Thermodynamics 

Introduction

Chemical thermodynamics deals with the relationship between various form of energy in a process. Thermodynamics deals with macroscopic properties. This chapter introduces a major subsidiary thermodynamic property, the Gibbs free energy which lets us express the spontaneity of a process in terms of the properties of the system. This chapter helps to explain why gases expand or diffuse.

System and Surrounding

(i) System : A specific portion of universe under study which is seperated from rest of the universe with a boundary is called system.

(ii) Surroundings : Rest of the universe which might be in a position to exchange energy and matter with the system is known as surrounding.

Types of System

(i) Open system : System can be open if it can exchange both energy and matter with surroundings.

(ii) Closed system : System can be closed if it can exchange energy but not matter with surroundings.

(iii) Isolated system : System can be isolated if it can neither exchange energy nor matter with surroundings.

Extensive Properties

The properties which depend upon mass of the substance is known as extensive properties i.e., mass, volume, internal energy, enthalpy etc.

Intensive Properties

The properties which are independent of mass of the substance is known as intensive properties i.e., temperature, pressure, density, refractive index.

Thermodynamic State of a System

A state is the condition of a system as specified by its physical properties. We can describe the state of a gas by quoting its pressure (p), volume (V), temperature (T), amount (n) etc. Variables like p, V, T are called state variables or state functions because their values depend only on the state of the system and not on how it is reached.

State Functions

The thermodynamic parameters which depends only on initial and final states of system is known as state function. i.e., internal energy(E), Enthalpy (H), entropy (S), Gibb's free energy (G).

Path Functions

The thermodynamic parameters where value does not depend merely on initial and final state but depends upon the path followed is known as path function. i.e., heat (q), work done (W).

Thermodynamic Process

The sequence followed to change one thermodynamic state of a system into another is called thermodynamic process. The types of thermodynamic processes are:

(a) Isothermal process: It is the process in which temperature is kept constant means temperature of initial and final state of system along with entire path of process is same.

(b) Isobaric process: It is the process in which pressure is kept constant for entire process.

(c) Isochoric process: It is the process in which volume is kept constant.

(d) Adiabatic process: The process in which heat transaction across boundary is not allowed.

(e) Reversible process and Irreversible process: In thermodynamics, a process is said to be reversible when energy change in each step of the process can be reversed by changing the variables such as pressure, volume or temperature acting on them. In such a process, the driving and opposing forces differ infinitesimally and the process can be reversed completely by increasing the opposing force by an infinitesimally small amount.

Any process which does not take place in the above mentioned manner is said to be an irreversible process. In an irreversible process the driving and opposing force differ by a large amount.

(f) Cyclic process: It is the process which run in close loop means process in which initial and final states are identical.

Internal Energy

Every substance is associated with definite amount of energy that is called internal energy. It is an extensive property and a state function. Internal energy of ideal gases is a function of temperature only.

Pressure-Volume Work

It is the work done when the gas expands or contracts against the external pressure. Consider a cylinder containing one mole of an ideal gas fitted with a frictionless and weightless piston having an area of cross-section A. The total volume of the gas is Vi and the initial pressure of the gas inside P.

Let the external pressure acting on the piston is pex. If the external pressure Pex is slightly greater than P piston moves downward till the pressure inside the cylinder becomes equal to Pex. Let this change be achieved in a single step and the final volume be Vf. During this compression, suppose the piston moves a very small distance Δl. Thus, the work done on the gas is given by,

• Work:- Work is said to be done when a force acting on a system displaces the body in its own direction.

dW = Fdx = PdV

W = P(Vf -Vi)

(a) If the gas expands, work is said to be done by the system. In this case Vf > Vi, therefore, W will be positive.

(b) If the gas is compressed, work is said to be done on the system. In this case Vf  < Vi, therefore, work done is negative.

Heat

The change in internal energy of a system can be brought about by the transfer of heat from the surroundings to the system or vice-versa. This exchange of energy between the system and surroundings is possible as a result of the temperature difference between them. This energy called heat is represented by Q.

First Law of Thermodynamics

First law of thermodynamics states the law of conservation of energy in a different manner. According to this law, whenever a quantity of one kind of energy disappears an equivalent amount of energy appears in some other form.

According to first law of thermodynamics,

ΔQ = ∆U + ∆W

Where, Q = Heat change
W = Work done
ΔU = Change in internal energy

ENTHALPY OF A SYSTEM

The quantity U + PV is known as the enthalpy of the system and is denoted by H. It represents the total energy stored in the system. Thus

H = U + PV

It may be noted that like internal energy, enthalpy is also an extensive property as well as a state function. The absolute value of enthalpy can not be determined, however the change in enthalpy can be experimentally determined.

ΔH = ΔU + Δ(PV)

Various kinds of processes:

(i) Isothermal reversible expansion of an Ideal gas: Since internal energy of an Ideal gas is a function of temperature and it remains constant throughout the process hence

ΔE = 0 and ΔH = ΔE + ΔPV

∵ΔE = 0

and P1V1 = P2V2 at constant temperature for a given amount of the gas

∴ ΔH= 0

Calculation of q and w:

∵ ΔE = q + w

For an Isothermal process, w = -q

This shows that in an Isothermal expansion, the work done by the gas is equal to amount of heat absorbed.

and w = - n RT ln(V2/V1) = - n RT ln(P1/P2).

(ii) Adiabatic Reversible Expansion of an Ideal gas:

∵ q = 0

∴ ΔE= -w.

Total change in the internal energy is equal to external work done by the system.

∴Work done by the system = ΔE= CvΔT.

and Cp-Cv = R

On dividing all the terms by Cv.

and ΔH = ΔE + PΔV.

Thus if T2>T1, w = +ve i.e. work is done on the system.

Thus if T2<T1, w = -ve i.e. work is done by the system.

Limitations of the first law. Need for the second law

A major limitation of the first law of thermodynamics is that its merely indicates that in any process there is an exact equivalence between the various forms of energies involved, but it provides no information concerning the spontaneity or feasibility of the process. For example, the first law does not indicate whether heat can flow from a cold end to a hot end or not.

The answers to the above questions are provided by the second law of thermodynamics.

Spontaneous and non – spontaneous process

If in the expansion of a gas the opposing pressure is infinitesimally smaller than the pressure of the gas, the expansion takes place infinitesimally slowly i.e. reversible. If however, the opposing pressure is much smaller than the pressure of the gas the expansion takes place rapidly i.e. irreversibly. Natural processes are spontaneous and irreversible.

Heat Capacity

Heat capacity is amount of heat require to raise the temperature of a system by unity. It is represented as "C". It is an extensive property and temperature dependent.

The increase in temperature is proportional to the heat transferred.

q = coeff. x ΔT

q = CΔT

Where, coefficient C is called the heat capacity.

C is directly proportional to the amount of substance.

Cm = C/n

It is the heat capacity for 1 mole of the substance.

• Molar heat capacity

It is defined as the quantity of heat required to raise the temperature of a substance by 1° (kelvin or Celsius).

• Specific Heat Capacity

It is defined as the heat required to raise the temperature of one unit mass of a substance by 1° (kelvin or Celsius).

q = C x m x ΔT

where m = mass of the substance

ΔT = rise in temperature.

• Relation Between Cp and Cv for an Ideal Gas

At constant volume heat capacity = Cv

At constant pressure heat capacity = Cp

At constant volume qv= CvΔT = ΔU

At constant pressure qp = Cp ΔT = ΔH

For one mole of an ideal gas

ΔH = ΔU + Δ (PV) = ΔU + Δ (RT)

ΔH = ΔU + RΔT

On substituting the values of ΔH and Δu, the equation is modified as

Cp ΔT = CvΔT + RΔT

or Cp-Cv = R

Hess’s Law

According to Hess's law, If a reaction takes place in several steps then its standard reaction enthalpy is the sum of the standard enthalpies of the intermediate reactions into which the overall reaction may be divided at the same temperature.

Ist method

C(s) + O2(g) ⟶ CO2(g) = ΔH

IInd method

C(s) + 1/2O2(g) ⟶ CO(g) = ΔH1

CO(s) + 1/2O2(g) ⟶ CO2(g) = ΔH2

According to Hess's law,

ΔH = ΔH1 + ΔH2

Application of Hess's Law

1. Calculation of enthalpy of formation.

2. Determination of standard enthalpies of reactions.

Bond Dissociation Energy

The energy required to break one mole bond of a particular type in gaseous molecule is known as bond dissociation energy. For example, we consider the dissociation of water,

H – OH(g) ⟶ H(g) + OH(g) = ΔH = 498 kJ/mol

Entropy

Entropy is a measure of degree of randomness or disorder in a system. Entropy is an extensive property and a state function.. Its value depends upon the amount of substance present in the system.

Free Energy (G)

Gibb's free energy is defined as,

ΔG = ΔH - TΔS

H is enthalpy, S is entropy and T is the temperature on Kelvin scale.

• Specific heat capacity of gases:- Specific heat capacity of a substance is defined as the amount of heat required to raise the temperature of a unit mass of substance through 1ºC.

(a) Specific heat capacity at constant volume (cv):- Specific heat capacity at constant volume is defined as the amount of heat required to raise the temperature of 1 g of the gas through 1ºC keeping volume of the gas constant.

Molar specific heat capacity, at constant volume (Cv), is defined as the amount of heat required to raise the temperature of 1 mole of gas through 1ºC keeping its volume constant.

Cv= Mcv

(b) Specific heat capacity at constant pressure (cp):- Specific heat capacity, at constant pressure, is defined as the amount of heat required to raise the temperature of 1 g of gas through 1ºC keeping its pressure constant.

Gram molecular specific heat capacity of a gas (Cp), at constant pressure, is defined as the amount of heat required to raise the temperature of 1 mole of the gas through 1ºC keeping its pressure constant.

Cp = Mcp

• Difference between two specific heat capacities – (Mayer’s formula):-

(a) Cp - Cv = R/J

(b) For 1 g of gas, cp - cv = r/J

(c) Adiabatic gas constant, γ = Cp/ Cv = cp/ cv

• Relation of Cv with energy:-

Cv= 1/m (dU/dT)

(a) Mono-atomic gas (3 degree of freedom):-

Total energy, U = mN 3 [(1/2) KT], Here m is the number of moles of the gas and N is the Avogadro’s number.

Cv = (3/2) R

Cp = (5/2) R

γ = Cp/ Cv = 5/3 = 1.67

(b) Diatomic gas:-

At very low temperature, Degree of Freedom (DOF) = 3

U = (3/2) mRT

Cv = (3/2) R,  Cp = (5/2) R

γ= Cp/ Cv = 5/3 = 1.67

At medium temperature, DOF = 5

U = (5/2) mRT

Cv = (5/2) R,  Cp = (7/2) R

γ = Cp/ Cv = 7/5 = 1.4

At high temperature, DOF = 7

U = (7/2) mRT

Cv = (7/2) R,  Cp = (9/2) R

γ = Cp/ Cv = 9/7 = 1.29

• Adiabatic gas equation:- PV γ = Constant

(a) Equation of adiabatic change in terms of T and V:- TV γ-1 = Constant

(b) Equation of adiabatic change in terms of P and T:- T γ P1-γ = Constant

• Comparison of slopes of an isothermal and adiabatic:-

(a) Slope of isothermal:- dP/dV = -P/V

(b) Slope of adiabatic:- dP/dV = -γP/V

(c) Adiabatic gas constant:- γ = Cp/Cv

As, Cp>Cv, So, γ>1

This signifies that, slope of adiabatic curve is greater than that of isothermal.

• Slope on PV diagram:-

(a) For isobaric process: zero

(b) For isochoric process: infinite

• Work done for isobaric process:- W = P(V2-V1)

• Work done for isochoric process:- W = 0

• Work done in isothermal expansion and compression:-

            W = 2.3026 RT log10Vf/Vi  (isothermal expansion)

W = - 2.3026 RT log10Vf/Vi  (isothermal compression)

• Work done during an adiabatic expansion:-

W = K/1-γ [Vf1-γ – Vi 1-γ] = 1/1-γ [P2V2-P1V1] = R/1- γ [T2-T1]

• Adiabatic constant (γ):- γ = Cp/Cv = 1+2/f, Here f is the degrees of freedom.

• Work done in expansion from same initial state to same final volume:-

• Wadiabatic < Wisothermal < Wisobaric

• Work done in compression from same initial state to same final volume:-

Wadiabatic < Wisothermal < Wisobaric

• Reversible process:- It is a process which can be made to proceed in the reverse direction by a very slight change in its conditions so that the system passes through  the same states as in direct process, and at the conclusion of which the system and its surroundings acquire the initial conditions.

Example:- All isothermal and adiabatic process when allowed to proceed slowly, are reversible, provided there is no loss of energy against any type of resistance. Friction, viscosity are other examples.

• Irreversible process:- A process which cannot be made to be reversed in opposite direction by reversing the controlling factor is called an irreversible process.

Example:-

(a) work done against friction

(b) Joule’s heating effect

(c) Diffusion of gases into one another

(d) Magnetic hysteresis

• Heat engine:- It is a device used to convert heat into mechanical energy

(a) Work done, W = Q1-Q2

(b) Efficiency:- Efficiency η of an engine is defined as the fraction of total heat, supplied to the engine which is converted into work.

η= W/ Q1 = [Q1- Q2]/ Q1 = 1-[Q2/Q1]

• Carnot engine – Carnot’s reverse cycle:-

(a) First stroke (isothermal expansion):- W1= RT1 loge[V2/V1]

(b) Second stroke (adiabatic expansion):- W2= R/γ-1 [T1-T2]

(c) Third stroke (isothermal compression):- W3= RT2 logeV3/V4

(d) Fourth stroke (adiabatic compression):- W4= R/γ-1 [T1-T2]

(e) Total work done in one cycle, W = W1+ W2+ W3+ W4 = R (T1-T2) loge (V2/V1)

• Efficiency of Carnot engine:- Efficiency η of an engine is defined as the ratio between useful heat (heat converted into work) to the total heat supplied to the engine.

η = W / Q1 = [Q1- Q2]/ Q1 = 1-[Q2/Q1] = 1- T2/T1

• Second law of thermodynamics:

• The second law of Thermodynamics helps us to determine the direction in which energy can be transformed. It also helps us to predict whether a given process or chemical reaction can occur spontaneously or not.
  (a) Clausius statement:- Heat cannot flow from a cold body to a hot body without the performance of work by some external agency.                                       (b) Kelvin’s statement:- It is impossible to obtain a continuous supply of energy by cooling a body below the coldest of its surroundings.                                 (c) Planck’s statement:- It is impossible to extract heat from a single body and convert the whole of it into work.
  Entropy Change: Entropy change is the state function and it is the ratio of heat change in a reversible process by the temperature.
  ΔS = qrev/T
  Thermodynamically irreversible process is always accompanied by an increase in the entropy of the system and its surroundings taken together while in a thermodynamically reversible process, the entropy of the system and its surroundings taken together remains unaltered.
  Physical Significance of Entropy: Entropy is the measure of disorderness because spontaneous processes are accompanied by increase in entropy as well as increase in the disorder of the system. Thus, increase in entropy implies increase in disorder.
  Some Other State Function: For a spontaneous process entropy change is positive and if it is zero, the system remains in a state of equilibrium. Two other functions are also there to decide the feasibility of the reactions like work function A and free energy change G.
  A = E – TS…….(i)
  G = H – TS…….(ii)
  And ΔA = ΔE - TΔS……(iii)
  ΔG = ΔH - TΔS………...(iv) (for a finite change at constant temperature)
  Since, ΔS = qrev./T Hence from eq. (i)
  ∴ΔA = ΔE – qrev………………..(v)
  and according to first law of Thermodynamics
  ΔE - qrev = wrev. …………….(vi)
  If during the change, work is done by the system, it would carry a negative sign,
  -wrev = ΔE – qrev…………….(vii)
  Comparing the equation (v) and (vii)
  -ΔA = wrev
  Since the process is carried out reversibly w represents the maximum work. It is thus clear that decrease in function A gives maximum work done that can be done by the system during the given change. The work function A is also called as Helmholtz function.
  From equation (iv)
  ΔG = ΔH - TΔS
  and ΔH = ΔE + PΔV
  ∴ΔG = ΔE + PΔV - TΔS
  Comparing it with eq. (iii)
  ΔG = ΔA + PΔV
  Since, ΔA is equal to – w, hence.
  ΔG = - w + PΔV.
  - ΔG = w- PΔV
  Hence decrease in free energy gives maximum work obtainable from a system other than that due to change of volume at constant temperature and pressure. This is called as Net Work.
  Net Work = w-PΔV = -ΔG
  The Net Work may be electrical work or chemical work.
  Criterion of spontaneity: For a spontaneous process ΔG should be -ve

• Refrigerator:- It is a device which is used to keep bodies at a temperature lower than that of surroundings.

• Coefficient of performance (β):- Coefficient of performance of a refrigerator is defined as the amount of heat removed per unit work done on the machine.

β = Heat removed/work done = Q2/W = Q2/[Q1- Q2] = T2/[T1- T2]

Coefficient of performance of a refrigerator is not a constant quantity since it depends upon the temperature of body from which the heat is removed.

For a perfect refrigerator, W = 0 or Q1= Q2 or β =∞

• Mean free path:- λ= 1/√2πd2ρn

Here ρn = (N/V) = number of gas molecules per unit volume

d = diameter of molecules of the gas.

• Heat added or removed:-

(a) For isobaric process:- Q = n CpΔT

(b) For isochoric process:- Q = n CvΔT

(c)For isothermal process:- Q = nRT loge (V2/V1)

(d) For adiabatic process: Q = 0

• Change in internal energy:-

(a) For isobaric process, ΔU = n CpΔT

(b) For isobaric process, ΔU = n CvΔT

(c) For isothermal process, ΔU = 0

(d) For adiabatic process, ΔU = -W = [nR (T2-T1)]/(γ-1)

• Mixture of gases:- n = n1+n2

M = n1M1+n2M2/ n1+ n2 = N1m1+N2m2/N1+N2

• Enthalpy (H):-

(a) H = U+PV

(b) At constant pressure:-

dH = dU + pdV

(c) For system involving mechanical work only:-

dH = QP (At constant pressure)

(d) For exothermic reactions:-

dH is negative

(e) For endothermic reactions:-

dH is positive

• Relation between dH and dU:-

dH = dU + dng RT

Here, dng = (Number of moles of gaseous products - Number of moles of gaseous reactants)

Summary

1. System : A part of universe which is under investigation.

2. Surroundings : The rest of the universe which is not a part of the system.

3. State of the system : The conditions of existence of a system when its macroscopic properties have definite values.

4. State functions : The thermodynamic quantities which depend only on the initial and final state of the system.

5. Energy is exchanged between the system and the surroundings as heat if they are at different temperatures.

6. The properties of the system whose value is independent of the amount of substance are called intensive properties. e.g., temperature, pressure, viscosity, surface tension, dielectric constant, specific heat capacity.

7. The properties of the system whose value depends upon the amount of substance present in the system are called extensive properties. e.g., mass, volume, surface area, energy, enthalpy, entropy, free energy, heat capacity.

8. Work is also a mode of transference of energy between system and the surroundings. Work done by the system on the surroundings is given by pΔV.

9. Internal energy (U) : The energy associated with the system at a particular conditions of temperature and pressure.

10. Enthalpy (H) : It is sum of internal energy and pressure-volume energy of the system at a particular temperature and pressure. It is also called heat content (H = E + pV).

11. Hess’s law : The enthalpy change in a particular reaction is the same whether the reaction takes place in one step or in a number of steps.

12. Bond enthalpy : The average amount of energy required to break one mole of the bonds of a particular type in gaseous molecules.

13. Entropy (S) : It is a measure of randomness or disorder of the system. Thus, the order is Gas > Liquid > Solid

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