Sunday 28 November 2010

Chemistry : Redox II

Balancing two ionic-half equation
The reaction occurs when a ionic-half equation at a higher position in E.C.S. goes as reverse, and another ionic-half equation at a lower position in E.C.S. is added together.
e.g. I2 + 2e- → 2I- is higher than MnO4- + 8H+ + 5e- → Mn2+ + 4H2O, so we write the first equation in reverse form: 2I- → I2 + 2e-, then we balance them by eliminating the electrons:
5x     2I- → I2 + 2e-  and  2x       MnO4- + 8H+ + 5e- → Mn2+ + 4H2O
Overall: 10I- + 2MnO4- + 16H+ → 5I2 + 2Mn2+ + 8H2O
Note that there shouldn’t be any electron in the balanced equation and the charge conserves.
Mentionable oxidizing and reducing agent:
1)       Chlorine as a strong oxidizing agent: Cl2 + 2e- → 2Cl-
i)                     Displacement on KBr and KI: Cl2 + 2X- → 2Cl- + X2, when Cl2 gas (pale green) is bubbled into the solution, Br2 (yellow-brown) / I2 (orange) solution is formed. When organic solvent (hexane) is added, Br and I become orange and violet respectively.
ii)                   Disproportionation against NaOH:
When it’s added to cold, dilute NaCl: Cl2 + NaOH → NaCl + NaOCl + H2O, O.N. of Cl change from 0 to +1 (in NaOCl) and -1(NaCl) respectively.
When it’s added to hot, conc., NaOH: 3Cl2 + 6NaOH → 5NaCl + NaClO3 + 3H2O, where O.N. of Cl in NaClO3 changed from 0 to +5.
The reaction where the same species is simultaneously reduced and oxidized is called disproportionation.

2)       Reactions with hot and conc. HNO3 (about 16M) gives brown and toxic gas NO2:
NO3- + 2H+ + e- → NO2 + H2O. e.g. Cu + 2NO3- + 4H+ → Cu2+ + 2NO2 + 2H2O
Reactions between non-metals and conc. HNO3:
C + 4HNO3 → CO2 + 4NO2 + 2H2O and S + 4HNO3 → SO2 + 4NO2 + 2H2O
Dilute HNO3 (about 2M) reacts to give NO: NO3- + 4H+ + 3e- → NO + 2H2O, but NO released will mix with oxygen in the air to give out brown NO2 (2NO + O2 → 2NO2), so when it’s mixed in a test tube, brown gas is only observed at the opening of the tube.
Very dilute HNO3 (less than 1M) only perform like a dilute acid which gives CO2.
3)       Reaction with dilute H2SO4: like a normal dilute acid, reacts with Zn but not Cu.
Hot and conc. H2SO4 oxidizes most metals: 2H2SO4 + 2e- → SO42- + SO2 + 2H2O
Reaction with hot and conc. H2SO4 with non-metals:
C + 4H2SO4 → CO2 + 4SO2 + 2H2O and S + 2H2SO4 → 3SO2 + 2H2O
Reaction with halides: for F and Cl, NaX + H2SO4 → NaHSO4 + HX
Bromides: NaBr + H2SO4 → NaHSO4 + HBr and 2HBr + H2SO4 → SO2 + Br2 + 2H2O, the overall reaction is : 2NaBr + 3H2SO4 → 2NaHSO4 + SO2 + Br2 + 2H2O
Iodides: NaI + H2SO4 → NaHSO4 + HI and 8HI + H2SO4 → H2S + 4I2 + 4H2O, the overall reaction is 8NaBr + 9H2SO4 → 8NaHSO4 + H2S + 4I2 + 4H2O
NaBr and NaI reacts in different behavior since HBr and HI is further oxidized by H2SO4.
4)       Aqueous sulphur dioxide (sulphurous acid, H2SO3) as a reducing agent is equivalent to SO32-. It can reduce other species and change to SO42-: SO32- + H2O → SO42- + 2H+ + 2e-
Test for reducing agent (especially SO2): filter paper soaked with KCr2O7/H+, if reducing agent presents, it reduces the filter paper from orange to green.
Test for oxidizing agent: filter paper soaked with (colourless) starch solution and KI, if oxidizing agent presents, it oxidizes 2I- to I2, and turn the starch solution to dark blue.
Test for HCl and HBr: white fumes formed with NH3(aq) is added (fumes from HCl is denser)
Test for H2S: turn lead(II) ethanoate paper black.

Electrochemistry is an extremely long topic and this is only one of the five chapters, doc. version won't be released until it's finished under the following sequences:
-Redox (completed)
-Chemical Cell
-Daily used cell

And I'm sorry that there's a little AL and non-DSE stuffs in the note.

Chemistry : Redox I

Definitions of reduction and oxidation:
In terms of
2Mg + O2 → 2MgO, Mg undergoes oxidation
CuO + H2 → Cu + H2O or N2H4 + O2 → N2 + 2H2O,
Cu and O is reduced.
Li+ + K → K+ + Li
K is oxidized.

The chemical species that oxidizes itself and reduce another species is called reducing agent, the chemical species that reduces itself and oxidize another species is called oxidizing agent.
Since number of e- conserves, reduction and oxidation must go in pairs.
Assignment of oxidation number (O.N.)
1)       Algebraical sum of oxidation number of each atom in a compound is equal to its charge of the ion (or equal to zero if it is neutral)
e.g., H2 is zero, so H in H2 is zero. K in K+ is +1.
2)       In a compound the element with higher electronegativity is assigned the O.N. first.
e.g., forCCl4, Cl was first assigned -1, then C is +4.
3)       Generally max. possible O.N. = group number, while the min. possible O.N. = 8 – group number, zero for metal.
Common O.N. of some elements:
H: +1 for H+, 0 for H2, -1 for H- (hydride)
O: -2 for O2-, 0 for O2, -1 for O22- (peroxide ion), -1/2 for O2- (superoxide ion)
Halogen: -1 for X-, 0 for X2; can have other O.N., e.g.: +1/3/5/7 for ClO- to ClO4-
Metal: 0 in neutral form and group number in ion form
Vanadium: +2 (V2+), +3 (V3+), +4(VO32+), +5(VO3+)
Stocking system to name ions: we use –ate(O.N.) to present the ion. e.g., SO42- is sulphate (VI) and Cr2O72- is dichromate (VI).
In a reaction, total decrease of O.N. is equal to the increase in O.N., increase in O.N. implies oxidation and decrease in O.N. implies reduction.
e.g., 2Mg + O2 → 2MgO, O.N. of Mg is raised from 0 to +2, while O decreased from 0 to -2.
We can balance the ionic-half equation by the following method
1)       If it is monatomic ion, then Xn+ + n e- → X (exceptional: Fe3+ + e- → Fe2+)
2)       If it is polyatomic ion (oxides), then a double amount of hydrogen ion is added and balance the charge with appropriate number of e-. e.g., Cr2O72- + 14H+ + 6e- → Cr3+ + 7H2O.
3)       Balance two ionic-half equation by multiply each of them so that the number of e- is balanced and can be eliminated completely.
The following is a electrochemical series (E.C.S.), where oxidizing power increase down the group, reducing power decrease down the group. Also, the LHS will be oxidized form while RHS will be the reduced form. i.e., oxidation goes as reverse and reduction go as written.
Li+ + e- → Li
Note that reactivity of halides decrease down the group, while reactivity of metals increase down the group, so Li and K is the strongest reducing agents while F2 is the strongest oxidizing agents.
Also, dichromate and permanganate ion is an oxidizing agent only if they’re in acidic solution (i.e. presence of H+), so they’re called acidified permanganate or dichromate solution.
Order of other metals is the same as metal reactivity series from K to Pb except Ca and Na is interchanged. Xn+ + n e- → X
2H+ + 2e- → H2
H2SO4 + 2H+ + 2 e- → SO2 + 2H2O
Cu2+ + 2e- → Cu
O2 + 2H2O + 4e- → 4OH-
I2 + 2e- → 2I-
Fe3+ + e- → Fe2+
Ag+ + e- → Ag
Br2 + 2e- → 2Br-
Cr2O72- + 14H+ + 6e- → Cr3+ + 7H2O
Cl2 + 2e- → 2Cl-
MnO4- + 8H+ + 5e- → Mn2+ + 4H2O
F2 + 2e- → 2F-

Saturday 27 November 2010


6:45AM, NUS High School
香城(?)有2間咖啡連鎖店,分別是Starbucks和Coffee Bean(為甚麼不是Pacific coffee? XD)
前提:人口平均分佈;等比於街道長度,另外一條街的market share會由最近這街的咖啡店拿到,如果一樣近則平分market share.
Q1 找出最好的開店策略,並在給定的地圖中找出用最少的店做出最高market share
Q2 給定一個資金,假沒開店費正比於街口的路口數目,用給定的資金做出最高market share
Q3 建立一些更真實的模型

-要佔領全市的market share?很簡單,每個街口開一間Coffee Beans!
-跟我們一起的那幾個星洲學生因為太無聊於是自成一角打起機來。他們在打GBA。嗯,是GBA無誤。而且是在玩遊戲王 XD

10:00PM, Day 2, 宿舍

Thursday 25 November 2010

Physics : Gravitation

Law of universal gravitation: F=GMm/r2, where M, m represents the mass of the two bodies (M is usually larger), r represents the distance, and G is the gravitation constant, g=6.67*10-11 with a unit of m3kg-1s-2.
Gravitational field strength is the gravitation force due to an object per unit mass. Mathematically g = F/m = GM/r2, with the unit of Nkg-1, the formula suggests that gravitational field strength is independent of the mass of object is feeling the gravitational force.
Gravitational field strength on Earth's surface is g0 = 9.81Nkg-1 (or 10), and the gravitational field strength above Earth's surface is given by g = g0r02/R2, where r0 is Earth's radius and R is the distance from the object to the center of Earth.
Gravitational field strength is often written in the form of g = GM/r2 = (GM/r0)(1+R/r0)-2 = g0(1+R/r0)-2, where this R is the distance from Earth's surface to the object. When R<<r0 (maybe a difference of thousands times), we estimate the field strength as g = g0(1-2R/r0).
It is suggested that outside the planet the field strength is proportional to r-2, but inside the planet the field strength is proportional to r (under the assumption of the perfectly mass-distributed and sphere planet) since M is proportional to V = kr3, then g = GM/r2 = Gkr.
Kepler's Laws on Planetary Motion
1.       The orbit of every planet is an ellipse with the Sun at one of the two foci.
2.       A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3.       The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis (r) of its orbit in the same solar system. i.e. T2 is proportional to r3.
The third law can be derived from circular motion, and it's suggested that T2 = 4π2r3/GM.
T2 = (2π/ω)2 = 4π22 = 4π2r/g = 4π2r3/GM.

Wednesday 24 November 2010

A short nice maths question

Today is the sports day and I managed to deal with a Putnam 2005 paper.

Putnam 2005 A1:
Show that every positive integer is a sum of one or more numbers of the form 2^r(3^s) where r and s are nonnegative integers and no summand divides another. e.g., 23=9+8+6.

1)What if "no summand divides another" is not nessary?
Then the answer is obvious: express them in binary form. e.g. 31 = 11111_2 = 16+8+4+2+1

Then the answer is not unique.

3)So any approach to attain "no summand divides another"?
Draw a square table, the horizontal row represents power of 3 and vertical row represents powers of 2.
e.g. A23 = 3^2 * 2^3 = 72

4)What's its significance?
If we say a=2^i 3^j and b=2^k 3^l is "one divdes another" (a>b), then i>= k and j >=l. Then in the table drawn, if the elements Amn is chosen then  all elements in the left-top and right-bottom quartile can't be chosen. Then we can eliminate our choice once a number is chosen.

5)What's your alogarithm?
-If it's odd, subtract it with the biggest possible powers of 3.
-If it's even, divide by 2.
-If it's power of 2, subtract by itself.
Repeat the above two step until it's zero.
21 - 9 => 12
12/2 => 6
6/2 => 3
3-3 => 0
Then we have 21=9+2(2(3)) = 9+12
77 - 27 = 50
50/2 = 25
25 - 9 = 16
77 = 27 + 2(9+16) = 27+18+32

6)Why this works?
Refer to the table, when it's multiplied by 2 and less power of three, it's the left-bottom quartile choice, so it is possible.
We see this alogarthim does not have any limitation on numbers, so we're done.

7)Any simplified representation like binary representation?
We don't see any simplified and possible representation since the sum is not ordered (by this alogarithm)
e.g. 77 = 27+18+32, it has decreasing powers of 3, but in the same power of 3 there may be more than one powers of 2.
e.g. 77 = 27 + 18+32, 61 = 27+18+16.

8) Is this unique?
No, 95 = 27+36+32 = 81 + 8 + 6.

Foods for thought:
If it isn't 2^r 3^s but other base, like 3^r 5^s, can it represents all numbers?

Pilot 2.3



很久沒寫,再回去寫覺得不夠成熟。 orz

Saturday 20 November 2010

Coase’s analysis on externalities

Assume there's two man owning two farmlands, one of them is for cattle rearing and another for wheat growing.
Externality exist when the cattle go to eat up some crops, where the cattle owner does not pay for it. In this case SC > PC.
There'll be four solutions:
1)       Forbidden the cattle to go to eat those crops.
2)       Integration among the two firms.
3)       Define private property rights to the cattle owner, i.e. they eat freely go to eat crops.
4)       Define private property rights to the crops owner, i.e., the cattle owner must pay for the crops eaten.
We will focus on case 2,3,4 where exchange still exist among the firms. Assume the market unit price of the crops is P.
Case 2: When crop owners' gain is also concerned by the cattle owner, there'll be no externalities anymore since cropland's gain is cattle owner's private benefit. Then it'll consume until MPB=MPC=P.
Case 3: When private property rights is defined to cattle owner, when the cattle eats the initial several units of crops, the farmer won't stop them since cattle owner's benefit > P, where it's not beneficial for the crop owner to beg the cattle owner to stop consuming the crops with a price higher than P.
When several consuming, MPB < P, then the crops owner will use a price of MPB (or a little bit higher) to beg the cattle owner stop consuming the crops. Since it's higher than MPB, the cattle owner will accept the exchange and the cattle owner can gain (P-MPB) from selling crops. In this case, they'll consume until MPB=P.
Case 4: When the private property is defined to the crop owners, externalities no longer exist also since the cattle owner must pay for the crops. Then it'll also consume MPB=P.
Our conclusion will be "under a well-defined private property rights, the market will emerge themselves to solve externalities.

Note: Pyhsics (Gas Law) and Economic (Efficiency and Equity) note is released on the note page.

Thursday 18 November 2010

Physics : Gas Law II

Avogadro’s law hypothesized that for any gas molecules, under same n, P, T, occupies the same V. Under s.t.p. (standard T = 273.15K and P = 1atm), 1 mole of gas occupies 22.4L (0.0224m3) of the spaces.
Ideal gas -- Macroscopic definition
Boyle’s law exactly true for all temperature and pressure, and PV is proportional to T.
Quantitative definition
-          Molecules collide perfectly elastically with the walls of container.
-          Time during collisions is negligible while comparing with time between collisions.
-          Size of molecules is negligible; and no intermolecular force among gas molecules exist.
Derivation of PV=Nmc2/3(Note: c2 is the sum of square root of the speed of molecules, and should be c-bar formally)
In a cubic container with side length L and N gas molecules inside, each of mass m, moving randomly in the container, with velocity c, with components vx, vy, vz in the three directions.
Consider one of the molecules in direction of x. Since it collides elastically, speed before and after collision unchanged, but with opposite direction. Hence Δp = -2mvx.
Time between successive collisions on the wall is given by Δt = 2L/vx.
Force exerted on the gas molecule Fg = Δp/Δt = -mvx2/L, then force exerted on the wall = Fw = -Fg = mvx2/L. Total force exerted on the wall = ΣFi = m(Σvix2)/L = Nmvx2/L (vx2 is the sum of square root of the components of velocity of the molecules)
Px = F/A = (Nmvx2/L)/L2 = Nmvx2/L3 = Nmvx2/V
When considering pressure on all direction, we know that pressure are the same for all direction, take average on force exerted on each wall:
P = (Px+Py+Pz)/3 = (Nmvx2/V+ Nmvy2/V+ Nmvz2/V)/3 = Nm(vx2+ vy2+ vz2)/3V
When the three components add up it’s the original velocity. Therefore arranging the terms gives PV=Nmc2/3, considering density ρ=Nm/V, P=ρc2/3.
Taking square root on c2gives the root-mean-square speed of the gas molecules, written as crms. It’s inversely proportional to square root of molar mass of the gas.
Considering the average transitional kinetic energy, KEavg=mc2/2, putting into PV=Nmc2/3,
PV=NkT= Nmc2/3 = N(KEavg)(2/3), therefore KEavg = mean ε = 2/3(kT).
For the internal energy of the whole system, ε = N(2kT/3) = PV(2/3).
The distribution of molecular speeds refers to the Maxwell-Boltzmann distribution, where the highest probability exist in √(2/3) crms, and the higher the temperature, the more disperse of the distribution. (Highest prob. decrease with temperature)
Note that real gas molecules can have other forms of energies such as rotational KE, vibrational energy and PE due to intermolecular attraction.
By KEavg = 2/3(kT), the molecular KE of the ideal gas become zero at 0K, which called the absolute zero where it’s the lowest possible temperature reached by any matter. However in real gas the molecular KE has a minimum value, which is the zero-point energy.

This is really hardcore orz

Wednesday 17 November 2010

Physics : Gas Law I

Gas Law and Kinetic Theory
Pressure is defined as perpendicular force per unit area. Mathematically P=F/A. It's a scalar, and has unit Nm-2 or Pascal (Pa), others including atm, mmHg or psi.
The pressure due to gas in a container is unique, and gas under higher pressure is pushed into region of lower pressure (since F=A*ΔP). Pressure can be measured by a Bourdon gauge.
Atmospheric pressure is the pressure due to the atmospheric environment (gas). It's about 101kPa (1atm)
Gas law – concerning P (pressure), V (volume), T (temperature), n (mass/amount of gas)
Note that T concerned is the absolute temperature, which implies the Kelvin scale. If a variable is directly proportional to absolute temperature, then it varies linearly with the temperature in Celsius scale.
1)       Boyle's law: P inversely proportional to V under fixed n, T. Mathematically P1V1=P2V2.
2)       Pressure law: P directly proportional to T under fixed n, V, Mathematically P1/T1=P2/T2.
3)       Charles' law: V directly proportional to T under fixed n, P, Mathematically V1/T1=V2/T2.
4)       General gas law : P1V1/T1=P2V2/T2. The gas is an ideal gas if it obeys this law for all temperature. Under room conditions, O, He, Ne behaves almost like an ideal gas. Gas behaves like ideal under very high T and very low P.
5)       Ideal gas law: PV=nRT, where n is number of moles of the gas, and R=8.31Jmol-1K-1 which is known as universal gas constant. There are different forms related to this law, such as PV=NkT, where N is number of molecules and k=R/NA, where NA is the Avogadro's number, 6.02*1023 mol-1, and k=1.38*10-21JK-1.
Kinetic theory
Properties related to kinetic theory:
-          Temperature of gas is the average KE due to random motion of the gas molecules.
-          Volume of gas spreads freely throughout the container.
-          Gas pressure is caused by collision of gas on the walls of container.
-          Particles moves in zigzag paths (Brownian motion), since they always collide and change direction.
Kinetic theory model
-          Boyle's law: weight added to piston (P), T unchanged, speed of molecules unchanged, it collides with the wall more frequently, exerting larger force, balancing weight of piston.
-          Pressure law: Voltage supply increase (T), speed of molecules, it hits the walls more vigorously and frequently, exerting a larger force on the piston. In order to maintain the same volume (height of piston), a weight is added to balance the force exerted by the gas.
-          Charles' law: Voltage supply increase (T), speed of molecules, it hits the walls more vigorously and frequently, exerting a larger force on the piston. P unchanged, therefore the weight of piston unchanged, then the piston moves upward to reduce frequency of collision, and reduce force exerted by the gas and balance the force.

Remaining part: Kinetic theory on ideal gas; internal energy.
doc. version of econ notes will be released soon.

Tuesday 16 November 2010

Economics - equity and inequality

Efficiency refers to the maximization of TSS in resource allocation, but equity refers to the fairness in society. There's trade off between efficiency and equity. Also, efficiency and inequality is positive statements while equity is normative and we don't have definite meaning for it.
Measures in income distribution
-          Income share: % of population VS different income groups
-          Decile groups: divide population into ten groups in increasing household income, e.g. the first decile group has the lowest 10% household income. If the household income is evenly distributed, then each group shares 10% of the total income.
-          Lorenz curve: cumulative % of population VS cumulative % of income
-          Gini coefficient Draw a line the 45-degree line in the graph, then
Gini coefficient = Area between Lorenz curve and 45-degree line / Area below 45-degree line
If Gini coefficient = 0, then the Lorenz curve is actually the 45-degree line, so it's absolute income inequality.
If it's 1, it's complete income disparity, where all income is occupied by one household unit.
If it's between 0 and 1, it's unequal income distribution, the larger coefficient, the larger income disparity.
Limitations in measuring income inequality
1)       Different in household size (e.g. $20000 for a 2-people family VS $30000 for a 8-people family)
2)       Imperfect market information / difficulty in collecting information
3)       Different definitions of income (some of the income is not recorded)
4)       Ignoring lifetime income (experienced worker was offered higher wages)
5)       It's calculated before transfer payments, exaggerating the income inequality.
Sources of income inequality
1)       Labour quality differs: better skilled labour has less supply but higher demand, hence the equilibrium price (wages) is higher.
2)       Working condition differs (e.g. policeman VS office lady)
3)       Difference in human capital (productivity)
4)       Geographical / age / sex difference or discrimination
5)       Unequal ownership of land, capital and business
6)       Monopoly and political power (e.g. professional membership as the entry barrier)
7)       Discrimination in labour, however it costs somehow, so discrimination is not common in the free market.
Solutions to income inequality
1)       Policies that equalizing income, such as transfer payments and legislation on minimum wages.
2)       Policies that equalizing opportunities, such as education and training, enhancing market competition, donations and foreign aid
What we have to consider about the policy is the different aspects / principles of equity, as well as balancing efficient and equity.

Up to now, the topic of "Efficiency and Equity" is completed, the next economic topic will be macroeconmics which concerns national income. If possible, I will try to put the topic into a deeper sight, with some light AL material (such as Coase's analysis) into a seperate set of notes.

Monday 15 November 2010

Economics note : Externalities

Government intervention corrects market failure and solves the problem of income inequality. Market failure means the market fails to allocate resource efficiently.
Reasons for market failure
1)       Lack of competition: Monopoly and oligopoly under cartel can rise P and decrease Q to earn more. Government can impose relegations to restrict them (promote competition).
2)       Provision of public goods makes the goods are consumed by all individuals concurrently, so that it's to costly to exclude non-payers from consuming, causing market failure.
Public goods refer to non-rival and non-excludable goods while private goods refer to rival and excludable goods.
Rivalry refers to one's consumption reduce availability in amount for other's consumption.
Excludability refers to difficulty to exclude specific individuals (non-payers) to consume it.
For Public goods, MC for an additional unit/serving additional consumer is zero, and it's too costly to exclude others to consume it.
Pure public goods refer to completely non-rival and non-excludable, which can be consumer by all individuals at the same time.
Impure public goods has a certain degree of rivalry and excludability but not compete at all.
Note that public goods provided by the government may be more efficient since it's too costly for private firms to exclude non-payers to use those goods, and it can bring greatest benefits to the society. However, goods provided by government may not be public good (e.g. water supply) and public good is not necessarily provided by the government.
3)       Externalities: when a party's action affect third party without any compensation or receiving payment.
Harmful (detrimental) and beneficial externalities refers to negative and positive externalities.
Private cost/benefit (PC/PB) is the cost/benefit of the economic agent taking that action.
External cost/benefit (EC/EB) refers to someone's action cause cost/benefit to third party without any compensation or payment.
Social cost/benefit (SC/SB) is the cost/benefit of the whole society by that action.
Mathematically SC = PC + EC, MSC = MPC + MEC, same as benefit.
Note that PB/SB is also called private/social product.
When external cost exists, the equilibrium quantity (ideal for individual considerations) is more than the efficient quantity, which is overproduced (or overconsumed). Deadweight loss is denoted by the bolded black lines. Note that MSC=MPC at Q=0.

When external benefit exists, the equilibrium quantity is less than the efficient quantity, which is underproduced. Deadweight loss is denoted by the bolded black lines.

 Since problem of externalities leads to inefficiency due to underproduction / overproduction which producers produces to MPB = MPC (equilibrium quantity) which maximizes his own gain, but the society's gain maximizes as MSC = MSB (efficient quantity), which differs from equilibrium quantity, taxation and subsidization shifts the MPC curve such that efficient quantity is equal to equilibrium quantity.
When there's divergence between MSC and MPC, we apply a unit tax of MEC at efficient quantity. As a result, MSC, new MPC, MB curve are concurrent at efficient quantity. The producers produce until MPB=MPC, but under taxation MSB=MPB=MPC=MSC, therefore producer's equilibrium quantity is equivalent to society's efficient quantity, and efficiency is attained.
When there's divergence between MSB and MPB, we apply a unit subsidy of MEB at efficient quantity, the MPC curve shifts downward (note that originally it was MC curve, but only MPC curve shifts only while MSC curve stays still), then the point of intersection of MPC'=MPB and MSC=MSB are both on efficient quantity. Therefore producers produce until MPC' = MPB, but at that equilibrium quantity, MSC = MSB is also attained, hence efficiency is attained.
-          Restriction or even removal on harmful externalities can be imposed by government.
-          When government takeover the firm, it concerns the gain of the whole society, then it'll produce to efficient quantity.
-          Defining and granting rights to scarce resources may emerge them to solve the problem of externalities. e.g., they can trade their polluting permission. (Coase's theorem)
-          Internalization or self-restraint (maybe engaged by government).
-          Integration between the firm and third party solve the externalities.
Note that if the transaction costs is larger than the benefit brought, or cost > benefit, we better not to do anything in the interests of the society. Also, the precondition for solutions to take place is the well-defined private property rights.
Government intervention is appropriate only if it's least costly and benefit > cost. The restriction of government intervention are imperfect information, lack of incentives (less efficient than private firm's solution), government official concern their own interests (corruption), and cost consideration.