7 equilibrium 2021.docx

1
7 Equilibrium
Chapter 7: Equilibrium
Student workbook
Name: ________________________________________________

2
7 Equilibrium
Table of Contents
7.1 Equilibrium............................................................................................................................... 3
Our classroom equilibrium........................................................................................................... 5
Equilibrium characteristics(link: page 8)................................................................................... 5
Quantifying the classroomequilibrium (link: page 10−11) .......................................................... 6
Is the classroom at equilibrium? (link: page 12−14)....................................................................7
Changes to a classroom at equilibrium: concentration (link: page 15+) .......................................8
Characteristics of a dynamic equilibrium ...................................................................................... 9
Relationship between equilibrium constants (Kc) for the same reaction at the same temperature . 12
The reaction quotient(Q).......................................................................................................... 13
Le Châtelier’s principle .............................................................................................................. 16
Applications of equilibrium shifts............................................................................................ 23

3
7 Equilibrium
7.1 Equilibrium
Essential idea: Many reactions are reversible. These reactions will reach a stateof equilibrium when
the rates of the forward and reverse reaction are equal. The position of equilibriumcan be
controlled by changing the conditions.
Understandings:
 A state of equilibrium is reached in a closed system when the rates of the forward and
reverse reactions are equal.
 The equilibrium law describes how the equilibrium constant (Kc) can be determined for
a particular chemical reaction.
 The magnitude of the equilibrium constant indicates the extent of a reaction at
equilibrium and is temperature dependent.
 The reaction quotient (Q) measures the relative amount of products and reactants
present during a reaction at a particular point in time. Q is the equilibrium expression
with non-equilibrium concentrations. The position of the equilibrium changes with
changes in concentration, pressure, and temperature.
 A catalyst has no effect on the position of equilibrium or the equilibrium constant.
Applications and skills:
 Deduction of the equilibrium constant expression (Kc) from an equation for a
homogeneous reaction.
 Determination of the relationship between different equilibrium constants (Kc) for the
same reaction at the same temperature.
 Application of Le Châtelier’s principle to predict the qualitative effects of changes of
temperature, pressure and concentration on the position of equilibrium and on the
value of the equilibrium constant.
Guidance:
 Physical and chemical systems should be covered.
 Relationship between Kc values for reactions that are multiples or inverses of one
another should be covered.
 Specific details of any industrial process are not required.
Nature of Science:
 Obtaining evidence for scientific theories—isotopic labelling and its use in defining
equilibrium. (1.8)
 Common language across different disciplines—the term dynamic equilibrium is used
in other contexts, but not necessarily with the chemistry definition in mind. (5.5)
International-mindedness:
 The Haber process has been described as the most important chemical reaction on
Earth as it has revolutionized global food production. However, it also had a large
impact on weaponry in both world wars.

4
7 Equilibrium
Theory of knowledge:
 Scientists investigate the world at different scales; the macroscopic and microscopic.
Which ways of knowing allow us to move from the macroscopic to the microscopic?
 Chemistry uses a specialized vocabulary: a closed system is one in which no matter is
exchanged with the surroundings. Does our vocabulary simply communicate our
knowledge; or does it shape what we can know?
 The career of Fritz Haber coincided with the political upheavals of two world wars. He
supervised the release of chlorine on the battlefield in World War I and worked on the
production of explosives. How does the social context of scientific work affect the
methods and findings of science? Should scientists be held morally responsible for the
applications of their discoveries?

5
7 Equilibrium
Our classroomequilibrium
Our classroomcontains6 people (let’ssaytheyare all students). The studentscanbe inone of two
states:sittingdownorstandingup. Thisisan example of aphysical equilibrium, andcanbe describedby
the equation:
Students(sitting) ⇌ Students(standing)
The two-directionalarrowindicatesthatthe reactioncango bothways:studentswhoare sittingcan
stand,and studentswhoare standingcan sit.
We define one of these directionsasthe forwards reaction (somewhatarbitrarily):
The opposite directionisthe reversereaction.
Students(standing) ⇌ Students(sitting)
Whenour classis at equilibrium:
 2 studentsare standing
 4 studentsare sitting
 and students are constantly switching between sitting and standing, so if someone sits, someone
else muststand(andvice versa)
Equilibrium characteristics (link: page 8)
Equilibriahave the followingcharacteristics,whichcanbe appliedtoour classroom:
In our classroomatequilibrium,this means… Characteristic
We closethe door at equilibrium;no-one enters or leaves,there
are always 6 students in the class
Equilibrium is achieved in a closed system
(e.g., flask)
People are constantly moving between sittingand standing Equilibrium is dynamic (constantly making
reactants and products)
The number of people sittingis constant(always 4). The number
of people standingis constant(always 2).
The concentrations of reactants and
products are constant at equilibrium
The total mass and volume of the class doesn’tchange; from a
distance,the appearanceis much the same (2 standers and 4
sitters)
At equilibrium, there is no change in
macroscopic properties (e.g., density or
colour)
It doesn’t matter if everyone is standingto begin with or everyone
is sitting;atequilibrium, 2 arestandingand 4 are sitting
Equilibrium can be reached from either
direction (100% reactants or 100%
products)

6
7 Equilibrium
Quantifying the classroom equilibrium (link: page 10−11)
The “equilibriumconstant”(Kc) isameasure of how likelythe reactionistogive the products – i.e., how
likelyitisthatany givenstudentwill be standing.
a) Write a Kc expressionforourclassroomatequilibrium.
________________________________________________________________________________
If Kc is large (> 1), the reactionissaidto be product-favoured– it makeslotsof products.
If Kc is small (<1), the reactionis reactant-favoured– it doesnot make a large quantityof products.
b) The equilibriumconstant, Kc,forourclassroomis0.50. Suggestwhetherourclassroomisreactant-
or product-favoured.
________________________________________________________________________________
The reverse reactionis:
Students(standing) ⇌ Students(sitting)
c) Write a Kc expressionforthe reversereaction(studentssitting)anddetermine the valueof Kc for
thisreaction.
________________________________________________________________________________
________________________________________________________________________________
d) Compare the Kc valuesforthe forwardand reverse reactions. Deduce how theyare mathematically
related.
________________________________________________________________________________

7
7 Equilibrium
Is the classroom at equilibrium? (link: page 12−14)
We use the ‘reactionquotient’, Q,to‘test’whetherthe classisat equilibrium.
Q =
[Students(𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔)]
[Students(𝑠𝑖𝑡𝑡𝑖𝑛𝑔)]
The expressionisthe same asthe Kc expression,butitiscalledthe reactionquotientbecause we are not
sure whetherthe classisat equilibriumornot!
e) Let’ssay 6 otherstudentssawthe lightandjoinedourclass. Upon enteringthe room, 3 of these
studentsstood,whilst3of themsat down. Calculate Qfor our new, 12-memberclass.
________________________________________________________________________________
________________________________________________________________________________
If we findthatQ = Kc,thenthe classisat equilibrium. If Q≠ Kc,the class is not at equilibrium, andtofix
thissome studentswill have toeithersitorstand,dependingonwhetherthe value ishigherorlower.
f) Identifywhetherthe value of Qforthe new 12-memberclass ishigheror lowerthan Kc.
________________________________________________________________________________
This means that too many students are __________________ .
To re-establish equilibrium, some students have to _________________ .
The reaction will have to shift to the _______________ in order to achieve equilibrium.
Rules:
If Q > Kc, the reaction shifts to the ____________ to achieve equilibrium.
If Q < Kc, the reaction shifts to the ____________ to achieve equilibrium.
g) If all studentsinthe school switchedintoourclass, thiswould addanother9 people. If theyall
came instanding,wouldthe classbe at equilibrium? Inwhichdirectionshouldthe reactionshiftto
achieve equilibrium? (Can equilibriumeven beachieved???)
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________

8
7 Equilibrium
Changes to a classroom at equilibrium: concentration (link: page 15+)
At the same temperature,the classroomatequilibriumwill alwayshave the same valueof Kc –
the Kc value ischaracteristic of a particularclassroom.
If students come intothe classroom(standing):
 Equilibriumisdisrupted(temporarilyatleast)
 To achieve equilibrium,twothirdsof studentscominginwouldhave tosit.
The classroom would
“shift towards sitting”
or
“shift to the left”
We saythat changesto the classroomcan affectthe positionof equilibrium– whetherthe reactionshifts
to favourreactants(left) orproducts(right).
 If too many studentsare standing,some have tositto achieve equilibriumagain –the reaction
shiftsto the left
 if too manystudentsare sitting,some have tostandto achieve equilibriumagain –the reaction
shiftsto the right
Thistendencyforthe systemtoalwaysrearrange itself toachieve the same equilibriumconstantisknown
as Le Châtelier’sprinciple:a systematequilibrium,when subjected to a change,will respond in such a
way asto minimise the effectof thechange.
State and explainwhatwouldhappentothe positionof equilibrium andthe equilibriumconstant(Kc) of
our classof 6 studentsif:
a) Two standingstudentsleave the class
Position of equilibrium: ____________________________________________________________
________________________________________________________________________________
Equilibrium constant: _____________________________________________________________
________________________________________________________________________________
b) Three students,whoare standing,enter the class
Position of equilibrium: ____________________________________________________________
________________________________________________________________________________
________________________________________________________________________________

9
7 Equilibrium
Characteristics of a dynamic equilibrium
Reactants ⇌ Products
Explainthe meaningof the two-directional equilibriumarrow.
______________________________________________________________________________________
Showhowan equilibriumarrowcanbe usedto indicate areactant-favouredorproduct-favouredreaction.
Reactant-favoured: ____________________________________________________________________
Product-favoured: _____________________________________________________________________
Outline three characteristicsof ahomogeneouschemical systemthatisina state of dynamicequilibrium.
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________

10
7 Equilibrium
1. A classicexample of chemical equilibriumisthatbetween
nitrogendioxide (abrowngas) anddinitrogentetroxide(a
colourlessgas).
State a balancedequationforthisreaction,includingstate
symbols.
________________________________________________________________________________
2. A sample of pure nitrogendioxide ismade byreactingcoppermetal
withnitricacid,and collectedinaflask.
Cu (s) + 4 HNO3 (aq)  Cu(NO3)2 (aq) + 2 NO2 (g) + 2 H2O(l)
As soonas itis collected,the gasinthe flaskisa dark browncolour.
Suggestand explainwhatwill happentothe colourof the gas mixture astime passes.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
3. Whenthe systemhas reached(isat) equilibrium, commentonthe following:
(a) The positionof equilibrium
________________________________________________________________________________
(b) The rate of the forwardand reverse reaction
________________________________________________________________________________
(a) The concentrationsof reactants and products
________________________________________________________________________________
4. Explainwhat will happentothe colourof the gas mixture afterequilibriumhasbeenreached.
________________________________________________________________________________
________________________________________________________________________________
NO2 N2O4

11
7 Equilibrium
The equilibriumconstant (Kc) and its temperature dependence
For the reactionaA + bB ⇌ cC+ dD:
𝐾c =
[𝐶]𝑐[𝐷]𝑑
[𝐴]𝑎[𝐵]𝑏
A reactionwill alwayshave the same value of Kc atthe same temperature (i.e.,the Kc value is
characteristicof – andconstant for– the reactionat that temperature).
Reaction Kc expression Value of Kc Reactant or product
favoured?
N2 (g) + 3 H2 (g) ⇌ 2 NH3 (g) 𝐾c =
[NH3]2
[H2]3[N2]
59.8
at 500 K
2 HI (g) ⇌ H2 (g) + I2 (g) 1.00×10−2
2 SO2 (g) + O2 (g) ⇌ 2 SO3 (g) 1.03×1012
N2O4 (g) ⇌ 2 NO2 (g)
7.7×10−4
at 0°C
𝐾c =
[H2O][CO]
[H2][CO2]
2.1
at 1200 K
H2 (g) + Cl2 (g) ⇌ 2 HCl (g)
2.40 × 1033
at 298 K
2 NOCl (g) ⇌ 2 NO(g) + Cl2 (g)
3.75 × 10−6
at 796°C

12
7 Equilibrium
Relationship between equilibrium constants (Kc) for the same reaction at the sametemperature
There are simple relationshipsbetween Kc valuesif the coefficientsof areactionchange,orthe reactionis
reversed.
aA + bB ⇌ cC + dD 2aA + 2bB ⇌ 2cC + 2dD cC + dD ⇌ aA + bB
𝐾c =
[𝐶]𝑐
[𝐷]𝑑
[𝐴]𝑎 [𝐵]𝑏
𝐾c =
[𝐶]2𝑐
[𝐷]2𝑑
[𝐴]2𝑎 [𝐵]2𝑏
𝐾c =
[𝐴]𝑎
[𝐵]𝑏
[𝐶]𝑐 [𝐷]𝑑
1. At a giventemperature,the followingreaction
CO2 (g) + H2 (g) ⇌ CO(g) + H2O(g)
has a value of Kc = 4.90 × 10−3
. Determine valuesof Kc forthe followingreactionsatthe same
temperature.
a. 2 CO2 (g) + 2 H2 (g) ⇌ 2 CO(g) + 2 H2O(g)
___________________________________________________________________________
b. CO(g) + H2O(g) ⇌ CO2 (g) + H2 (g)
___________________________________________________________________________
c. ½ CO2 (g) + ½ H2 (g) ⇌ ½ CO(g) + ½ H2O(g)
___________________________________________________________________________
2. At 1000 K,the methanationreaction
CO(g) + 3 H2 (g) ⇌ CH4 (g) + H2O(g)
has a Kc value of 254. Determine valuesof Kc forthe followingreactionsatthe same
temperature.
a. CH4 (g) + H2O(g) ⇌ CO(g) + 3 H2 (g)
___________________________________________________________________________
b. 2 CH4 (g) + 2 H2O(g) ⇌ 2 CO(g) + 6 H2 (g)
___________________________________________________________________________
c. 3 CO(g) + 9 H2 (g) ⇌ 3 CH4 (g) + 3 H2O(g)
___________________________________________________________________________

13
7 Equilibrium
The reaction quotient (Q)
Q measuresthe relative amountof productsandreactantspresentduringareactionat a particularpoint
intime. Q isthe equilibriumexpressionwithnon-equilibriumconcentrations.The positionof the
equilibriumchangeswithchangesinconcentration,pressure,andtemperature.
If Q > Kc, then the reaction ___________________________________________________
If Q < Kc, then the reaction ___________________________________________________
If Q = Kc, then the reaction ___________________________________________________
1. At 440°C, 0.0500 mol H2 is reactedwith0.0500 mol I2 ina 1.00 dm3
flask.
H2 (g) + I2 (g) ⇌ 2 HI(g) Kc = 49.4
The concentrationsattime t are foundto be:
[H2] = 0.0275 mol dm−3
[I2] = 0.0275 mol dm−3
[HI] = 0.450 mol dm−3
Determine whetherthe reactionisatequilibrium attime t. If it isnot,state andexplaininwhich
directionthe reactionwill proceed.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________

14
7 Equilibrium
2. For the followingreaction, Kc = 1.42 × 10−2
at a particulartemperature.
2 NOBr(g) ⇌ 2 NO(g) + Br2 (g)
1.0 mol NOBr,0.20 mol NOand 0.80 mol Br2 are introducedintoanevacuated1.0 dm3
flask.
Determine the value of Qbefore the reactionbegins. State andexplaininwhichdirectionthe
reactionwill proceedinordertoestablishequilibrium.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
3. At 373 K, 0.250 mol of SO2Cl2 (g),0.0175 mol SO2 (g) and0.0400 mol Cl2 (g) are injectedintoanempty
2.50 dm3
flask.
SO2Cl2 (g) ⇌ SO2 (g) + Cl2 (g) Kc = 7.8 × 10−2
Determine the value of Qbefore the reactionbegins. State andexplaininwhichdirectionthe
reactionwill proceedinordertoestablishequilibrium.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________

15
7 Equilibrium
4. At a particulartemperature,0.20mol N2 (g) and0.20 mol H2 (g) are placedina 1.0 dm3
closed
container.
N2 (g) + 3 H2 (g) ⇌ 2 NH3 (g) Kc = 16
Aftertime t,the flaskisfoundto contain0.22 g H2 (g),4.76 g N2 (g) and1.02 g NH3 (g).
Determine whethertheseamountsrepresentan equilibriummixtureandif not,state inwhich
directionthe reactionwill proceed.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________

16
7 Equilibrium
Le Châtelier’s principle
Allowsus topredictthe qualitative effectsof changes inconcentrationonthe positionof equilibriumand
on the value of the equilibriumconstant.
Predictand explainthe effectof eachchange on the equilibrium position and the equilibriumconstant of
the followingreactions.
1. Br2 (l) ⇌ Br2 (g)
A small ampoule containingbromineisaccidentallydropped,anditsmashesopenonthe
floorof a large classroom.
Equilibrium position: ______________________________________________________________
2. heat + [Cu(H2O)6]2+
(aq) + 4 Cl−
(aq) ⇌ CuCl4
2−
(aq) + 6 H2O(l)
a. The temperature isincreasedfrom roomtemperature to75°C
________________________________________________________________________________
________________________________________________________________________________
b. The systemisdilutedwithwater
________________________________________________________________________________
________________________________________________________________________________
c. Chloride ionsare removedbyadditionof Ag+
ions(forminginsolubleAgCl)
________________________________________________________________________________
________________________________________________________________________________

17
7 Equilibrium
3. N2 (g) + 3 H2 (g) ⇌ 2 NH3 (g) ∆H0
= –92 kJ
a. The temperature isincreasedfrom150°C to 300°C
________________________________________________________________________________
________________________________________________________________________________
b. Ammoniagasis removedfromthe systemasitforms
________________________________________________________________________________
________________________________________________________________________________
c. Iron metal isaddedas a catalyst
________________________________________________________________________________
________________________________________________________________________________

18
7 Equilibrium
4. CO (g) + 2 H2 (g) ⇌ CH3OH (g) ∆H0
= –103 kJ
a. The systemiscooledfromroom temperature to77 K.
________________________________________________________________________________
________________________________________________________________________________
b. Pressure is decreased.
________________________________________________________________________________
c. More hydrogengasis injectedintothe system.
________________________________________________________________________________
________________________________________________________________________________

19
7 Equilibrium
5. Br2 (g) + Cl2 (g) ⇌ 2 BrCl (g) ∆H0
= +14 kJ mol–1
a. The reactionflaskis heated
________________________________________________________________________________
________________________________________________________________________________
b. The pressure isincreased
________________________________________________________________________________
________________________________________________________________________________
c. A small amountof productis removed
Equilibrium position:
________________________________________________________________________________
________________________________________________________________________________

20
7 Equilibrium
6. 2 SO2 (g) + O2 (g) ⇌ 2 SO3 (g) + heat ∆H0
= –197 kJ
a. The reaction is heatedto700 K
________________________________________________________________________________
________________________________________________________________________________
b. The volume isdecreased,atconstanttemperature
________________________________________________________________________________
________________________________________________________________________________
c. SO3 (g) is removedasitisformed
________________________________________________________________________________
________________________________________________________________________________
d. A catalyst,V2O5,isadded
________________________________________________________________________________
________________________________________________________________________________

21
7 Equilibrium
7. 2 NO(g) + Cl2 (g) ⇌ 2 NOCl (g) Endothermicreaction
a. The reactionflask,originallyatroomtemperature,isplacedinice
________________________________________________________________________________
________________________________________________________________________________
b. Pressure isincreased
________________________________________________________________________________
c. Chlorine isremovedfromthe system
________________________________________________________________________________

22
7 Equilibrium
Predictand explainthe effectof the followingchangesonthe equilibriumconcentrationof product(s).
1. CO(g) + 2 H2 (g) ⇌ CH3OH(g) ∆H0
= − 90 kJ
a. The temperature isincreased
Equilibrium concentration of CH3OH: ________________________________________________
Explanation: _____________________________________________________________________
________________________________________________________________________________
b. More CO(g) is introduced
Equilibrium concentration of CH3OH: ________________________________________________
Explanation: _____________________________________________________________________
________________________________________________________________________________
2. PCl5 (g) ⇌ PCl3 (g) + Cl2 (g) ∆H0
= +92.5 kJ
a. The temperature isdecreased
Equilibrium concentrations of PCl3 and Cl2: ____________________________________________
Explanation: _____________________________________________________________________
________________________________________________________________________________
b. The volume isdecreased(atconstant temperature)
Explanation: _____________________________________________________________________
________________________________________________________________________________
c. A catalystisadded
Explanation: _____________________________________________________________________

23
7 Equilibrium
Applications of equilibrium shifts
The followingreactionsinvolvecoloured substances. Observationsof the colourchangesof the
equilibriummixture canbe usedtoconfirmthe effectof a particularchange onthe equilibriumposition.
1. Iron(III) ions(whichare pale yellow) reactwiththiocyanateions(colourlessinsolution) to
make brightred Fe(SCN)2+
.
Fe3+
(aq) + SCN−
(aq) ⇌ Fe(SCN)2+
(aq)
Deduce the Kc expressionforthisreaction.
________________________________________________________________________________
Predictandexplainthe colourchange that will be observedwhenthe followingchangesare
made to the reactionsystemat equilibrium.
a. More Fe3+
ionsare added
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
b. Silverions,whichreact irreversibly withthiocyanate ions,are added
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
c. SodiumhydroxideisaddedtoforminsolubleFe(OH)3
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
d. A catalystisadded
___________________________________________________________________________
___________________________________________________________________________

24
7 Equilibrium
2. Solutionscontainingchromate(VI)ions(yellow) canbe acidifiedtoproduce orange
dichromate(VI) ions,accordingtothe equationbelow.
CrO4
2−
(aq) + H+
(aq) ⇌ Cr2O7
2−
(aq) + H2O(l)
________________________________________________________________________________
a. Ammoniumdichromate isadded
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
b. Excessacidis added
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
c. A small amountof base is added
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

25
7 Equilibrium
3. Hydrogenchloride reactswithoxygento make steamandchlorine gas,whichisyellow.
4 HCl (g) + O2 (g) ⇌ 2 H2O(g) + 2 Cl2 (g)
________________________________________________________________________________
a. Steamisremoved
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
b. The pressure of the systemisincreased atconstantvolume
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
c. A catalystisadded
___________________________________________________________________________
___________________________________________________________________________

26
7 Equilibrium
4. Nitrogendioxide,adark browngas,dimerisestomake
dinitrogentetroxide,whichiscolourless. The equationfor
the reactionisgivenbelow.
2 NO2 (g) ⇌ N2O4 (g)
The photographshowstwoequilibriummixturesof NO2
and N2O4. The flaskon the leftisat a highertemperature
than the flaskonthe right.
Usingthis information,deduce andexplainwhetherthe
reactionabove isexothermicorendothermic.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
Representationsof the Lewisstructuresof NO2 andN2O4 are shownbelow. Use Table 11 of the
Data Booklettodetermine the expectedenthalpyof reaction, includingitssign,andstate whether
thisisconsistentwithyourdeductionabove.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________

7 equilibrium 2021.docx

Recommended

Recommended

More Related Content

Similar to 7 equilibrium 2021.docx

Similar to 7 equilibrium 2021.docx (20)

Recently uploaded

Recently uploaded (20)

7 equilibrium 2021.docx