80347

Solutions of electrolytes. Features of Solutions of Salts, Acids, and Bases

Лекция

Химия и фармакология

Solutions of salts, acids, and alkalis greatly deviate from the laws mentioned above. The osmotic pressure, depression of the vapour pressure, and the changes in the boiling and freezing points are always greater for them than they ought to be for the given concentration of the solution.

Английский

2017-02-21

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2 чел.

Solutions of electrolytes

Features of Solutions of Salts, Acids, and Bases

Solutions of salts, acids, and alkalis greatly deviate from the laws mentioned above. The osmotic pressure, depression of the vapour pressure, and the changes in the boiling and freezing points are always greater for them than they ought to be for the given concentration of the solution.

To extend this equation to solutions with an "abnormal" osmotic pressure, van't Hoff introduced the correction coefficienti(the isotonic coefficient) into it. This coefficient shows how many times the osmotic pressure of a given solution is greater than the "normal" value:

p = 1000icRT

Since the osmotic pressure and the changes in the freezing and boiling points are proportional to the number of solute particles in a solution, the coefficienti can be expressed by the relations

where symbol “¢ ” indicatesp,Dtfr, andDtb quantities not obeying van't Hoff's and Raoult's laws. The values of the coefficienti found by van't Hoff for 0.2N solutions of selected salts according to the depression of their freezing points are given inTable 3.

Table 3. Values of coefficienti for 0.2 N Solutions of selected salts

Salt

Formula

Depression of freezing point

Dt'fr

t =¾¾

Dtfr

observed

(Dt'fr)

calculated by Raoult's formula (Dtfr)

Potassium chloride

KC1

0.673

0.372

1.81

Potassium nitrate

KNO3

0.664

0.372

1.78

Magnesium chloride

MgCl2

0.519

0.186

2.79

Calcium nitrate

Ca(NO3)2

0.461

0.186

2.48

The data of Table 3 show that the coefficientivaries for different salts. For all salts formed by monovalent metals and monobasic acids, with sufficient dilution of their solutions, the coefficientiapproaches 2; for salts formed by divalent metals and monobasic acids, it approaches 3. How can this phenomenon be explained?

It was natural to assume that the solute molecules also decompose into smaller particles so that the total number of particles in the solution grows. When this number increases, the osmotic pressure also grows. Such an assumption was advanced for the first time in 1887 by the Swedish scientist S. Arrhenius.

Aqueous solutions of salts, acids, and bases have another feature— theyconduct an electric current. Anhydrous solid salts and bases, and also anhydrous acids do not conduct a current; pure water also does not practically conduct a current. We shall see below that these changes consist in thedissociation of the relevant substances into ions (ionization).

Substances decomposing in solutions or melts into ions are called electrolytes. They include salts, acids, and bases.

The theory of electrolytic dissociation

According totheory of electrolytic dissociation, when electrolytes dissolve in water, they decompose (dissociate) into positively charged ions (cations) and negatively charged ions (anions). The process of electrolytic dissociation the dissociation of HC1 is expressed by the equation

HCl = H+ + Cl-

The dissociation of electrolytes into ions (ionisation) explains the deviations from van't Hoff's and Raoult's laws. We showed as an example the depression of the freezing point of a solution of BaCI2.

In fact, the osmotic pressure in a very dilute solution of barium chloride, dissociating according to the equation

BaCI2 = Ba2+ + 2Cl-

isthree times greater than that computed according to van't Hoff's law.

Thus, the features of aqueous solutions of electrolytes that at first sight contradicted van't Hoff's and Raoult's laws were explained on the basis of these same laws.

Arrhenius's theory, however, treated ions as free particles not depending on the solvent molecules.Arrhenius's theory was opposed to the chemical, or hydrate theory of solutions advanced by Mendeleev that is based on the notion of solvent-solule interaction. The development of this idea later resulted in combination of Arrhenius's and Mendeleev's theories.

The dissociation process

The dissociation of the solute proceeds differently depending on its structure in the anhydrous state. Two cases are the most typical. One of them is the dissociation of dissolving salts, i.e.crystals having an ionic structure, and the second is dissociation when acids, i.e.substances consisting of polar molecules, dissolve.

When a crystal of a salt, for instance, potassium chloride, gets into water, the ions on its surface attract the polar molecules of the water (ion-dipole interaction,figure 1. The dissociation of polar molecules occurs in a different way. The water molecules attracted to the ends of a polar molecule (dipole-dipole interaction) cause its poles to move apart  —they polarize the molecule. This leads in the long run to ionization of the polar molecule. For instance, when hydrogen chloride dissolves in water, a process occurs that can be shown schematically by the equation

H2O + HCl = H3O+ + Cl-

The ions that have passed into the solution remain bound to the water molecules and formion hydrates; these compounds are also called ion solvates.

Degree of Dissociation. Strength of Electrolytes

Already van't Hoff established that the coefficientiis expressed by fractionsthat grow when the solution is diluted, approaching integers. Arrhenius explained this by the fact that only a part of an electrolyte dissociates into ions in a solution, and introduced the concept of the degree of dissociation.

Bythe degree of dissociation (a)of an electrolyte is meant the ratio of the number of its molecules that have broken up into ions in the given solution to the total number of its molecules in the solution.

It was later found that electrolytes could be divided into two groups— strong and weak ones.

Strong electrolytes dissociate virtually completely in aqueous solutions.The concept of the degree of dissociation cannot in essence be applied to them, and the deviation of the isotonic coefficient i from integral values is explained by other reasons. Weak electrolytes dissociate only partly in aqueous solutions.

Virtually all salts belong to strong electrolytes. Among the most important acids and bases, the following are strong electrolytes:

HNO3, HCIO4, HCl, HBr, HI, KOH, NaOH, Ba(OH)2, and Ca(OH)2.

Weak electrolytes include most of theorganic acids, and the following important inorganic compounds:

H2CO3, H2S, HCN, H2SiO3 and NH40H

The degree of dissociation is designated by the Greek lettera and is customarily expressed either in fractions of unity or as a percentage. For instance, for a 0.1N solution of CH3COOH,a= 0.013 (or 1.3%), and for a 0.1N solution of HCN,a == 10-4 (or 0.01%).

Dissociation Constant

The equilibrium constant corresponding to the disssociation of a weak electrolyte is known as the dissociation (orionization) constant.

For instance, for the dissociation of acetic acid CH3COOHD CH3COO- + H+the equilibrium constant has the form

Herethe numerator of the fraction contains the concentration of the ions that are the dissociation products and thedenominatorcontains the concentrationof the undissociated molecules.

The value ofK depends on the nature of the electrolyte and solvent and also on the temperature, but does not depend on the concentration of the solution. It characterises the ability of a given acid or given base to break up into ions: the greater is K, the more easily does the electrolyte dissociate.

Polybasic acids, and also bases of di- and polyvalent metals dissociate in steps. For example, carbonic acid dissociates in two steps:

H2CO3D H++ HCO3-

HCO3-D H++ CO32-

Thedissociation in the first stepis characterised by the dissociation constantK1 anddissociation in the second stepis characterised by the dissociation constantK2:

and

The summary dissociation constantK corresponds to the summary equilibrium H2CO3D 2H++ CO32-:

The quantities K1and K2 are related to one another by the expression

K =K1K2

Similar relations characterise the stepwise dissociation of bases of polyvalent metals.

In the stepwise dissociation of substances, the decomposition in a following step always occurs to a smaller degree than in the preceding one:

K1 > K2>K3. . .

If we denote the concentration of an electrolyte dissociating into two ions by c, and the degree of its dissociation in the given solution bya, then the concentration of each of the ions will be ca, and the concentration of the undissociated molecules will be c (1 —a). Hence, the equation of the dissociation constant acquires the form

This equation expressesW. Ostwald's dilution law.

For solutions in which the dissociation of an electrolyte is very small, the equation of Ostwald's law is simplified. Since in these casesa << 1, we may disregard the quantitya in the denominator of the right-hand side of the equation. Hence, the equation acquires the form

This equation clearly shows the relationship existing between the concentration of a weak electrolyte and the degree of its dissociation:

the degree of dissociation increases upon dilution of a solution.

Table 2 gives the values of the dissociation constants for selected weak electrolytes.

Table 2.Dissociation constants of selected weak electrolytes in aqueous solutions at 25 °C.

Electrolyte name

Formula

Dissociation constant

Acetic acid

Ammonium hydroxide

Carbonic acid

CH3COOH

NH4OH H2CO3

2×10-5

2× 10-5

K1= 4.5×10-7

K2= 4.7×10-11

Hydrogen

Hydrogen fluoride

Hydrogen peroxide

HCN

HF

H2O2

8× 10-10

7× 10-4

k1 = 10-12

Nitrous acid

Orthophosphoric acid

HNO2

H3PO4

4× 10-4

K1=8× l0-3

K2=6×10-8

K3=1×10-12

Strong Electrolytes

Incomplete dissociation of strong electrolytes was explained quantitatively in 1923 byPeter Debye and E. Hückel in theirtheory of interionic attraction. The main points of their theory are:

1. Ionic crystals dissociate in water and yield solutions of ions.

2. Strong interionic attractions between ions in solution result in a clustering which decreases the effective concentration of individual ions.

3. The interionic attraction forces decrease as the solution is diluted because the ions are farther apart.

4. The magnitude of the interionic attraction and the degree of clustering actions are related to the charge on the ions.

Strong electrolytes are usually completely dissociated in aqueous solutions of small concentration. However the forces of interionic attraction and repulsion are quite great inconcentratedsolutions, the ions arenot entirely free, their motion is hampered by their mutual attraction. Owing to this attraction, each ion is surrounded with a spherical cluster of oppositely charged ions that has been named an"ionic atmosphere".

The values of the degree of dissociation of potassium chloride calculated for 18 °C according to the electrical conductivity of its solutions show thata diminishes with a growth in the concentration:

The drop in the degree of dissociation, however, is explained not by the formation of molecules, but by an increase in the retarding action of the ionic atmosphere. In this connection, the value of the degree of dissociation of strong electrolytes determined from the electrical conductivity (or by other methods) is called the apparentdegree of dissociation.

All the properties of an electrolyte solution depending on the concentration of the ions manifest themselves as if the number of ions in the solution were smaller than in the complete dissociation of the electrolyte.

The state of the ions in a solution is assessed with the aid of a quantity called the activity.By the activity of an ion is meant its effective, conditional concentration according to which it acts in chemical reactions. The activity of an ion,a, equals its concentration c multiplied by the activity coefficient f:

a = fc

The activity coefficient is usually less than unity for concentrated solutions, and it approaches unity in very dilute solutions.

Table 3. Dissociation constants of selected strong acids in aqueous solutions at 25 °C

Acid

Formula

Dissociation constant K

Hydrobromic

Hydrochloric

Hydriodic

Nitric

HBr

HC1

HI

HNO3

109

107

1011

43.6

Permanganic

Sulphuric

HMn04

H2SO4

200

K1=1000, K2=10-2

Net Ionic Equations

Reaction of neutralisation.When any strong acid is neutralised with any strong base, about 57.6 kJ of heat are evolved per mole of water formed:

HCl + NaOH = NaCl + H2O  + 57.53 kJ

HNO3 + KOH = KNO3 + H2O   + 57.61 kJ

Let us rewrite its equation with strong electrolytes in the ionic form and the weak ones in the molecular form because the latter in the solution mainly in the form of molecules:

H+ + Cl- + Na+ + OH- = Na+ + Cl-+ H2O

Examining the equation obtained, we see that the Na+ and C1- ions have remained unchanged in the course of the reaction. We shall therefore rewrite the equation again:

H+ + OH- == H2O

Thus, the reaction of neutralisation of any strong acid with any strong base consists in the same process—in the formation of water molecules from hydrogen and hydroxide ions. It is clear that the heat effects of these reactions must also be the same.

We shall see below, however, that water is a very weak electrolyte and dissociates only to a negligibly small extent. In other words, equilibrium between the water molecules and the ions shifts greatly in the direction of molecule formation. For this reason, a reaction of neutralisation of a strong acid with a strong base goes practically to completion.

Reaction of double exchange.When a solution of a silver salt is mixed with hydrochloric acid or a solution of any of its salts, a characteristic white curdy precipitate of silver chloride is formed:

AgNO3 + HCl = AgCl + HNO3

Ag2SO4 + CuCI2 = 2AgCl + CuSO4

To obtain its net ionic equation, let us rewrite, for example, the equation of the first reaction, putting the strong electrolytes, as in the preceding example, in the ionic form, and the substance in the precipitate in the molecular one:

Ag+ +NO3- + H+ + Cl-= AgCl¯ + H+ + NO3-

Here it must also be borne in mind that the silver chloride precipitate is in equilibrium with the Ag+ and Cl- ions in the solution, so that the process expressed by the last equation is reversible:

Ag+ + Cl-D AgCl

Owing to the low solubility of silver chloride, however, this equilibrium shifts very greatly to the right. We can therefore consider that the reaction of AgCl formation from ions goes virtually to completion.Hence, the C1- ion can be a reagent for the Ag+ ion, and the Ag+ ion a reagent for the C1- ion.

In the following, we shall widely use the net ionic form of writing equations of reactions with the participation of electrolytes. To compile net ionic equations, we must know which salts are soluble in water and which are virtually insoluble. A general characteristic of the solubility of the most important salts in water is given in Table 4.

Table 4. Solubility Rules for Ionic Compounds in Water

Soluble Salts

1. The Na+, K+, and NH4+ ions formsoluble salts. Thus, NaCl, KNO3, (NH4)2SO4, Na2S, and (NH4)2CO3 are soluble.

2. The nitrate (NO3-) ion formssoluble salts. Thus, Cu(NO3)2 and Fe(NO3)3 are soluble.

3. The chloride (Cl-), bromide (Br-), and iodide (I-) ions generally formsoluble salts. Exceptions to this rule include salts of the Pb2+, Hg22+, Ag+, and Cu+ ions. ZnCl2 is soluble, but CuBr is not.

4. The sulfate (SO42-) ion generally formssoluble salts. Exceptions include BaSO4, SrSO4, and PbSO4, which are insoluble, and Ag2SO4, CaSO4, and Hg2SO4, which are slightly soluble.

Insoluble Salts

1. Sulfides (S2-) are usuallyinsoluble. Exceptions include Na2S, K2S, (NH4)2S, MgS, CaS, SrS, and BaS.

2. Oxides (O2-) are usuallyinsoluble. Exceptions include Na2O, K2O, SrO, and BaO, which are soluble, and CaO, which is slightly soluble.

3. Hydroxides (OH-) are usuallyinsoluble. Exceptions include NaOH, KOH, Sr(OH)2, and Ba(OH)2, which are soluble, and Ca(OH)2, which is slightly soluble.

4. Chromates (CrO42-) are usuallyinsoluble. Exceptions include Na2CrO4, K2CrO4, (NH4)2CrO4, and MgCrO4.

5. Phosphates (PO43-) and carbonates (CO32-) are usuallyinsoluble. Exceptions include salts of the Na+, K+, and NH4+ ions.

Shift of ionic equilibria

Equilibrium in solutions of electrolytes, like any chemical equilibrium, remains unchanged until the conditions determining it change. A change in the conditions leads to violation of equilibrium.

The introduction of ions in common with one of the electrolyte ions into a solution of a weak electrolyte lowers the degree of dissociation of the electrolyte. Thus, the solubility of an electrolyte diminishes as a result of the introduction of common ions into the solution.

The above examples allow us to arrive at the following general conclusion.

An essential condition for reactions between electrolytes to go on is the removal from the solution of some species of the ions, for example owing to the formation of poorly dissociating substances, or substances liberated from the solution as a precipitate or gas. In other words, reactions in solutions of electrolytes always proceed in the direction of formation of the least dissociated or least soluble substances.

It follows from this conclusion, in particular, that strong acids displace weak ones from solutions of their salts. For instance, when sodium acetate reacts with hydrochloric acid, the reaction proceeds virtually to the end with the formation of acetic acid:

CH3COONa + HC1 = CH3COOH + NaCI

or in the net ionic form:

CH3COO- + H+= CH3COOH

Dissociation of Water. Ion product of water. pH

Pure water conducts an electric current very poorly, but nevertheless it has a measurable electrical conductivity that is explained by the slight dissociation of water into hydrogen and hydroxide ions:

H2OD H+ + OH-

The concentration of hydrogen and hydroxide ions in water at 25oC is     10-7 mol/l. Let us write an expression for the dissociation constant of water:

We can rewrite this equation as follows:

K[H2O]= [H+][OH-]

Since the degree of dissociation of water is very low, the concentration of undissociated molecules of H2O in water virtually equals the total concentration of water, i.e. 55.55 mol/1 (1 litre contains 1000 grams of water, i.e. 100:18.02 = 55.55 moles). Therefore, replacing the product  [H2O]K in the last equation with the new constantKw, we have:

[H+][OH-] = Kw

The obtained equation shows that for water and dilute aqueous solutions at a constant temperature, the product of the hydrogen ion and hydroxide ion concentrations is a constant quantify. The latter is called the ion product of water.

It is not difficult to obtain its numerical value by introducing into the last equation the concentrations of the hydrogen and hydroxide ions. For pure water at 25 °C, we have [H+] = [OH-]=1×10-7 mol/l. Hence, for this temperature:

Solutions in which the concentrations of the hydrogen ions and hydroxide ions are the same are called neutral solutions.

At 25 °C, as we have already indicated, the concentration of both hydrogen ions and hydroxide ions in neutral solutions is 10-7 mol/1. If, for instance, we add so much acid to pure water that the hydrogen ion concentration grows to 10-3 mol/l, then the hydroxide ion concentration will diminish so that the product [H+][OH-] remains equal to 10-14. Consequently, in this solution, the hydroxide ion concentration will be

These examples show that both the degree of acidity and the degree of alkalinity of a solution can be characterised quantitatively by the hydrogen ion concentration:

The acidity or alkalinity of a solution can be expressed in another, more convenient way: instead of the hydrogen ion concentration, its common logarithm is taken with the reverse sign. The latter quantity is named the hydrogen ion index, but most chemists call it the pH-value or simply the pH1

:

For instance, if [H+] = 10-5 mol/1, then pH = 5, if [H+] = 10~9 mol/1, then pH = 9, and so on. It is thus clear that for a neutral solution ([H+] =                10-7mol/1), pH = 7 (at a temperature of 25 °C). For acid solutions, pH < 7, and diminishes with an increasing acidity of the solution. Conversely, for alkaline solutions, pH > 7, and grows with an increasing alkalinity of the solution.

The pH is of great significance for many processes. For example, plants can grow normally only when the pH of the soil solution is within a definite interval characteristic of a given species of plant. The properties of natural water, particularly its corrosion activity, depends greatly on the pH.

Indicators

There are different ways of measuring the pH. The reaction of a solution can be determined approximately with the aid of specialreagents called indicators whose colour changes depending on the hydrogen ion concentration. The most widespread indicatorsaremethyl orange, methyl red, and phenolphthalein.Table 5 characterises the most important indicators.

Table 5.Most Important Indicators

Hydrolysis of Salts

Byhydrolysisis meant the reaction of a substance with water in which the constituent parts of the substance combine with the constituent parts of water.

An example of hydrolysis is the reaction of phosphorus trichloride PCl3 with water. It results in the formation of phosphorous acid H3PO3 and hydrochloric acid:

Compounds of various classes are subjected to hydrolysis. We treat one of the most important cases — thehydrolysis of salts.

Salts formed by a weak acid and a weak base, or by a weak acid and a strong base, or by a weak base and a strong acid, enter into reactions of hydrolysis.Salts formed by a strong- acid and a strong base do not undergo hydrolysis,here neutralisation consists in the process

H+ + OH- = H2O

Let us consider the hydrolysis of a salt formed by a monobasic acid and a monovalent metal. Ina salt formed by a weak base and a strong acid, the cation of the salt is hydrolysed, and the reaction is attended by the formation of H+ ions; for example

Accumulation of the H+ ion leads to diminishing of the OH ion concentration. Thus,solutions of salts formed by a weak base and a strong acid are acidic.

Similarly,solutions of salts formed by a weak acid and a strong base are basic.

The fraction of a substance subjected to hydrolysis—the degree of hydrolysis-depends on the constant of this equilibrium, and also on the temperature and the salt concentration.

Let us write the equation of hydrolysis in the general form. Let HA be an acid and MOH a base. The salt they form is MA. The equation of hydrolysis will thus have the form

The following constant corresponds to this equilibrium

The concentration of water in dilute solutions is virtually a constant quantity. Introducing the symbol

The quantityKhis known as the hydrolysis constant of a salt, Its value characterises the ability of the given salt to be hydrolysed. An increase inKh is attended by a greater (at the same temperature and salt concentration) degree of hydrolysis.

The degree of hydrolysis is determined by the nature of a salt, its concentration, and temperature. The nature of a salt manifests itself in the value of the hydrolysis constant. The dependence on the concentration expresses itself in that the degree of hydrolysis grows with dilution of a solution. Indeed, suppose we have a solution of potassium cyanide. The following equilibrium sets in it:

KCN + H2O = HCN + KOH

which the constant corresponds to.

Let us dilute the solution 10 times. At the first moment, the concentrations of all the substances—KCN, HCN, and KOH—diminish to one-tenth their initial values. As a result, the numerator in the right-hand side of the equation of the hydrolysis constant decreases 100 times, and the denominator only 10 times. But the hydrolysis constant, like any equilibrium constant, does not depend on the concentrations of the substances. Therefore, equilibrium in the solution will be violated. For it to set in again, the numerator of the fraction must grow and the denominator diminish, i.e. a certain amount of the salt must additionally hydrolyse. As a result, the concentrations of HCN and KOH will grow, and that of KCN will drop.Thus, the degree of hydrolysis of the salt will increase.

We now considerthe hydrolysis of salts formed by a weak poly-basic acid. The hydrolysis of such salts proceeds in steps. Thus, the first step of hydrolysis of sodium carbonate proceeds according to the equation

or in the net ionic form:

The acid salt hydrolysis, in turn (the second step of hydrolysis):

It can be seen that in hydrolysis in the first step the ion HCO3-is formed. Hydrolysis in the first step always proceeds to a greater degree than in the second one. In addition, the ions formed in the first-step hydrolysis (in the example being considered—the OH- ions) facilitate the shifting of equilibrium in the second step to the left, i.e. also suppress the second-step hydrolysis.

The hydrolysis of salts formed by a weak base of a polyvalent metal proceeds similarly. For example, the hydrolysis of copper(II) chloride proceeds in the first step with the formation of copper chloride hydroxide:

or in the net ionic form:

Hydrolysis in the second step occurs to a negligibly small degree:

or

Especially great isthe hydrolysis of salts formed by a weak acid and a weak base. An example is the hydrolysis of aluminium acetate that proceeds to the basic salts—aluminium acetate hydroxide and dihydroxide:

Lets us consider the hydrolysis of the cation, and the hydrolysis of the anion separately for the given case. These processes are expressed by the net ionic equations

We see that in the hydrolysis of the cation the H+ ion is formed, and in the hydrolysis of the anion—the OH- ion. These ions cannot exist in considerable concentrations; they combine to form molecules of water. The result is the shifting of both equilibria to the right. In other words, the hydrolysis of the cation and the hydrolysis of the anion in this case

amplify each other.

The Solubility Product Expression

Consider the slightly soluble in water salt in equilibrium with water solution likefigure 6 with CaF2.

Silver chloride is so insoluble in water that a saturated solution contains only about 1.3×10-5 moles of AgCl per liter of water.

H2O

AgCl(s)

D

Ag+(aq)

+

Cl-(aq)

The [AgCl] term has to be translated quite literally as the number of moles of AgCl in a liter of solid AgCl.The concentration of solid AgCl can be calculated from its density and the molar mass of AgCl.

This quantity is a constant, however. The number of moles per liter in solid AgCl is the same at the start of the reaction as it is when the reaction reaches equilibrium.

Since the [AgCl] term is a constant, which has no effect on the equilibrium, it is built into the equilibrium constant for the reaction.

[Ag+][Cl-] =Kc [AgCl]

This equation suggests that the product of the equilibrium concentrations of the Ag+ and Cl- ions in this solution is equal to a constant. Since this constant is proportional to the solubility of the salt, it is called thesolubility product equilibrium constant for the reaction, orKsp.

Ksp = [Ag+][Cl-]

TheKsp expression for a salt is the product of the concentrations of the ions, with each concentration raised to a power equal to the coefficient of that ion in the balanced equation for the solubility equilibrium.

Buffers and pH of buffers

A buffer is a solution that resists changes in pH. A buffer is made with a weak acid and a soluble salt of this weak acidor a weak Some examples of buffer material pairs are: acetic acid and sodium acetate, phosphoric acid and potassium phosphate, oxalic acid and lithium oxalate, carbonic acid and sodium carbonate, or ammonium hydroxide and ammonium nitrate.

A buffer is most effective in solutions of pH at or close to the pKA2


 

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49139. Трехзвенный Г-образный фильтр верхних частот 667 KB
  Переходная харатеристика Техническое задание Электрическая принципиальная схема Задание: Расчет АЧХ ФЧХ и переходной характеристики трехзвенного Гобразного фильтра. Расчет Рис.
49140. Полосовой фильтр 24.46 MB
  Получить Амплетудно–Частотную, Фаза –Частотную характеристики, переходную характеристику и построить их графики Задание Расчет стационарных характеристик цепи Таблицы и графики АЧХ и ФЧХ...
49141. ИСПОЛЬЗОВАНИЕ АКУСТООПТИЧЕСКОГО ЭФФЕКТА ДЛЯ ИЗМЕРЕНИЯ ФИЗИЧЕСКИХ ВЕЛИЧИН 2.4 MB
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