Sorensen, a Danish biochemist developed a scale to measure the acidity in terms of concentrations of \[{{H}^{+}}\] in a solution. As defined by him, “pH of a solution is the negative logarithm to the base 10 of the concentration of H+ ions which it contains.”
\[pH=-\log [{{H}^{+}}]\] or \[pH=\log \frac{1}{[{{H}^{+}}]}\]
Just as pH indicates the hydrogen ion concentration, the pOH represents the hydroxyl ion concentration, i.e.,
\[pOH=-\log [O{{H}^{-}}]\]
Considering the relationship, \[[{{H}^{+}}][O{{H}^{-}}]={{K}_{w}}=1\times {{10}^{-14}}\]
Taking log on both sides, we have
\[\log [{{H}^{+}}]+\log [O{{H}^{-}}]=\log {{K}_{w}}=\log (1\times {{10}^{-14}})\]or
\[-\log [{{H}^{+}}]-\log [O{{H}^{-}}]=-\log {{K}_{w}}=-\log (1\times {{10}^{-14}})\]
or \[pH+pOH=p{{K}_{w}}=14\]
A solution whose pH is not altered to any great extent by the addition of small quantities of either sirong acid (H+ ions) or a sirong base (OH– ions) is called the buffer solution. It can also be defined as a solution of reserve acidity or alkalinity which resists change of \[pH\] upon the addition of small amount of acid or alkali.
(1) Types of buffer solutions : There are two types of buffer solutions,
(i) Solutions of single substances : The solution of the salt of a weak acid and a weak base.
Example : ammonium acetate \[(C{{H}_{3}}COON{{H}_{4}})\], \[N{{H}_{4}}CN\] act as a buffer.
(ii) Solutions of Mixtures : These are further of two types,
(a) Acidic buffer : It is the solution of a mixture of a weak acid and a salt of this weak acid with a strong base.
Example : \[C{{H}_{3}}COOH+C{{H}_{3}}COONa\]
(b) Basic buffer : It is the solution of a mixture of a weak base and a salt of this weak base with a strong acid.
Example : \[N{{H}_{4}}OH+N{{H}_{4}}Cl\]
(2) Buffer action : Buffer action is the mechanism by which added H+ ions or OH– ions are almost neutralised; so that pH practically remains constant. Reserved base of buffer neutralises the added \[{{H}^{+}}\] ions while the reserved acid of buffer neutralises the added OH– ions.
(3) Examples of buffer solutions
(i) Phthalic acid + potassium hydrogen phthalate
(ii) Citric acid + sodium citrate.
(iii) Boric acid + borax (sodium tetraborate).
(iv) Carbonic acid \[({{H}_{2}}C{{O}_{3}})\]+ sodium hydrogen carbonate \[(NaHC{{O}_{3}})\]. This system is found in blood and helps in maintaining \[pH\] of the blood close to 7.4 (\[pH\] value of human blood lies between 7.36 – 7.42; a change in pH by 0.2 units may cause death).
(v) \[Na{{H}_{2}}P{{O}_{4}}+N{{a}_{3}}P{{O}_{4}}\]
(vi) \[Na{{H}_{2}}P{{O}_{4}}+N{{a}_{2}}HP{{O}_{4}}\]
(vii) Glycerine + \[HCl\]
(viii) The \[pH\] value of gastric juice is maintained between 1.6 and 1.7 due to buffer system.
(4) Henderson - Hasselbalch equation : \[pH\] of an acidic or a basic buffer can be calculated by Henderson- Hasselbalch equation.
For acidic buffers, \[pH=p{{K}_{a}}+\log \frac{[salt]}{[acid]}\]
When \[\frac{[salt]}{[acid]}=10\], then, \[pH=1+p{{K}_{a}}\] and
when \[\frac{[salt]}{[acid]}=\frac{1}{10}\], then, \[pH=p{{K}_{a}}-1\]
So weak acid may be used for preparing buffer solutions having \[pH\] values lying within the ranges \[p{{K}_{a}}+1\] and \[p{{K}_{a}}-1\]. The acetic acid has a \[p{{K}_{a}}\] of about 4.8; it may, therefore, be used for making buffer solutions with \[pH\] values lying roughly within the range 3.8 to 5.8.
For basic Buffers, \[pOH=p{{K}_{b}}+\log \frac{[salt]}{[base]}\]
Knowing \[pOH\], \[pH\] can be calculated by the application of formula, \[pH+pOH=14\]
\[pH\] of a buffer solution does not change with dilution but it varies with temperature because value of \[{{K}_{w}}\] changes with temperature.
(5) Buffer capacity : The property of a buffer solution to resist alteration in its pH value is known as buffer capacity. It has been found that if the ratio \[\frac{[salt]}{[acid]}\] or \[\frac{[salt]}{[base]}\] is unity, the pH of a particular buffer does not change at all. Buffer capacity is defined quantitatively as number of moles of acid or base added in one more...
It is the reaction of the cation or the anion or both the ions of the salt with water to produce either acidic or basic solution. Hydrolysis is the reverse of neutralization.
(1) Hydrolysis constant : The general equation for the hydrolysis of a salt (BA),\[\underset{\text{salt}}{\mathop{BA}}\,+{{H}_{2}}O\] \[\rightleftharpoons \] \[\underset{\text{acid}}{\mathop{HA}}\,+\underset{\text{base}}{\mathop{BOH}}\,\]
Applying the law of chemical equilibrium, we get
\[\frac{[HA][BOH]}{[BA][{{H}_{2}}O]}=K\], where K is the equilibrium constant.
Since water is present in very large excess in the aqueous solution, its concentration \[[{{H}_{2}}O]\] may be regarded as constant so,
\[\frac{[HA][BOH]}{[BA]}=K[{{H}_{2}}O]={{K}_{h}}\]
where \[{{K}_{h}}\] is called the hydrolysis constant.
(2) Degree of hydrolysis : It is defined as the fraction (or percentage) of the total salt which is hydrolysed at equilibrium. For example, if 90% of a salt solution is hydrolysed, its degree of hydrolysis is 0.90 or as 90%. It is generally represented by ‘\[h\]’.
\[h=\frac{\text{Number of moles of the salt hydrolysed}}{\text{Total number of moles of the salt taken}}\]
An indicator is a substance, which is used to determine the end point in a titration. In acid-base titrations, organic substance (weak acids or weak bases) are generally used as indicators. They change their colour within a certain \[pH\] range. The colour change and the \[pH\] range of some common indicators are tabulated below
Colour changes of indicators with pH
(1) A gas may be liquefied by cooling or by the application of high pressure or by the combined effect of both. The first successful attempt for liquefying gases was made by Faraday.
(2) Gases for which the intermolecular forces of attraction are small such as \[{{H}_{2}}\], \[{{N}_{2}}\], Ar and \[{{O}_{2}}\], have low values of \[{{T}_{c}}\] and cannot be liquefied by the application of pressure are known as “permanent gases” while the gases for which the intermolecular forces of attraction are large, such as polar molecules \[N{{H}_{3}}\], \[S{{O}_{2}}\] and \[{{H}_{2}}O\] have high values of \[{{T}_{c}}\] and can be liquefied easily.
(3) Methods of liquefaction of gases : The modern methods of cooling the gas to or below their \[{{T}_{c}}\] and hence of liquefaction of gases are done by Linde's method and Claude's method.
(i) Linde's method : This process is based upon Joule-Thomson effect which states that “When a gas is allowed to expand adiabatically from a region of high pressure to a region of extremely low pressure, it is accompained by cooling.”
(ii) Claude's method : This process is based upon the principle that when a gas expands adiabatically against an external pressure (as a piston in an engine), it does some external work. Since work is done by the molecules at the cost of their kinetic energy, the temperature of the gas falls causing cooling.
(iii) By adiabatic demagnetisation.
(4) Uses of liquefied gases : Liquefied and gases compressed under a high pressure are of great importance in industries.
(i) Liquid ammonia and liquid sulphur dioxide are used as refrigerants.
(ii) Liquid carbon dioxide finds use in soda fountains.
(iii) Liquid chlorine is used for bleaching and disinfectant purposes.
(iv) Liquid air is an important source of oxygen in rockets and jet-propelled planes and bombs.
(v) Compressed oxygen is used for welding purposes.
(vi) Compressed helium is used in airships.
(5) Joule-Thomson effect : When a real gas is allowed to expand adiabatically through a porous plug or a fine hole into a region of low pressure, it is accompanied by cooling (except for hydrogen and helium which get warmed up).
Cooling takes place because some work is done to overcome the intermolecular forces of attraction. As a result, the internal energy decreases and so does the temperature.
Ideal gases do not show any cooling or heating because there are no intermolecular forces of attraction i.e., they do not show Joule-Thomson effect.
During Joule-Thomson effect, enthalpy of the system remains constant.
Joule-Thomson coefficient. \[\mu ={{(\partial T/\partial P)}_{H}}\].
For cooling, \[\mu =+ve\] (because \[dT\] and \[dP\] will be \[-ve\])
For heating \[\mu =-ve\](because \[dT=+ve,\ dP=-ve)\].
For no heating or cooling \[\mu =0\] (because \[dT=0)\].
(6) Inversion temperature : It is the temperature at which gas shows neither cooling effect nor heating effect i.e., Joule-Thomson coefficient \[\mu =0\]. Below this temperature, it shows cooling effect and above this temperature, it shows heating effect.
Any gas like \[{{H}_{2}},\ He\] etc, whose inversion temperature is low would show heating effect at room temperature. more...
(1) Specific heat (or specific heat capacity) of a substance is the quantity of heat (in calories, joules, kcal, or kilo joules) required to raise the temperature of 1g of that substance through \[{{1}^{o}}C\]. It can be measured at constant pressure \[({{c}_{p}})\] and at constant volume \[({{c}_{v}})\].
(2) Molar heat capacity of a substance is the quantity of heat required to raise the temperature of 1 mole of the substance by \[{{1}^{o}}C\].
\ Molar heat capacity = Specific heat capacity ´ Molecular weight, i.e.,
\[{{C}_{v}}={{c}_{v}}\times M\] and \[{{C}_{p}}={{c}_{p}}\times M\].
(3) Since gases upon heating show considerable tendency towards expansion if heated under constant pressure conditions, an additional energy has to be supplied for raising its temperature by \[{{1}^{o}}C\] relative to that required under constant volume conditions, i.e.,
\[{{C}_{p}}>{{C}_{v}}\]or \[{{C}_{p}}={{C}_{v}}+\text{Work done on expansion, }P\Delta V(=R)\]
where, \[{{C}_{p}}=\] molar heat capacity at constant pressure; \[{{C}_{v}}=\] molar heat capacity at constant volume.
(4) Some useful relations of Cp and Cv
(i) \[{{C}_{p}}-{{C}_{v}}=R=2\,calories=8.314J\]
(ii) \[{{C}_{v}}=\frac{3}{2}R\] (for monoatomic gas) and \[{{C}_{v}}=\frac{3}{2}+x\] (for di and polyatomic gas), where x varies from gas to gas.
(iii) \[\frac{{{C}_{p}}}{{{C}_{v}}}=\gamma \] (Ratio of molar capacities)
(iv) For monoatomic gas \[{{C}_{v}}=3\,calories\] whereas, \[{{C}_{p}}={{C}_{v}}+R=5calories\]
(v) For monoatomic gas, \[(\gamma )=\frac{{{C}_{p}}}{{{C}_{v}}}=\frac{\frac{5}{2}R}{\frac{3}{2}R}=1.66\].
(vi) For diatomic gas \[(\gamma )=\frac{{{C}_{p}}}{{{C}_{v}}}=\frac{\frac{7}{2}R}{\frac{5}{2}R}=1.40\]
(vii) For triatomic gas \[(\gamma )=\frac{{{C}_{p}}}{{{C}_{v}}}=\frac{8R}{6R}=1.33\]
(1) A state for every substance at which the vapour and liquid states are indistinguishable is known as critical state. It is defined by critical temperature and critical pressure.
(2) Critical temperature (Tc) of a gas is that temperature above which the gas cannot be liquified however large pressure is applied. It is given by, \[{{T}_{c}}=\frac{8a}{27Rb}\]
(3) Critical pressure (Pc) is the minimum pressure which must be applied to a gas to liquify it at its critical temperature. It is given by, \[{{P}_{c}}=\frac{a}{27{{b}^{2}}}\]
(4) Critical volume (Vc) is the volume occupied by one mole of the substance at its critical temperature and critical pressure. It is given by, \[{{V}_{c}}=3b\]
(5) Critical compressibility factor (Zc) is given by, \[{{Z}_{c}}=\frac{{{P}_{c}}{{V}_{c}}}{R{{T}_{c}}}=\frac{3}{8}=0.375\]
A gas behaves as a Vander Waal?s gas if its critical compressibility factor \[({{Z}_{c}})\] is equal to 0.375. A substance is the gaseous state below \[{{T}_{c}}\]is called vapour and above \[{{T}_{c}}\]is called gas.
(1) To rectify the errors caused by ignoring the intermolecular forces of attraction and the volume occupied by molecules, Vander Waal (in 1873) modified the ideal gas equation by introducing two corrections,
(i) Volume correction (ii) Pressure correction
(2) Vander Waal's equation is obeyed by the real gases over wide range of temperatures and pressures, hence it is called equation of state for the real gases.
(3) The Vander Waal's equation for n moles of the gas is,
\[\underset{\begin{smallmatrix} \text{Pressure correction} \\ \text{for molecular attraction} \end{smallmatrix}}{\mathop{\left( P+\frac{{{n}^{2}}a}{{{V}^{2}}} \right)}}\,\underset{\begin{smallmatrix} \\\text{Volume correction for } \\ \text{finite size of molecules} \end{smallmatrix}}{\mathop{[V-nb]}}\,=nRT\]
a and b are Vander Waal's constants whose values depend on the nature of the gas. Normally for a gas \[a>>b\].
(i) Constant a : It is a indirect measure of magnitude of attractive forces between the molecules. Greater is the value of a, more easily the gas can be liquefied. Thus the easily liquefiable gases (like \[S{{O}_{2}}>N{{H}_{3}}>{{H}_{2}}S>C{{O}_{2}})\] have high values than the permanent gases (like \[{{N}_{2}}>{{O}_{2}}>{{H}_{2}}>He)\].
Units of 'a' are : atm.\[{{L}^{2}}\,mo{{l}^{-2}}\] or atm.\[{{m}^{6}}mo{{l}^{-2}}\] or \[N\,{{m}^{4}}\,mo{{l}^{-2}}\](S.I. unit).
(ii) Constant b : Also called co-volume or excluded volume,
\[b=4{{N}_{0}}v=4{{N}_{0}}\left( \frac{4}{3}\pi {{r}^{3}} \right)\]
It's value gives an idea about the effective size of gas molecules. Greater is the value of b, larger is the size and smaller is the compressible volume. As b is the effective volume of the gas molecules, the constant value of b for any gas over a wide range of temperature and pressure indicates that the gas molecules are incompressible.
Units of 'b' are : \[L\,mo{{l}^{-1}}\] or \[{{m}^{3}}\,mo{{l}^{-1}}\](S.I. unit)
(iii) The two Vander Waal's constants and Boyle's temperature \[({{T}_{B}})\] are related as,
\[{{T}_{B}}=\frac{a}{bR}\]
(4) Vander Waal's equation at different temperature and pressures
(i) When pressure is extremely low : For one mole of gas,
\[\left( P+\frac{a}{{{V}^{2}}} \right)\,(V-b)=RT\] or \[PV=RT-\frac{a}{V}+Pb+\frac{ab}{{{V}^{2}}}\]
(ii) When pressure is extremely high : For one mole of gas,
\[PV=RT+Pb\]; \[\frac{PV}{RT}=1+\frac{Pb}{RT}\] or \[Z=1+\frac{Pb}{RT}\]
where Z is compressibility factor.
(iii) When temperature is extremely high : For one mole of gas,
\[PV=RT\].
(iv) When pressure is low : For one mole of gas,
\[\left( P+\frac{a}{{{V}^{2}}} \right)\,(V-b)=RT\] or \[PV=RT+Pb-\frac{a}{V}+\frac{ab}{{{V}^{2}}}\]
or \[\frac{PV}{RT}=1-\frac{a}{VRT}\] or \[Z=1-\frac{a}{VRT}\]
(v) For hydrogen : Molecular mass of hydrogen is small hence value of 'a' will be small owing to smaller intermolecular force. Thus the terms \[\frac{a}{V}\] and \[\frac{ab}{{{V}^{2}}}\] may be ignored. Then Vander Waal's equation becomes,
\[PV=RT+Pb\] or \[\frac{PV}{RT}=1+\frac{Pb}{RT}\]
or \[Z=1+\frac{Pb}{RT}\]
In case of hydrogen, compressibility factor is always greater than one.
(5) Merits of Vander Waal's equation
(i) The Vander Waal's equation holds good for real gases upto moderately high pressures.
(ii) The equation represents the trend of the isotherms representing the variation of PV with P for various gases.
(iii) From the Vander Waal's equation it is possible to obtain expressions of Boyle's temperature, critical constants and inversion temperature in terms of the Vander Waal's constants 'a' and 'b'.
(iv) Vander Waal's equation is useful in obtaining a 'reduced equation of state' which being a general equation of state has the advantage that a single curve can be obtained for all gases when more...
(1) Gases which obey gas laws or ideal gas equation \[(PV=nRT)\] at all temperatures and pressures are called ideal or perfect gases. Almost all gases deviate from the ideal behaviour i.e., no gas is perfect and the concept of perfect gas is only theoretical.
(2) Gases tend to show ideal behaviour more and more as the temperature rises above the boiling point of their liquefied forms and the pressure is lowered. Such gases are known as real or non ideal gases. Thus, a “real gas is that which obeys the gas laws under low pressure or high temperature”.
(3) The deviations can be displayed, by plotting the P-V isotherms of real gas and ideal gas.
(4) It is difficult to determine quantitatively the deviation of a real gas from ideal gas behaviour from the P-V isotherm curve as shown above. Compressibility factor Z defined by the equation,
\[PV=ZnRT\] or \[Z=PV/nRT=P{{V}_{m}}/RT\]
is more suitable for a quantitative description of the deviation from ideal gas behaviour.
(5) Greater is the departure of Z from unity, more is the deviation from ideal behaviour. Thus, when
(i) \[Z=1\], the gas is ideal at all temperatures and pressures. In case of \[{{N}_{2}}\], the value of Z is close to 1 at \[{{50}^{o}}C\]. This temperature at which a real gas exhibits ideal behaviour, for considerable range of pressure, is known as Boyle's temperature or Boyle's point \[({{T}_{B}})\].
(ii) \[Z>1\], the gas is less compressible than expected from ideal behaviour and shows positive deviation, usual at high P i.e. \[PV>RT\].
(iii) \[Z<1\], the gas is more compressible than expected from ideal behaviour and shows negative deviation, usually at low P i.e. \[PV<RT\].
(iv) \[Z>1\] for \[{{H}_{2}}\] and He at all pressure i.e., always shows positive deviation.
(v) The most easily liquefiable and highly soluble gases \[(N{{H}_{3}},\,S{{O}_{2}})\] show larger deviations from ideal behaviour i.e. \[Z<<1\].
(vi) Some gases like \[C{{O}_{2}}\] show both negative and positive deviation.
(6) Causes of deviations of real gases from ideal behaviour : The ideal gas laws can be derived from the kinetic theory of gases which is based on the following two important assumptions,
(i) The volume occupied by the molecules is negligible in comparison to the total volume of gas.
(ii) The molecules exert no forces of attraction upon one another. It is because neither of these assumptions can be regarded as applicable to real gases that the latter show departure from the ideal behaviour.
(1) At any particular time, in the given sample of gas all the molecules do not possess same speed, due to the frequent molecular collisions with the walls of the container and also with one another, the molecules move with ever changing speeds and also with ever changing direction of motion.
(2) According to Maxwell, at a particular temperature the distribution of speeds remains constant and this distribution is referred to as the Maxwell-Boltzmann distribution and given by the following expression,
\[\frac{d{{n}_{0}}}{n}=4\pi {{\left( \frac{M}{2\pi RT} \right)}^{3/2}}.{{e}^{-M{{u}^{2}}/2RT}}.{{u}^{2}}dc\]
where, \[d{{n}_{0}}=\] Number of molecules out of total number of molecules n, having velocities between c and \[c+dc\], \[d{{n}_{0}}/n=\]Fraction of the total number of molecules, M = molecular weight, T = absolute temperature. The exponential factor \[{{e}^{-M{{u}^{2}}/2RT}}\] is called Boltzmann factor.
(3) Maxwell gave distribution curves of molecular speeds for \[C{{O}_{2}}\] at different temperatures. Special features of the curve are : (i) Fraction of molecules with two high or two low speeds is very small.
(ii) No molecules has zero velocity.
(iii) Initially the fraction of molecules increases in velocity till the peak of the curve which pertains to most probable velocity and thereafter it falls with increase in velocity.
(4) Types of molecular speeds or Velocities
(i) Root mean square velocity (urms) : It is the square root of the mean of the squares of the velocity of a large number of molecules of the same gas.
\[{{u}_{rms}}=\sqrt{\frac{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}+.....u_{n}^{2}}{n}}\]
\[{{u}_{rms}}=\sqrt{\frac{3PV}{(m{{N}_{0}})=M}}=\sqrt{\frac{3RT}{(m{{N}_{0}})=M}}\]\[=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3kT}{m}}=\sqrt{\frac{3P}{d}}\]
where k = Boltzmann constant \[=\frac{R}{{{N}_{0}}}\]
(a) For the same gas at two different temperatures, the ratio of RMS velocities will be, \[\frac{{{u}_{1}}}{{{u}_{2}}}=\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\]
(b) For two different gases at the same temperature, the ratio of RMS velocities will be, \[\frac{{{u}_{1}}}{{{u}_{2}}}=\sqrt{\frac{{{M}_{2}}}{{{M}_{1}}}}\]
(c) RMS velocity at any temperature \[{{t}^{o}}C\] may be related to its value at S.T.P. as, \[{{u}_{t}}=\sqrt{\frac{3P(273+t)}{273d}}\].
(ii) Average velocity \[({{v}_{av}})\] : It is the average of the various velocities possessed by the molecules.
\[{{v}_{av}}=\frac{{{v}_{1}}+{{v}_{2}}+{{v}_{3}}+......{{v}_{n}}}{n}\]; \[{{v}_{av}}=\sqrt{\frac{8RT}{\pi M}}=\sqrt{\frac{8kT}{\pi m}}\]
(iii) Most probable velocity \[({{\alpha }_{mp}})\]: It is the velocity possessed by maximum number of molecules of a gas at a given temperature.
\[{{\alpha }_{mp}}=\sqrt{\frac{2RT}{M}}=\sqrt{\frac{2PV}{M}}=\sqrt{\frac{2P}{d}}\]
(5) Relation between molecular speeds or velocities,
(i) Relation between \[{{u}_{rms}}\] and \[{{v}_{av}}\]: \[{{v}_{av}}=0.9213\times {{u}_{rms}}\]
or \[{{u}_{rms}}=1.085\times {{v}_{av}}\]
(ii) Relation between \[{{\alpha }_{mp}}\] and \[{{u}_{rms}}\]: \[{{\alpha }_{mp}}=0.816\times {{u}_{rms}}\]
or \[{{u}_{rms}}=1.224\times {{\alpha }_{mp}}\]
(iii) Relation between \[{{\alpha }_{mp}}\] and \[{{v}_{av}}\]: \[{{v}_{av}}=1.128\times {{\alpha }_{mp}}\]
(iv) Relation between \[{{\alpha }_{mp}}\], \[{{v}_{av}}\] and \[{{u}_{rms}}\]
\[{{\alpha }_{mp}}\] : \[{{v}_{av}}\] : \[{{u}_{rms}}\]
\[\sqrt{\frac{2RT}{M}}\] : \[\sqrt{\frac{8RT}{\pi M}}\] : \[\sqrt{\frac{3RT}{M}}\]
\[\sqrt{2}\] : \[\sqrt{\frac{8}{\pi }}\] : \[\sqrt{3}\]
1.414 : 1.595 : 1.732
1 : 1.128 : 1.224 i.e., \[{{\alpha }_{mp}}<{{v}_{av}}<{{u}_{rms}}\]
(1) Hydrolysis constant : The general equation for the hydrolysis of a salt (BA),\[\underset{\text{salt}}{\mathop{BA}}\,+{{H}_{2}}O\] \[\rightleftharpoons \] \[\underset{\text{acid}}{\mathop{HA}}\,+\underset{\text{base}}{\mathop{BOH}}\,\]
Applying the law of chemical equilibrium, we get
\[\frac{[HA][BOH]}{[BA][{{H}_{2}}O]}=K\], where K is the equilibrium constant.
Since water is present in very large excess in the aqueous solution, its concentration \[[{{H}_{2}}O]\] may be regarded as constant so,
\[\frac{[HA][BOH]}{[BA]}=K[{{H}_{2}}O]={{K}_{h}}\]
where \[{{K}_{h}}\] is called the hydrolysis constant.
(2) Degree of hydrolysis : It is defined as the fraction (or percentage) of the total salt which is hydrolysed at equilibrium. For example, if 90% of a salt solution is hydrolysed, its degree of hydrolysis is 0.90 or as 90%. It is generally represented by ‘\[h\]’.
\[h=\frac{\text{Number of moles of the salt hydrolysed}}{\text{Total number of moles of the salt taken}}\]
(4) It is difficult to determine quantitatively the deviation of a real gas from ideal gas behaviour from the P-V isotherm curve as shown above. Compressibility factor Z defined by the equation,
\[PV=ZnRT\] or \[Z=PV/nRT=P{{V}_{m}}/RT\]
is more suitable for a quantitative description of the deviation from ideal gas behaviour.
(5) Greater is the departure of Z from unity, more is the deviation from ideal behaviour. Thus, when
(i) \[Z=1\], the gas is ideal at all temperatures and pressures. In case of \[{{N}_{2}}\], the value of Z is close to 1 at \[{{50}^{o}}C\]. This temperature at which a real gas exhibits ideal behaviour, for considerable range of pressure, is known as Boyle's temperature or Boyle's point \[({{T}_{B}})\].
(ii) \[Z>1\], the gas is less compressible than expected from ideal behaviour and shows positive deviation, usual at high P i.e. \[PV>RT\].
(iii) \[Z<1\], the gas is more compressible than expected from ideal behaviour and shows negative deviation, usually at low P i.e. \[PV<RT\].
(iv) \[Z>1\] for \[{{H}_{2}}\] and He at all pressure i.e., always shows positive deviation.
(v) The most easily liquefiable and highly soluble gases \[(N{{H}_{3}},\,S{{O}_{2}})\] show larger deviations from ideal behaviour i.e. \[Z<<1\].
(vi) Some gases like \[C{{O}_{2}}\] show both negative and positive deviation.
(6) Causes of deviations of real gases from ideal behaviour : The ideal gas laws can be derived from the kinetic theory of gases which is based on the following two important assumptions,
(i) The volume occupied by the molecules is negligible in comparison to the total volume of gas.
(ii) The molecules exert no forces of attraction upon one another. It is because neither of these assumptions can be regarded as applicable to real gases that the latter show departure from the ideal behaviour. 
(4) Types of molecular speeds or Velocities
(i) Root mean square velocity (urms) : It is the square root of the mean of the squares of the velocity of a large number of molecules of the same gas.
\[{{u}_{rms}}=\sqrt{\frac{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}+.....u_{n}^{2}}{n}}\]
\[{{u}_{rms}}=\sqrt{\frac{3PV}{(m{{N}_{0}})=M}}=\sqrt{\frac{3RT}{(m{{N}_{0}})=M}}\]\[=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3kT}{m}}=\sqrt{\frac{3P}{d}}\]
where k = Boltzmann constant \[=\frac{R}{{{N}_{0}}}\]
(a) For the same gas at two different temperatures, the ratio of RMS velocities will be, \[\frac{{{u}_{1}}}{{{u}_{2}}}=\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\]
(b) For two different gases at the same temperature, the ratio of RMS velocities will be, \[\frac{{{u}_{1}}}{{{u}_{2}}}=\sqrt{\frac{{{M}_{2}}}{{{M}_{1}}}}\]
(c) RMS velocity at any temperature \[{{t}^{o}}C\] may be related to its value at S.T.P. as, \[{{u}_{t}}=\sqrt{\frac{3P(273+t)}{273d}}\].
(ii) Average velocity \[({{v}_{av}})\] : It is the average of the various velocities possessed by the molecules.
\[{{v}_{av}}=\frac{{{v}_{1}}+{{v}_{2}}+{{v}_{3}}+......{{v}_{n}}}{n}\]; \[{{v}_{av}}=\sqrt{\frac{8RT}{\pi M}}=\sqrt{\frac{8kT}{\pi m}}\]
(iii) Most probable velocity \[({{\alpha }_{mp}})\]: It is the velocity possessed by maximum number of molecules of a gas at a given temperature.
\[{{\alpha }_{mp}}=\sqrt{\frac{2RT}{M}}=\sqrt{\frac{2PV}{M}}=\sqrt{\frac{2P}{d}}\]
(5) Relation between molecular speeds or velocities,
(i) Relation between \[{{u}_{rms}}\] and \[{{v}_{av}}\]: \[{{v}_{av}}=0.9213\times {{u}_{rms}}\]
or \[{{u}_{rms}}=1.085\times {{v}_{av}}\]
(ii) Relation between \[{{\alpha }_{mp}}\] and \[{{u}_{rms}}\]: \[{{\alpha }_{mp}}=0.816\times {{u}_{rms}}\]
or \[{{u}_{rms}}=1.224\times {{\alpha }_{mp}}\]
(iii) Relation between \[{{\alpha }_{mp}}\] and \[{{v}_{av}}\]: \[{{v}_{av}}=1.128\times {{\alpha }_{mp}}\]
(iv) Relation between \[{{\alpha }_{mp}}\], \[{{v}_{av}}\] and \[{{u}_{rms}}\]
\[{{\alpha }_{mp}}\] : \[{{v}_{av}}\] : \[{{u}_{rms}}\]
\[\sqrt{\frac{2RT}{M}}\] : \[\sqrt{\frac{8RT}{\pi M}}\] : \[\sqrt{\frac{3RT}{M}}\]
\[\sqrt{2}\] : \[\sqrt{\frac{8}{\pi }}\] : \[\sqrt{3}\]
1.414 : 1.595 : 1.732
1 : 1.128 : 1.224 i.e., \[{{\alpha }_{mp}}<{{v}_{av}}<{{u}_{rms}}\]
