Chemistry Podcast

Thursday, 15 January 2015

BSc2Year Chemistry of Elements of First Transition Series

Chemistry of Elements of First Transition Series

There are four types of orbital i.e. s, p, d and f. On the basis of electronic configuration-
·         s-block elements : last electron goes in s orbital.
·         p-block elements : last electron goes in p orbital.
·         d-block elements : last electron goes in d orbital.
·         f-block elements : last electron goes in f orbital.

Transition Elements: -  

                              The elements of d-block which act as bridge or transit point between s and p-block elements is known as transition elements.
d-block elements have four series :

1.      First transition series :- 

                                      This series contains elements from atomic number 21 (scandium) to atomic number 30 (Zinc). It is called 3d series because last electron goes in 3d orbital. And it is present in fourth period of periodic table.
Elements of First Transition Series

2.      Second transition series :- 

                                           This series contains elements from atomic number 39 (Yttrium) to atomic number 48 (cadmium). It is called 4d series because last electron goes in 4d orbital. And it is present in fifth period of periodic table.

3.      Third transition series :- 

                                           This series contains elements 57 (Lanthanum) and from atomic number 72 (Hafnium) to atomic number 80 (Mercury). It is called 5d series because last electron goes in 5d orbital. And it is present in sixth period of periodic table.

4.      Fourth transition series :- 

                                      This series contains Actinium (89) and element with atomic  number 104 (Rutherfordium) and all above series elements. It is called 6d series. And it is present in sixth period of periodic table. All the elements other then Actinium in this series are synthesized.

Characteristic Properties of d-block elements : 

                                                               Main characteristics of d-block elements are as follow:

Physical state and metallic properties : 

                                                     All d-block elements are solid except Mercury. Mercury is present in liquid state. Atoms of d-block elements have maximum 2 electrons in outermost shell so they show metallic character. Unlike s-block elements these are rigid, malleable and ductile. D-block elements are good conductor of heat and electricity. and have metallic luster.

Melting point and Boiling point :  

                                               These have high Melting and high Boiling point due to strong bond between elements. Zn, Cd and Hg have low Melting and low Boiling point due to completed sub-orbit.

Atomic radius :  

                      In periods of d-block elements, atomic radius generally decreases with increase in atomic no. It is due to increase in nuclear charge of atoms in a period.

Ionic radius : 

             d-block elements form cations which are smaller than its corresponding atoms. Ionic radius generally decreases across a period with increase in atomic no.

Atomic volume :  

                     Atomic volume of the d-block elements is low as compared to near s and p block elements. Atomic volume decreases with increase in atomic number in a period, but after acquiring a minimum volume it increases due to increased screening effect.

Density : 

          In a period of d block elements gradual increase in density take place with increase in atomic no. And density decreases after acquiring maximum value due to increase in atomic radius and atomic volume.

Standard Electrode Potential : 

                                          Standard Electrode Potential of hydrogen is assumed to be zero. Standard Electrode Potential of other electrodes is determined relative to Standard Electrode Potential of hydrogen.

Ionization Potential :  

                              Ionization Potential of d-block elements is intermediate between s and p block elements. Ionization Potential value increases with increase in atomic number across a period.

Electropositive character : 

                                      d-block elements are electropositive, but as compared to s-block elements these are less electropositive. So form electrovalent compound with more difficulties as compared to s-block elements.

Electronegativity : 

                         Electronegativity of transition elements increases with increase in nuclear charge. Last element of each period have complete d sub-shell so have low electronegativity value because screening effect of complete d sub-shell is more than incomplete d sub-shell.

Oxidation states : 

                        d-block elements show variable oxidation state. Electrons in d orbital are responsible for variable oxidation state. +2 oxidation state occur on removal of two s electron from outermost shell. Other oxidation state (i.e. more than +2) require removal of electron of d orbital of penultimate shell.

Complex formation : 

                          Due to incomplete d sub-shell of d-block elements they are able to form complex compounds. Central ion is capable to accept lone pair of electron donated by ligands to form complex. Empty orbital in a atom adjust these lone pair of electrons. According to Pauling, Transition elements either have empty orbitals or they produced empty orbital when surrounded by the ligands.

Catalytic character :  

                          d-block elements or their compounds are used as a catalyst in many chemical reactions. Generally Fe, Cr, Pt, Ni, V2O5, Mn etc., are used as a catalyst. A essential property of a catalyst is to form an unstable intermediate. Due to various oxidation state transition elements form intermediate easily. Good catalyst have free valency on its surface. Catalyst are used in finely divided form to increase surface area to attain increased number of free valency.

Alloy formation : 

                       Due to approximately equal size of d-block elements they form alloy.

Reactivity : 

                d-block elements are less reactive due to higher ionization potential (because of smaller atomic size), so hydration of cations of d-block elements is difficult and have high heat of sublimation.

Formation of Interstitial or non-stoichiometric compounds :  

                                                                                              Compounds which do not follow valency rule are known as non-stoichiometric compounds. These type of compound are formed due to entrance of the non-metallic atoms into the interatomic spaces of metal atoms. Eg. TiH1.7, VSe0.98, FeO0.94, etc.

Coloured Ions : 

                   Compounds or Ions of transition elements appear to be coloured due to these reasons-
(i)                 d-d transition
(ii)               charge transfer

(i)                 d-d transition : 

                                          In a compound of d-block element colour depend on transition of electron from lower energy level to higher energy level and in ion of d-block elements, due to unpaired electron in d orbital, it get split into two parts at the time of complex formation.  These two orbital differ in their energies. Electron absorb radiations in visible region and transition of electron occur from lower energy level to higher. So colour of transition elements depends on d-d transition. Also If no of d-d transition is higher then colour of ion is darker.

(ii)               charge transfer : 

                                          d-d transition is not possible in PbO2, MnO4-, Cr2O72-, Sn2+ and Sn4+ etc. No unpaired d electrons are found in them. In these ions transition of electrons occur from orbital of one atom to orbital of another atom by absorbing radiation to produced dark colour. This transition occur in UV region (1800  - 4000 ) is known as charge transfer transition.

Magnetic Properties : 

                              On the basis of magnetic behavior they are classified into five categories
(i)                 Diamagnetic
(ii)               Paramagnetic
(iii)             Ferromagnetic
(iv)             Antiferromagnetic
(v)               Ferrimagnetic

(i)                 Diamagnetism: 

                                      The type of substance which when placed in a magnetic field the intensity of magnetic field decreases as compared to the intensity in vacuum the substances are known as diamagnetic substances and this property is called diamagnetism.
Diamagnetism is due to the presence of paired electrons hence found in all substances except hydrogen. Magnetic lines of force tends to move away from the substances so these substances are repelled by magnetic field and these substances align themselves at right angle to magnetic field. Diamagnetism is occurs due to the presence of paired electrons. In this type of substances magnetic moment produced by one electron is cancelled by another which is equal and opposite to first one. So no magnetic moment present in substances having paired electrons.
Diamagnetism increases with increase in atomic number.

(ii)               Paramagnetic : 

                                      It is found in substances which have unpaired electrons like transition elements. This type of substances having permanent magnetism. When this type of substances placed in external magnetic field, they aligns themselves in the direction of magnetic field. Para-magnetism occurs due to motion and spin of electrons. Paramagnetism decreases with increase in temperature. These types of substances are attracted in magnetic field. If number of unpaired electrons in a substance is n then magnetic moment.
                   Number of unpaired electrons in substances is calculated by the magnetic.

(iii)             Ferromagnetism: 

                                        Substances which have very high paramagnetic character are known as ferromagnetic substances. And found in some alloys or compounds of Fe, Co, Ni, Mn. These types of substances even remain magnetic characters after removing from the external magnetic field. This type of substances contains tiny magnets in them which arrange randomly. On placing this type of substances in magnetic field these tiny magnets arrange them in one direction so these show very high magnetism.

(iv)             Antiferromagnetism: 

                                      These types of substances do not show magnetism (Paramagnetism) even they have unpaired electrons like MnF2, MnO.

 Properties of elements of first transition series:

Elements from Sc (21) to Zn (30) are known as elements of first transition series (i.e. 3d transition series). In the atoms of first transition series last electron goes in 3d sub-sell.

Binary Compounds

The compounds which are formed by two types of elements and ions are known as binary compounds. Elements of the first transition series react with so many non metallic elements like carbon, oxygen, phosphorus, Sulphur and nitrogen etc. to form binary compounds. Oxides, halides, sulphides, carbides are main binary compounds of first transition series.

Oxides: -  

          When element of first transition series heated with oxygen at high temperature metal oxides are formed. Important oxides of first transaction series are as follows.
Acidic  oxides : V205, CrO3, MnO3.
Basic Oxides : Sc2O3, TiO, Ti2O3, VO, V2O3, MnO, CrO, FeO, Fe2O3, Fe3O4, CoO, NiO, Cu2O.
Amphoteric Oxides : TiO2, VO2, Cr2O3, CrO2, Mn3O4, Mn2O3, MnO2, CuO, ZnO.
Main properties of oxides :
1.      Acidic, Basic or Amphoteric nature : As the oxidation No. of metal increases its acidic nature of oxides also increases.

Oxides of Vanadium              VO                  V2O3              VO2                V2O5
Oxidation No. of vanadium   +2                    +3                    +4                    +5
Nature of oxides                     Basic               Basic        Amphoteric            Acidic

2.      Solubility : Amphoteric and basic oxides are soluble in acids which do not act as oxidants. Acidic oxides form oxy acids in water and oxy salts in bases to get dissolved.
3.      Reducing nature of oxides : Electron donor substances act as  reductant. Substances (atoms, ions and molecules) which donate their electrons easily have higher reducing character.
Halides : Elements of first transition series (3d series) react with halogens at high temperature to form halides. Order of reactivity of halogens with the metals is as given below.
                                                F2 >Cl2> Br2> I2
Generally fluorides formed in higher oxidation states. Formation of halides require high activation energy so this reaction occurs at high temperature.
Properties of halides
1.      Transition metal halides are less volatile and more susceptible to hydrolysis. Metal halides in higher oxidation states have high tendency to undergo hydrolysis.
TiCl4 + 2H2O ® TiO2 + 4HCl
2.      In lower oxidation states more stable oxides are formed.
Eg. :ZnCl2, VCl2 etc.
3.      Fluorides are ionic in nature. Chlorides, Bromides and iodides have both ionic and covalent character.
Fluoride > Chloride > Bromide > Iodide

Sulphides : 

                Sulphides are obtained on heating metal with sulphur. Metal sulphides are also produced on reacting aqueous solution of metal salts with Na2S or H2S.
Properties of sulphides :
1.      First transition metal sulphides are mainly dark colored or black.
CuS – Black
NiS – Black
CoS – Black
2.      Sulphides are insoluble in water.
3.      They get oxidised to metal sulphates on oxidation.
4.      Some sulphides such as CoS, NiS and FeS behave as an alloy or exhibit the semi-metallic character.
5.      FeS2, CoS2 contain discrete S2 units with S-S bonding.
Carbides : carbides are produced on heating transition metals or metal oxides with carbon at very high temperature about 2000-2200°C.
Carbides formed by first transition series elements are of two types-
(a)    Salt like carbides : These carbides are also known as electrovalent carbides or ionic carbides. Metals like Sc, Cu, Zn etc. form this type of carbides.
(b)   Interstitial Carbides : These carbides are also known as metallic carbides. Metals like Ti, V, Mn, Fe, Co form such type of carbides. These type of carbides are obtained on heating a carbon and metal..
Properties of interstitial carbides
1.      interstitial carbides are extremely hard.
2.      interstitial carbides have high melting point.
3.      interstitial carbides have high electrical conductivity.
4.      interstitial carbides show inertness to chemical reactions.

BSc2Year Thermodynamics


BSc Chemistry Notes on Thermodynamics


                           study of the inter-relations of various forms of energy in a system is called thermodynamics.

Limitation of thermodynamics

        i.            laws of thermodynamics are not applicable to small particles like individual atoms or molecule, but laws can be applied to macroscopic system or very large system.
      ii.            Thermodynamics does not gives information about rate at which a given chemical reaction/process may proceed and also time for this change.

Thermodynamic system

                                  any specified portion of the universe or matter, real or imaginary, separated from the rest of the universe, which is selected for the thermodynamic treatment is called a system.


Leaving the system the rest of the universe, which may exchange matter or energy or both with the system is called surroundings.

Types of system

1.      Open system:-a system which can exchange energy as well as matter with its surroundings.
Ex:- water in a open beaker.
2.      Closed system:- when a system can exchange only energy and not matter with its surroundings.
Ex:- a chemical reaction taking place in a closed vessel can exchange only heat with surrounding
3.      Isolated system:-  a system which can neither exchange matter nor energy with its surrounding.
Ex:- a reaction in closed vessel which can’t exchange heat or matter.

Homogeneous system:-   

                                  A system which is uniform throughout i.e. for chemicals it must have same composition throughout. Homogeneous system consists of only one phase.
Ex:- Glucose dissolved in water.

Heterogeneous system:- 

                                   A system which is not uniform throughout i.e. it may consists two or more phases in equilibrium. Its phases are separated from one another by bounding surfaces.
EX:- ice in water.

Macroscopic system:-   

                          A system which consist of a large no. of atoms, particles, molecules, radicals.

Macroscopic properties:-  

                                   properties of macroscopic system is known as macroscopic properties. EX:- pressure, temperature, volume, composition, density, viscosity, surface tension, etc.
Change of any macroscopic property changes the state of the system or vice-versa.

State of a system:-  

                         It is defined by the macroscopic properties. When the macroscopic properties a of a system have specific or definite value it is said that the system is in definite state.

Thermodynamic Equilibrium:-  

                                   If macroscopic properties like temperature, pressure, volume composition etc. do not change with time.
Types of thermodynamic equilibrium

1.      Thermal equilibrium:-  

                              A system whose temperature do not change along with the temperature of the surroundings.

2.      Mechanical equilibrium:-   

                                           A system which do not perform any mechanical work.

3.      Chemical equilibrium:-  

                                    A system whose chemical composition does not change with time (remains same throughout).

Physical properties of the system 

                                                 Physical properties of the system are of two types-

1.      Extensive property

                                   This property depends on quantity or amount of matter present in the system.
Ex:- Mass, energy, no. of moles, enthalpy, entropy etc.

2.      Intensive property

                                    This property do not depends on quantity or amount of matter present in the system.
Ex:- temperature , pressure, density, viscosity, surface tension etc.

State function:- 

                    It is the property of the thermodynamic system whose value is definite for a particular state of the system. When a change is brought about in this particular state of system, change in state function also occurs. It depends only on initial and final state of the system.
Ex:- pressure, temperature, volume, energy are state function.

Path function:-  

                   When a system passes from one state A to another state B depends on the nature of the path followed, not on initial and final state.
Ex:- work done is path function.

Thermodynamic process

                                  If a thermodynamic system changes from one state to another state the operation is known as thermodynamic process.

Types of process-

1.      Isothermal process:- in this process temperature of the system remains constant throughout the process i.e. dT=0
2.      Adiabatic process:- in this process no heat enters or leaves the system during any stage of the process i.e. dH=0
3.      Isobaric process:-  in this process pressure of the system remains constant throughout the process i.e. dP=0
4.      Isochoric process:- in this process volume of the system remains constant throughout the process i.e. dV=0

Cyclic process or cycles:- 

                                    When a system return to its initial state after completing the process in various stages, that is system has completed one cycle and process is known as cyclic process.

Reversible process:

                              If a thermodynamic process is carried out infinitesimally slowly so that at every stage of it, the system in temperature and pressure remains in equilibrium with surrounding, This type of process is called reversible process.

Irreversible process:-

                              If a thermodynamic process is not carried out infinitesimally slowly so that at every stage of it, the system do not remains in equilibrium with surrounding, This type of process is called irreversible process.

Endothermic process: -

                              The process in which amount of energy is absorbed to carried out a reaction, known as endothermic process.

Exothermic process:

                             The process in which amount of energy is evolved to carry out a reaction, known as endothermic process.
Heat: - Tt is a way to transfer of energy from one body to another by the difference in the temperature between these bodies.
Work:- Tt is a link between the system and its surroundings for the transfer of heat energy.

Signs for heat and work

Ø  q is positive when heat is evolved
Ø  q is negative when heat is absorbed
Ø  w is positive when work is done by the system
Ø  w is negative when work is done on the system

First law of Thermodynamics

First law of thermodynamics states that “Energy can neither be created nor destroyed but can only be transformed from one form to another.” This law is also known as law of conservation of energy.
Let, a system achieves a state B from A along the path 1. Let in this process heat absorbed by the system is Q1 joules and work done by the system is W1 joules.
Difference in energy = Q1 – W1
Again,  a system achieves a state B from A along the path 2. Let in this process heat absorbed by the system is Q2 joules and work done by the system is W2 joules.
Difference in energy = Q2 – W2
Similarly for path 3 & 4
Difference in energy = Q3 – W3
Difference in energy = Q4 – W4         and so on…..
We get,
             Q1 – W1 = Q2 – W2 = Q3 – W3 = Q4 – W4
From it we come to know difference in energies for all path connecting A & B is same so we can write as (Q-W)
 Where,                        Q = energy absorbed by the system
                        W = Energy consumed by the system in doing work
                        Q - W = change or increase in internal energy of the system when it changes from state A to B and is independent of the path followed 
 If U = internal energy
Then, UA and UB shows energies at state A & B.
            DU = UB-UA = Q-W            [mathematical representation of First law of thermodynamics]

First law of thermodynamics in relation with work and heat

Suppose, a system be given q heat. It is used in raising the internal energy of the system from initial state (UA) to final state (UB) and doing work W by the system on the surroundings
Then,   from first law of thermodynamics
            q = (UB-UA) +w
            q = DU + w
It is mathematical statement of the first law of thermodynamics.
If changes in energy is infinitesimally small then
dU = dq - dw
dU = dq – PdV
in a cyclic system, when system returns to its initial state i.e. UB= UA or DU = 0, there will be no change in internal energy
then,    q-w = DU
or,        q-w = 0
or,        q = w

Heat changes at constant volume: - 

                                                  When the process is carried out at constant volume there will be neither expansion nor contraction in volume of gas. At this condition no work is done by the system i.e. w=0; put it in First law of Thermodynamics, we get
            q = DU + w
            qv = (DU)v
It means at constant volume, Heat absorbed is utilised in increasing the Internal Energy of the system.

Heat changes at constant pressure:-  

                                               Let at constant pressure P, a change of state of a system is brought about from initial state 1 to final state 2 by the absorption of qp amount of heat.
In this change, volume increase from V1 to V2
Then,   increase in volume is given by V2 – V1 = DV     ……………………………………………(1)
Then,   work done by the system in expansion is given by      W = P(V2 – V1)………….(2)
Now,    According to first law of thermodynamics
qp = (DU)p + W    ………………………………………………………………………..(3)
qp = (DU)p + P(V2 – V1)    ……………………………………………………………….(4)
qp = (DU)p + P(DV)       ……………………………………………………………….(5)
if U1 & U2 are values of internal energies in initial and final states of system resp.
then,    change in internal energy is given by   U2 – U1 = (DU)p    ………………………….(6)
from eq. (4) & (6),   we get     
                        qp = (U2 – U1) + P(V2 – V1)
                         qp = (U2 + PV2) – (U1 + PV1)   …………………………………………………….(7)
Since,   P & V are definite properties of state of system and U is also a definite property. it is clear that like internal energy U,  (U+ PV) is also a definite property of state  and depends only on states of system and not on path by which this state is achieved. This thermodynamic property is denoted by H, i.e. (H= U + PV) and is known as enthalpy, Total energy or Heat content at constant pressure of the system. From eq. (vii) & (H= U + PV) we get,
                        qp = H2 – H1 = (DH)p    …………………………………………………………(8)

Relation between DH and DU :- From eq. (8) & (5) we get,
                        qp = (DU)p + P(DV)
                        qp = H2 – H1 = (DH)p
                        (DH)p = (DU)p + P(DV)    ……………………………………………………(9)

Enthalpy of vaporisation:- 

                                 It is defined as change in enthalpy (DH) when liquid evaporates into vapour state or vapours condenses into liquid state.

Enthalpy of fusion:- 

                        It is defined as change in enthalpy (DH) when solid melts into liquid state or liquid freezes into solid state.

Heat Capacity:-   

                   The amount of heat required to raise the temperature of known quantity of substance or system by 1°C. If q is heat added to the system to raise the temperature from T1  to T2 then, heat capacity, C of the system between T2  & T1 is given by
For very small quantity of heat dq to be added to the system, it rises a small raise in temperature by dT. Then
When, the amount of the substance is 1 gram molecule then heat capacity is known as the molar heat capacity.

Unit of heat capacity:-   

                                   In general
                                   In SI,

Heat capacity at constant volume:-  

                                               The amount of heat required to raise the temperature of known quantity of substance or system by 1°C at constant volume is known as Heat capacity at constant volume (CV).
From first law of thermodynamics
dq = dU + PdV
at constant volume dV = 0 then,                               

Heat capacity at constant pressure:-  

                                                The amount of heat required to raise the temperature of known quantity of substance or system by 1°C at constant pressure is known as Heat capacity at constant pressure (Cp).
Cp is always larger then Cv by an amount equal to P-V work done so,
                        Cp = Cv + external work
Since,  work done in expansion = P (V/T)
 Putting this value in eq. (xvi) we get
But,                       H = U + PV
Differentiate this w.r.t. T at constant pressure
By comparing this eq. with (xvii) we get,

Relation Between Cp & Cv :- 

                                              As we know Cp is always greater than Cv.
Therefore, we shall find Cp-Cv
As we know                
But                                   H = U + PV
Differentiating this eq. w.r.t. temperature at constant pressure
Combining eq. (20) and (21) we get
Now we have to relate 1st and 3rd terms of eq. (22). For this we consider V and T as independent variables out of P, V and T, then
                                    U = f (T, V)
Dividing both sides by dT, and consider pressure constant, we get
Now this eq. is substituted in eq. (22) we get

For an ideal gas            PV = RT
Differentiated with respect to T at constant pressure then
For ideal gas        
Hence, with the help of eq. (27), eq. (26) will become
                        CP – CV = R   ………………………………………………..(28)

Joule’s Law or Joule-Thomson Effect :- 

                                                          J.P. Joule and W. Thomson in year 1852-1862 made Joule-Thomson Law or effect.
According to this law when a gas is made to expand adiabatically from high pressure to a extremely low pressure cooling is produced, i.e. gas gets cooled. This phenomenon is known as Joule-Thomson Effect or Joules law.
All gases behaves like this except Hydrogen and Helium i.e. they get heated instead of cooling is produced.
The cooling effect in Joule-Thomson effect is due to decrease in kinetic energy of the gas molecules because a part of this energy is used in overcoming the forces of attraction existing between the molecules of the gas in expansion. For ideal gases there is no force of attraction between the gas molecules and therefore on expansion in vacuum through the porous plug, neither cooling nor heating is produced. i.e., neither absorption nor evolution of heat takes place and therefore no external work to separate the molecules and so,
                                                Q = 0, w = 0, and therefore DU = 0

Joule-Thomson Coefficient:- 

                                          Enthalpy is a definite property depending upon the state of the system. Hence , dH is complete differential. Suppose that P and T are variables, then
The enthalpy remains constant (dH = 0) in adiabatic expansion of the real gases therefore, in above eq. put dH = 0

Inversion Temperature: -

                                    As per joule-Thomson coefficient
In above eq. when value of 2a/RT > b, value of mJ.T. is +ve.It means joule Thomson effect will be cooling the gas or substance.
When 2a/RT = b, the value of mJ.T.=0 so joule Thomson effect will be nil. When 2a/RT < b. value of mJ.T. is -ve. It means joule Thomson effect will be warming the gas or substance.
We know the value of a, b and R are constants so mJ.T­ is depend on temperature only.
The temperature at which joule Thomson coefficient changes its sign from +ve to –ve or vice versa is known as inversion temperature. At this temperature mJ.T =0.
Or                                            2a/RTi = b
Or                                            Ti = 2a/Rb
Where Ti = inversion temperature.
            a and b are vander walls constants
Work done in the expansion of ideal gases under isothermal conditions for reversible process:-
Let an ideal gas is enclosed in a cylinder fitted with weightless and frictionless piston
Also suppose that cylinder is placed in thermostat so its temperature will remain constant throughout the process of expansion therefore the gas is in thermal equilibrium with surroundings. if external pressure on piston be P equal to the pressure of the gas within the cylinder in the beginning so therefore the piston is at rest. Pressure P is lowered by infinitesimally small amount dP. The new pressure is now (P-dP). So gas expands by infinitesimally small volume dV. Then the volume of the gas now becomes (V+dV). so piston is pushed up until it comes to rest (when internal and external pressure becomes equal).
In this process gas does infinitesimally small work on the piston.
If all the above process will be repeated again. Then infinitesimally small work done on the piston again second time.
(We know for isothermal expansion q = w i.e. work is done by the gas is equal to heat absorbed by the gas from the surroundings)
If the all above process are continued then
                        q = w = (P-dP) ´ dV
                           = PdV – dPdV
Neglecting the very small term dPdV we get w = PdV.
The total work done by the gas In the process of expansion will be the sum of a continuous series of PdV terms, as volume increase from V1 initial state to final state V2.
Since for a reversible process the external pressure is always only infinitesimally lower then the pressure of the gas itself, ideal gas pressure P = (RT/V) can be substituted for P in above eq.
If the gas is an ideal gas then at constant temperature P1V1 = P2V2 the above equation becomes
Where P1 and P2 are the external pressure in initial and final state of the gas respectively.
For the isothermal reversible expansion of n moles of an ideal gas above equation becomes
                        q = +W = nRT ln V2/V1 = nRT ln P1/P2
if expansion occurs (increase in volume) V2>V1 then work is done by the system is +ve.
if compression occurs (decrease in volume) V1>V2 then work is done on the system is -ve.


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