule.
(ii) The phase rule is applicable only to heterogeneous equilibrium systems.
7. What are the advantages of phase rule?
(i) The phase rule takes no account of the nature or amount of substances.
(ii) Phase rule is applicable only to macroscopic systems. Information about molecular
structure is not required.
8. What is phase diagram? What is its significance?
Phase diagram is a graph obtained by plotting one degree of freedom against another. If
temperature is plotted against pressure, it is known as temperature-pressure diagram. If
temperature is plotted against composition, it is known as temperature–composition
diagram.
Significance: Phase diagrams help in studying and controlling the various processes such
as phase separation, solidification of metals, and change of structure during heat
treatment like annealing, quenching, tempering etc.
9. State reduced phase rule (or) condensed phase rule.
For solid-liquid alloy system, the variable factor pressure is ignored due to the absence
of gaseous phase. For such a system the phase rule equation is reduced to F = C – P + 1
and it is known as reduced phase rule.
10. State the conditions under which two substances can form a simple eutectic.
(i) The two solids should be completely miscible in liquid state and completely
immiscible in solid state.
(ii) They should not react chemically with each other.
11. Mention the differences among melting point, triple point and eutectic point.
Melting point: It is the temperature at which a pure metal starts melting.
Triple point: It is the temperature at which any 3 phases are simultaneously in
equilibrium.
Eutectic point: It is the temperature at which a mixture of two solids melts together.
12. Eutectic is a mixture and not a compound. Explain.
Since the two solids are completely miscible in liquid state and are completely
immiscible in solid state, eutectic is a mixture of two solids and not a compound.
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
(ii) Consider a one component system consisting of one phase: water vapour: P = 1;
C = 1; and F = C – P + 2 = 1 – 1 + 2 = 2; to define such a system, two variable factors
(temperature and pressure) are needed. Hence the system is bivariant.
(iii) Consider a one component system consisting of three phases: ice ↔ water ↔ water
vapour: P = 3; C = 1; and F = C – P + 2 = 3 – 1 + 2 = 0; to describe this equilibrium
system, no need to specify any variable factors, because all the three phases can
occur in equilibrium only at a particular temperature and pressure. Hence this system
does not have any degree of freedom (invariant or zero variant).
5.1.2 Phase Diagram of Water System
2. Draw and explain the phase diagram of ice-water–water vapour system.
The water system is an example of one component system. It consists of three phase’s
namely solid ice, liquid water and water vapour.
Temperature
Pressure
Solid Ice
Liquid Water
Water Vapour
O
B A
C
A'
273
o
C 0.
o C 100
o C 374
o C
4.58mm
1atm
218atm
Vapourization curve
Sublimation curve
Fusion curve
0
o
C
Salient features of phase diagram of water system:
S. No. Salient
Features
Description of the System Degrees of Freedom
1. Curves Curve OA : Vaporization curve
because it separates water and water
vapour.
The curve OA ends at the point A ,
which corresponds to the critical
temperature (
° C) and critical
pressure (218.5 atm) of water.
F = C–P+
F = 1–2+2 = 1
(Monovarient)
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
S. No. Salient
Features
Description of the System Degrees of Freedom
Curve OB : Fusion curve or melting
curve because it separates ice and
water phases.
The curve OB is slightly inclined
towards pressure axis. This shows
that melting point of ice decreases
with the application of pressure.
F = C–P+
F = 1–2+2 = 1
(Monovarient)
Curve OC : Sublimation curve
because it separates ice and water
vapour.
At the lower limit, the curve OC
terminates at absolute zero (−
° C)
where no vapour can be present and
only ice exists.
F = C–P+
F = 1–2+2 = 1
(Monovarient)
2. Triple Point The three curves O A, OB , and OC
meet at a point O , at which solid ice,
liquid water and water vapour are
simultaneously in equilibrium. This
is called triple point.
The triple point of water system
corresponds to a temperature of
°
C and a pressure of 4.58 mm
mercury.
F = C–P+
F = 1–3+2 = 0
(Zero variant)
3. Areas The curves are divided into three
areas: area AOB, area BOC and area
COA.
Each area represents a single phase.
F = C–P+
F = 1–1+2 = 2
(Bivariant)
4. Meta-stable
Equilibrium
Curve
The curve OA’ represents the vapour pressure curve of super
cooled water.
The process of cooling of water below its freezing point
without the separation of solid is known as super cooling.
The super cooled water is highly unstable. It can be converted
into solid by a slight disturbance. Hence the system along the
curve OA’ is said to be in “meta stable equilibrium”.
5.1.3 Thermal Method of Analysis
3. Explain the construction of eutectic phase diagram by thermal analysis.
Thermal analysis involves the study of cooling curve of various compositions of two
solids during solidification.
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
a
b
c
e
d
f
g
Liquid melt
Solid
A
Liquid melt
Solid
B
Liquid melt
Solid A + Solid B
Temperature
Time
100%A 100%B
Temperature
Composition
1 2 3 4 5 6 7
a
b
c
d
e
f
g
1. 100%A 2. 80%A+20%B 3. 70%A+30%B 4. 60%A+40%B 5. 45%A+55%B 6. 30%A+70%B 7. 100%B
5.1.4 Construction of Simple Eutectic System (Lead-Silver System)
4. Draw the phase diagram of lead-silver system and explain the salient features and
its application?
O
Liquid melt
Solid Ag +
Liquid melt
Solid Pb + Solid Ag
100%Pb (^) 100%Ag
Composition wt%
Temperature
A
B
Solid Pb +
Liquid melt
303
o C
961
o
C
303
o
C
327
o C
97.4%Pb
a
b
Salient features of phase diagram of lead-silver system:
S. No. Salient
Features
Description of the System Degrees of
Freedom
1. Curves Curve AO: Pure Ag melts at 961
° C.
Curve AO is the freezing point curve of silver
because addition of Pb lowers the freezing point
of Ag along the curve AO.
Along AO , solid Ag and liquid melt are in
equilibrium.
F = C–P+
F = 2–2+1 = 1
(Monovarient)
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
Curve BO: Pure Pb melts at 327°C.
Curve BO is the freezing point curve of lead
because addition of Ag lowers the freezing point
of Pb along the curve BO.
Along BO , solid Pb and liquid melt are in
equilibrium.
F = C–P+
F = 2–2+1 = 1
(Monovarient)
2. Eutectic
Point
The curve AO and BO intersect at the point O,
called the eutectic point.
At the eutectic point, three phases are in
equilibrium (Liquid melt ↔ Solid Ag + Solid
Pb).
Below the point O, both silver and lead exist in
the solid state.
Eutectic point is the lowest temperature at which
a mixture of two solids melts (eutectic = easy
melting).
The eutectic point of Pb-Ag system corresponds
to a temperature 303
° C and composition
97.4%Pb + 2.6%Ag.
F = C–P+
F = 2–3+2 = 0
(Zero variant)
3. Areas The area above AOC has a single phase (molten
Pb and Ag ).
F = C–P+
F = 2–1+1 = 2
(Bivariant)
The area below AO (solid Ag + liquid melt),
below BO (solid Pb + liquid melt) and below the
point O (solid Ag + solid Pb ) have two phases
each.
F = C–P+
F = 2–2+1 = 1
(Monovariant)
Application of Pb-Ag System (Pattison’s Process)
The process of recovery of silver from argentiferous lead is called as desilverisation.
Desilverisation of lead is based on the formation of eutectic mixture.
Argentiferous lead consisting of a very small amount of silver (0.1%).
The ore is heated to a temperature well above its melting point, so that it exists as
liquid melt (point a in Pb-Ag system).
When it is allowed to cool slowly, the temperature of the melt decreases along the
line ab.
As soon as point b is reached, lead is crystallized out.
On repeating the process of melting and cooling, more and more lead is separated
along the line BO.
At the point O, an eutectic mixture consisting of 2.6%Ag and 97.4%Pb is obtained.
The eutectic alloy is then chemically treated for the recovery of silver.
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
7. Define octane number of petrol. How can it be improved?
Octane number is defined as “the percentage of isooctane in isooctane-n-heptane mixture,
which matches same amount of knocking as the standard binary mixtures when tested in
standard engine and standard conditions”
Octane number of petrol is improved by adding additives like tetra ethyl lead (TEL).
Petrol to which TEL is added is called leaded petrol.
8. What are the advantages and disadvantages of leaded petrol?
Advantage: Leaded petrol prevents knocking in SI engine.
Disadvantage: Leaded petrol when used in SI engines produces lead bromide which
causes atmospheric pollution.
9. Why is ethylene dibromide added, when TEL is used as antiknock?
Lead particles produced by TEL may deposit in spark plug of SI engines. To prevent this
deposit, ethylene dibromide is added. It reacts with lead particles to form lead dibromide
(PbBr 2 ) and goes out with exhaust gases.
10. How does TEL improve the octane number of petrol?
TEL decomposes to form (i) lead metal particles (ii) ethyl free radicals.
Lead particles act as oxidation centres and thereby making combustion
homogeneous and uniform.
Ethyl free radicals convert straight chain particles into iso-paraffin.
Sometimes ethyl free radicals combine with the free radicals of knocking process
and terminate the chain growth.
11. Select the compound which possesses highest octane number and cetane
number out of n-heptane, n-hexadecane, n-octane and isooctane.
Isooctane possesses highest octane number while n-hexadecane possesses highest cetane
number.
12. Arrange n-octane, naphthalene, isooctane in increasing order of their knocking
tendency.
n - octane → Isooctane → Naphthalene
13. Which is having higher octane number, natural gasoline or cracked gasoline?
Cracked gasoline has higher octane number than natural gasoline. This is because
cracking produces more of isoparaffins and aromatics.
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
14. Define cetane number of diesel. How can it be improved?
Cetane number is defined as “the percentage of cetane in cetane - α-methyl naphthalene
mixture that has same ignition delay as the standard mixture tested in standard engine
and standard conditions”.
Cetane number of diesel oil can be increased by adding certain additives called dopes.
Examples: Isoamyl nitrate, Ethyl nitrate.
15. Define cetane number. Mention the cetane number of two hydrocarbons.
Cetane number is defined as “the percentage of cetane in cetane - α-methyl naphthalene
mixture that has same ignition delay as the standard mixture tested in standard engine
and standard conditions”.
CH
3
CH
2
CH
3
n-Cetane
(Cetane Number = 100 )
14
CH
3
(Cetane Number = 0)
o - Methyl naphthalene
5.2.2 Determination of Calorific Values of a Solid Fuel / Liquid Fuel
16. Explain a gross and net calorific value? How calorific value of a liquid fuel is
determined by bomb calorimeter?
Determination of Calorific Value of a Solid (or) Liquid Fuel
Principle: A known weight of the fuel is burnt completely and the quantity of heat
liberated is absorbed in water and measured.
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
5.2.3 Manufacture of Synthetic Petrol by Fischer-Tropsch Method
17. Explain the manufacture of synthetic petrol by Fischer-Tropsch method.
The raw materials used in Fischer-Tropsch process are hard coke and steam to
produce water gas. i.e.: water gas is obtained by passing steam over red hot coke.
C + H
2
O CO + H
2
Water Gas
o
C
CO + H 2
Mixture of
Hydrocarbons
Mixer
Catalyst Chamber
Gasoline
Heavy Oil
Fractionating Column
Compressor
Gasoline
Purifying
Chamber,
Fe 2
O 3
Na 2
CO 3
Cracking
The first step in this process is purification of water gas. To remove H 2 S, the gas is
passed through Fe 2 O 3 and to remove organic sulphur compounds, the gas is again
passed through a mixture of Fe 2 O 3 and Na 2 CO 3.
The purified gas is compressed to 5-25 atmospheres over a catalyst containing oxides
of Th, Co and Mg on Kieselguhr at 250
°
C. The reaction products mainly contain
straight chain paraffins and olefins.
nCO + (2n+1)H
2
C
n
H
2n+
+ nH
2
O
nCO + 2nH
2
C
n
H
2n
+ nH
2
O
n-Paraffins
Olefins
Since the reactions are exothermic, the vapours leaving the vessel are condensed in
the condenser to give petroleum. It is fractionally distilled to yield petrol and heavy
oil. Heavy oil is used for cracking to get more amount of gasoline.
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
5.2.4 Knocking in IC Engines
18. Explain the causes, mechanism and prevention of knocking in I.C. engines?
Definition of Knocking:
Knocking is “a kind of mild explosion which occurs in IC engines due to sudden
increase of pressure developed by spontaneous combustion of fuel and air mixture.
Knocking in SI engine is due to pre-ignition or pre-mature ignition of fuels.
Knocking in CI engines is due to delayed ignition or ignition lag.
Causes of Knocking:
Knocking in internal combustion engine causes mechanical damage in cylinder and
reduction of power output.
Chemical Structure and Knocking:
The knocking tendency in IC engines depends upon the compactness of molecules,
double bonds and cyclic structure. Knocking tendency in SI engines decreases in the
following order while knocking tendency in CI engines increases in the same order_._
n - Paraffins→Iso-paraffins→Olefin →Naphthenes→Aromatics
It means that the presence of maximum quantity of aromatics and minimum quantity
of n-paraffins in petrol is desirable. It also indicates that the presence of maximum
quantity of n-paraffins and minimum quantity of aromatics in diesel is desirable.
Measurement of knocking in SI engines (or) Definition of Octane Number:
The quality of petrol is expressed in terms of octane number. Octane number is
defined as “the percentage of isooctane in isooctane + n-heptane mixture, which has
the same amount of knocking as a standard binary mixture when tested in standard
engine under standard conditions”.
Leaded petrol (or) Octane number Enhancers:
The octane number of gasoline may be improved by adding about 1-3ml of ethyl
fluid (60% Tetra ethyl lead + 26% Ethylene bromide + 9% Ethylene chloride + 2%
Red dye) to 1 gallon of petrol. Petrol containing TEL is known as leaded gasoline.
Measurement of knocking in CI engines (or) Definition of Cetane Number:
The quality of diesel is expressed in terms of cetane number. Cetane number is
defined as “the percentage of cetane in cetane + -methyl naphthalene mixture
which has same ignition delay as a standard binary mixture when tested in a standard
engine under standard conditions”.
Improvement of antiknock value (or) Cetane number Enhancers:
Cetane number of diesel oil can be improved through the addition of isoamyl nitrite,
ethyl nitrate.
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
7. What are precursors are used for the synthesis of silver nanoparticles by
chemical reduction method?
In chemical reduction method, silver nitrate (AgNO 3 ) is used as starting material,
trisodium citrate (C 6 H 5 O 7 Na 3 ) is used as the reducing agent and ascorbic acid (C 6 H 8 O 6 )
is used as surfactant to prevent the nanoparticles from agglomerating.
8. What are precursors are used for the synthesis of gold nanoparticles by chemical
reduction method?
In the sodium borohydride reduction reduction method, NaBH 4 is used as reduction
agent, the trisodium citrate acts as a stabilizing agent and tetrachloroauric acid (HAuCl 4 )
as the precursor.
9. Write any two applications of nanomaterials in medicine.
(i) Drug Delivery: CNTs can be widely used as drug carriers for drug delivery, as they
can easily adapt themselves and enter the nuclei of the cell. The most important use
of CNT in drug delivery is that they can be equipped with miniaturized video
cameras, which help in colonoscopy and endoscopic procedures.
(ii) Cancer Therapy: Silver nanoparticles can be used as a promising candidate in
chemotherapy, photosensitizers and/or radiosensitizers, bio diagnostics, bioimaging,
transfection vectors, and antiviral agents.
10. What are the characteristic properties of nanoparticles?
As the particle size decreases, surface to volume ratio increases. This enhances
catalytic activity of Nanoparticles.
Size reduction also influences thermal properties like melting point. Melting point
decreases with the particle size reduction.
11. Mention the various properties which get altered by size reduction.
The various properties, which get tremendously altered due to the size reduction are:
Chemical properties: Reactivity; Catalysis.
Thermal property: Melting point temperature.
Electronic properties: Electrical conduction.
Optical properties: Absorption and scattering of light.
5.3.2 Size Dependent Properties of Nanomaterials
12. Discuss the size dependent properties of nanomaterials. 1. Chemical Properties
It has been observed that the surface area to volume ratio increases with decrease of
particle size diameter. Therefore, size reduction in nanomaterials leads to more number
of surface atoms with enhanced surface area. This effect leads more reactivity and
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
catalytic activity of materials in nanoform in turn increases the reaction rate and
efficiency of various chemical reactions.
Examples: Au and Ag in bulk form are chemically inert. But they become good catalyst
in nanoform.
Particle diameter, nm
Reaction rate , mole m
day
(^0 10 ) 30 40 50
1x
7.5x
5x
2.5x
1x
2. Thermal Properties
In bulk materials all the atoms are compactly packed and eletrovalently bonded and hence
they exhibit higher melting points.
Radius of particle size, nm
Temperature, K
(^0 10 2030 )
1350
1325
1300
1275
1250
Bulk
In Nanomaterials, surface atoms are not bonded in direction normal to surface plane and
hence surface atoms will have more freedom to move. Hence the melting point of
Nanomaterials is size dependent and it decreases with decrease of particle size diameter.
Examples: Bi (in bulk form: 660K): Bi (6nm: 500K); Bi (4nm: 480K) and Bi (2nm:
400K).
3. Electronic Properties:
In bulk materials, conduction of electron is delocalized (i.e: the electrons can move freely
in all directions). When the scale is reduced to nanoscale, quantum confinement
dominates. For 0-D nanomaterials, all the dimensions are at nanoscale and hence
electrons are confined in 3-D space. Therefore, no delocalization occurs. For 1-D
nanomaterials, electron confinement occurs in 2-D space and electron delocalization
occurs along the axis of Nanowires / Nanotubes / Nanorods.
Examples: Si and Ge are well known semiconductors in single crystal bulk form. But in
nanoform, due to electron confinement, they do not show semiconducting properties.
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
5.3.4 Synthesis of Silver Nanoparticles
14. Discuss the synthesis of silver nanoparticles by chemical reduction method.
In chemical reduction method, silver nitrate (AgNO 3 ) is used as starting material,
trisodium citrate (C 6 H 5 O 7 Na 3 ) is used as the reducing agent and ascorbic acid
(C 6 H 8 O 6 ) is used as surfactant to prevent the nanoparticles from agglomerating.
During this process, the solution becomes greenish in colour and gradually changes
into light yellow colour indicates the growth of the Ag nanoparticles.
4Ag
**+
- C 6 H 5 O 7 Na 3 + 2H 2 O → 4Ag**
**o
**+
**+
5.3.5 Applications of Nanomaterials in Medicine
16. Write a brief account on the applications of nanomaterials in medicine.
(iii) Drug Delivery: CNTs can be widely used as drug carriers for drug delivery, as they
can easily adapt themselves and enter the nuclei of the cell. The most important use
of CNT in drug delivery is that they can be equipped with miniaturized video
cameras, which help in colonscopic and endoscopic procedures.
(iv) Cancer Diagnosis and Cancer Therapy: Silver nanoparticles can be used as a
promising candidate in chemotherapy, photosensitizers and/or radiosensitizers,
biodiagnostics, bioimaging, transfection vectors, and antiviral agents.
Gold nanoparticles have become an interesting research area in cancer diagnosis,
imaging and especially treatment. The gold nanoparticles that are located at a site of
cancer cells can be irradiated with infrared to heat them up and destroy the nearby
cancer cells.
(v) Tissue Engineering Applications: Nanocomposite of graphene, carbon nanotubes,
molybdenum disulfide and tungsten disulfide is used as reinforcing agents to
fabricate mechanically strong biodegradable polymeric nanocomposites for bone
tissue engineering applications.
(vi) Dental Diagnostics: The iron nanoparticles conjugated with polysaccarides and
gold nanoparticles conjugated with peptides are used for specific cellular biomarkers
in dental diagnostic. They are also used as optical/magnetical contrasting agents in
medical imaging techniques for early detection of oral cancer, and for identifying
and differentiating infectious pathogens.
(vii) Cell labelling and Bio-imaging: Fluorescent CdSe/ZnS quantum dots could
produce a higher contrast image, when used in combination with MRI (magnetic
resonance imaging) and hence they are used for tumor targeting, tumor angiogenesis
imaging, and metastasis cell tracking.
SCYA1101: Engineering Chemistry UNIT 5 : Engineering Materials
5.4 PROBLEMS BASED ON PHASE RULE
1. How many number of phases, components and degrees of freedom are available
in the following systems:
(i) BaCO 3 (s) BaO (s) + CO 2 (g)
(ii) MgCO 3 (s) MgO (s) + CO 2 (g)
(iii) CaCO 3 (s) CaO (s) + CO 2 (g)
All the above systems have 2 solid phases and one gaseous phase.
Number of phases, P = 3
Number of components, C = 2
Number of degrees of freedom, F = C – P + 2
F = 2 – 3 + 2
F = 1(univariant)
2. How many phases, components, and degrees of freedom are present in:
(i) Water at 0.
° C and 4.58mm Hg
(ii) Water) Water vapour at 30
° C
(iii) Pb–Ag alloy system at constant pressure [composition 2.6%Ag and
303
° C]
(i) This indicates triple point of water system,. At the triple point, solid ice, liquid water
and water vapour are simultaneously in equilibrium.
Number of phases, P = 3
Number of components, C = 1
Number of degrees of freedom, F = C – P + 2
F = 1 – 3 + 2
F = 0 (invariant)
(ii) Water ↔ Water vapour at 30
°
C.
Number of phases, P = 2
Number of components, C = 1
Number of degrees of freedom, F = C – P + 2
F = 1 – 2 + 2
F = 1(univariant)
(iii)This indicates eutectic point of Pb-Ag system. At the eutectic point, solid Ag, solid
Pb and liquid melt are in equilibrium.
Number of phases, P = 3
Number of components, C = 2
Number of degrees of freedom, F = C – P + 1
F = 2 – 3 + 1
F = 0 (invariant)
