Subject Chemistry Paper No and Title 10: Physical

CHEMISTRY

Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics)

Module No. 6: Thermochemistry and Hess’s law

Subject Chemistry

Paper No and Title 10: Physical Chemistry- III (Classical

Thermodynamics, Non-Equilibrium

Thermodynamics, Surface chemistry, Fast kinetics)

Module No and

Title

6, Thermochemistry and Hess’s law

Module Tag CHE_P10_M6

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TABLE OF CONTENTS

1. Learning outcomes

2. Introduction

3. Change in thermodynamic quantities during a chemical reaction

3.1 Change in internal energy of a chemical reaction

3.2 Change in enthalpy of a chemical reaction

4. Relation between the enthalpy at constant volume and enthalpy at constant pressure

5. Enthalpy of a chemical reaction

6. Determination of enthalpies of a reaction

7. Kirchhoff equation: Variation of enthalpy of reaction with temperature

8. Flame and explosion temperatures

9. Hess’s law

10. Extension of Hess’s law

11. Application of Hess’s law

12. Summary

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1. Learning Outcomes

After studying this module you shall be able to:

Depict the change in thermodynamic quantities during a chemical reaction

Derive the relation between enthalpy at constant pressure and enthalpy at

constant volume

Know about Kirchhoff equation

Learn about Flame and Explosion temperature

Learn about Hess’s law of constant heat summation

Know how Hess’s law is extended to find entropy and free energy change

Know the applications of Hess’s law

2. Introduction

The branch of chemistry which deals with the energy changes involved in chemical reaction

is called thermochemistry. It is the study of energy and heat associated with chemical

reactions. Thermochemistry focuses on the energy changes, primarily on the system’s energy

exchange with its surroundings. Thermochemistry can predict the quantities of reactants and

products during the course of the reaction. It is also useful in prediction of the spontaneity of

the reaction. It merges the concepts of thermodynamics with the concept of energy in the

form of chemical bonds. The quantities like enthalpy, heat capacity, entropy, free energy,

heat of formation are mainly calculated through this. According to thermochemistry, the

change in energy which occurs in chemical reaction is mainly because of the change of bond

energy, i.e., it results from the breaking of bonds in the reactants and formation of new bonds

in products.

Thermochemistry is based on two laws:

Lavoisier and Laplace’s law: This law states that the change in energy

accompanying any transformation is equal and opposite to change in energy

accompanying the reverse process.

Hess’s law: This law states that the change in energy accompanying any

transformation is same whether the process occurs in one step or many steps.

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Lavoisier, Laplace and Hess have also done investigation on specific heat and latent

heat.

3. Change in thermodynamic quantities during a chemical reaction

3.1 Change in internal energy of a chemical reaction

Consider a chemical reaction during which the temperature and volume is kept constant, i.e.,

dV=0. Thus, work done (w) is also equal to zero as w=PdV. Therefore, equation of the First

law (viz., U = q + w) becomes:

U = vq …(1)

where vq stands for the heat exchanged at constant volume.

Let UR be the internal energy of the reactants and UP be the internal energy of the products,

thus change in internal energy will be

ΔU= UP – UR = qv …(2)

3.2 Change in enthalpy of a chemical reaction

The heat exchanged at constant pressure is known as the enthalpy change. Suppose pq be the

heat exchanged during a chemical reaction which is occurring at constant pressure. Therefore,

pH q …(3)

Let HR be the enthalpy of the reactants and HP be the enthalpy of the products, thus change in

enthalpy will be

ΔH = HP – HR = qP …(4)

Thermochemistry enables us to predict the amount of heat that would be evolved or absorbed

in a process without actually performing a tedious set of experiments in the laboratory. The

energy changes for the processes which are not feasible experimentally can also be calculated

through thermochemistry.

Sign convention:

Reactions in which heat is absorbed by the system are called endothermic reactions. In

such reactions HP > HR, so ΔH is positive. Since the energy of the system also increases by

the absorption of heat thus ΔU is also positive in endothermic reactions. While the reactions

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in which heat is evolved, are called exothermic reactions. In such reactions HP < HR so ΔH

is negative. In such reactions ΔU is also negative.

4. Relation between enthalpy at constant volume (qv) and enthalpy at

constant pressure (qp)

The relation between ΔH and ΔU is given by:

H U P V …(5)

where ΔV is the volume change taking place in a reaction.

Since vq U and pq H , therefore we can write equation (5) as

p vq q P V …(6)

Now writing the above equation in simplified manner;

For n moles of an ideal gas,

PV = nRT …(7)

Suppose n1 be the number of moles for gaseous reactants and n2 be the number of moles of

gaseous products. Let n2 > n1. Thus increase in the number of moles is given by 2n – 1n =

gn . The corresponding increase in volume ( V ) will be given by (V/n) gn . Therefore,

g gP V P(V / n) n RT n …(8)

Thus, gP V RT n …(9)

Substituting equation (8) in equation (6),

p v gq q n RT …(10)

In the above equation gn stands for the difference between the number of moles of

gaseous products and gaseous reactants.

5. Enthalpy of a chemical reaction

Standard Enthalpy change of a reaction is defined as the enthalpy change of reaction

determined at 25ºC and at 1atm pressure and is denoted by ΔHº

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Considering various enthalpy changes:

(a) Enthalpy of formation

The enthalpy of formation can be defined as the amount of heat exchanged at constant

temperature and pressure during the formation of one mole of the substance from its

constituent elements in their standard states. It is represented by ΔfH. Its unit is kJ mol−1.

For example, the enthalpy of formation of CO2 is equal to the enthalpy change for the

following reaction

C(s) + O2(g) → CO2(g) ΔfH = −393.5 kJ mol−1

𝐻2(𝑔) + 1

2𝑂2(𝑔) → 𝐻2𝑂(𝑙) ΔfH = -285.830 kJ mol-1

𝐻2𝑂(𝑙) → 𝐻2(𝑔) +1

2𝑂2(𝑔) ΔfH = +285.830 kJ mol-1

(b) Enthalpy of combustion

It is defined as the enthalpy change that takes place when one mole of a substance is burnt

completely in the presence of oxygen at a given temperature and pressure. It is denoted by

ΔcH and the unit is kJ mol−1. The combustion is always an exothermic process.

For example, combustion of methane

CH4(g) + 2O2(g) → CO2(g) + 2H2O(l) ΔcHө(298 K)= −890 kJ mol−1

𝐶2𝐻6(𝑔) +7

2𝑂2(𝑠) → 2𝐶𝑂2(𝑔) + 3𝐻2𝑂(𝑙) ∆𝐻°(298 𝐾) = −1560 𝑘𝐽 𝑚𝑜𝑙−1

(c) Enthalpy of solution

The amount of heat exchanged when 1 mole of solute is dissolved in a sufficient amount of

solvent at a specified temperature and pressure is known as enthalpy of solution.

For example,

HCl (g) + 10 H2O(l) → HCl.10H2O(aq) ΔH = −69.01 kJ mol−1

HCl (g) + 40 H2O(l) → HCl.40H2O(aq) ΔH = −72.79k J mol−1

These values of ΔH show the general dependence of the heat of solution on the amount of the

solvent. As more and more solvent is used the value of heat of solution changes. As the

amount of the solvent increases the resulting solution becomes more dilute and ultimately it

becomes so dilute that further addition of solvent produces no enthalpy change. This solution

is known as infinitely dilute solution.

(d) Enthalpy of Sublimation

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It is the amount of enthalpy change to convert one mole of a solid to vapor state at a given

temperature and pressure.

H2O (l) → H2O(g) ΔsubHө(298 K)= 50.0 kJmol−1

(e) Enthalpy of Fusion

It is the change in enthalpy to convert one mole of a solid to its liquid state at a given

temperature and pressure.

H2O (s) → H2O(l) ΔfusHө(298 K)= 6.0 kJ mol−1

(f) Enthalpy of Atomization

It is the amount of heat required to convert one mole of a substance into its constituent atoms

in the gaseous state.

C(graphite) natomizatio

C(g) ∆aH(C) =716.68 kJmol−1

H2(g) natomizatio

2H(g) ∆aH(H)= 436 kJ mol−1

6. Determination of Enthalpies of reactions

Enthalpies of reactions at 25ºC can be determined if ΔHºf values of the reactants and products

involved in the reactions are known as

ΔHº = ΣΔHºf (products) - ΣΔHºf (reactants) …(11)

By convention, ΔHºf values for the elements in their standard states are taken as zero.

7. Kirchhoff Equation: Variation of Enthalpy of reaction with

Temperature

The change in enthalpy of any physical or chemical process varies with temperature at

constant pressure. The effect of temperature on the enthalpy can be understood as follows:

Consider a reaction,

aA + bB cC + dD

The enthalpy change for the above reaction will be:

products reac tan ts C D A BH H H (cH dH ) (aH bH ) …(12)

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Differentiating equation (12) with respect to temperature, keeping pressure constant

C D A B

P P P PP

H H H H( H)c d a b

T T T T T

…(13)

Since, P PC ( H / T)

Therefore, equation (13) can be written as

[𝜕(∆𝐻)

𝜕𝑇]

𝑃= ∑ 𝐶𝑃 (𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠) − ∑ 𝐶𝑃(𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠)

P, C P, D P, A P, B PP

( H)cC dC aC bC C

T

…(14)

where PC = Sum of heat capacities of products – Sum of heat capacities of reactants.

Equation (14) is known as Kirchhoff equation. This equation states that the variation of H

of a reaction with a temperature at constant pressure is equal to PC of the system, i.e.,

P P[ ( H) / T] C …(15)

Rearranging the above equation,

Pd( H) C dT …(16)

Similarly, the dependence of enthalpy on temperature at constant volume is given by,

V V[ ( H) / T] C or Vd( U) C dT …(17)

If the temperature range is small, then change in heat capacity is given by, (assuming

heat capacities are not dependent on temperature)

T T T2 2 2

P PT T T1 1 1

d( H) C dT C dT or 2 1 P 2 1H H C (T T )

…(18)

Similarly,

2 1 V 2 1U U C (T T ) …(19)

If the temperature range is not small then the heat capacities will vary with temperature.

Thus it is convenient to express the heat capacity as a power series in Temperature (T) i.e.

2PC T T …(20)

where , and are constants for a given species. Similarly,

PC = = 2T T ...... …(21)

Substituting equation (21) in equation (15) and integrating between T1 and T2, we get

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T T2 2 2

T T1 1

d( H) ( T T )dT …(22)

Or,

2 2 3 32 1 2 1 2 1 2 1

1H H (T T ) (1 / 2) (T T ) (T T )

3 …(23)

Equation (23) is the integrated Kirchhoff equation.

8. Flame and Explosion Temperatures

The combustion of a gaseous fuel in air occurs so rapidly that the heat produced during

combustion does not get enough time to dissipate into the surroundings. Thus, combustion

process is found to be equivalent to an adiabatic process. The entire amount of heat produced

is used up to heat the gases which are produced during combustion. Maximum flame

temperature is defined as the maximum temperature attained by the flame zone (containing

the resultant gases) due to the heat evolved by the combustion of the fuel under adiabatic

conditions at constant pressure. On the other hand, if the combustion is carried out under

adiabatic conditions at constant volume, the maximum temperature attained is called

maximum explosion temperature.

Kirchhoff equation is used to calculate the maximum flame temperature for an isobaric

adiabatic process. This is done as follows,

Pd( H) / dT C or Pd( H) C dT …(24)

Integrating the above equation gives,

Tf

PTi

d( H) C dT or P f iH C (T T ) …(25)

In the above equation PC is assumed to be constant that is why it is taken outside the

integral sign. Thus, if the values of H , PC and the initial temperature iT are known then

the final temperature fT (maximum flame temperature ) can be calculated.

9. Hess’s law

Hess’s law was established by the Russian chemist German H. Hess in 1840. This law is

known as Hess’s law of constant heat summation. This law states that the amount of heat

evolved and absorbed in a process, including a chemical change, is the same whether the

process takes place in one or several steps, i.e., total change in enthalpy does not change

during the course of the reaction. Thus Hess’s law is also known as principle of conservation

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of energy. The change in enthalpy does not depend on the path taken from the initial to the

final state ( i.e. enthalpy is a state function). Reaction enthalpy changes can be determined by

calorimetry for many reactions. It is of particular utility in calculations of the heat of

reactions which are difficult for practical calorimetric measurements. The overall energy

needed for a chemical reaction can be determined by Hess’s law.

Hess’s law states that the enthalpy change (i.e., heat of reaction at constant pressure) in a

chemical reaction does not depend on the path between the initial and final states of the

system. That is the overall change in enthalpy is same during a chemical change of a reaction

regardless of the number of steps through which the reaction has been taken place. For

example, for a change from reactant to product that can take place in four steps or a single

step, the total enthalpy change will be same.

Single step process:

Reactant → Product ΔH

Multiple step process:

Reactant →A ΔH1

A →B ΔH2

B → C ΔH3

C→ Product ΔH4

According to Hess’s law

ΔH = ΔH1 + ΔH2 + ΔH3 + ΔH4

This can also be shown by following diagram:

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This generalization means that enthalpy of the reaction depends only on the initial reactants

and the final products and not at all on the intermediate products that can be formed.

Thus, enthalpy change which cannot be measured directly is calculated by Hess’s law. If the

net enthalpy change of the reaction is negative, then the reaction is said to be exothermic;

positive value for enthalpy change corresponds to endothermic reactions.

Hess’s law states that changes in enthalpy are additive. Thus for a single reaction change in

enthalpy ΔH is given by:

𝛥𝐻𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛º = ∑ 𝛥𝐻𝑓(𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠)

º − ∑ 𝛥𝐻𝑓(𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠)º

where ΔHf stands for the enthalpy of formation and superscripts º represent standard state

values. The above equation is the combination of two reactions. These are:

Reactants → Elements

𝛥𝐻º = − ∑ ∆𝐻𝑓(𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠)°

Elements → Products

∆𝐻° = ∑ ∆𝐻𝑓(𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠)°

10. Extension of Hess’s law

The changes in entropy and in Gibbs free energy can also be calculated by applying the

concepts of Hess’s law. The Bordwell thermodynamic cycle is an example of such an

extension which takes advantage of easily measured equilibria and redox potentials to

determine experimentally inaccessible Gibbs free energy values.

Thus the change in free energy can be determined by:

∆𝐺𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛° = ∑ ∆𝐺𝑓(𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠)

° − ∑ ∆𝐺𝑓(𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠)°

But entropy can be measured as an absolute value thus entropy of formation is not required,

simply absolute values of entropy are used.

∆𝑆𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛° = ∑ ∆𝑆𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠

° − ∑ ∆𝑆𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠° Ext

11. Applications of Hess’s law

Hess’s law of constant heat summation is useful in the determination of enthalpies of the

following:

Calculation of enthalpies of reactions

Determination of enthalpy changes of slow reactions

Calculation of enthalpies of formation

https://en.wikipedia.org/wiki/Bordwell_thermodynamic_cycle

https://en.wikipedia.org/wiki/Chemical_equilibrium

https://en.wikipedia.org/wiki/Redox_potential

https://en.wikipedia.org/wiki/Gibbs_free_energy

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Enthalpy of formation of reactive intermediates

It helps in determining the lattice energies of ionic substances by building Born-Haber

cycles if the electron affinity to form the anion is known

Exercise:

The heat of dissociation per mole of a gaseous water at 18º C and 1 atm is 241750 J,

calculate its value at 68º C. Data given are:

𝐶𝑃(𝐻2𝑂) = 33.56; 𝐶𝑃(𝐻2) = 28.83; 𝐶𝑃(𝑂2) = 29.12 𝐽𝐾−1𝑚𝑜𝑙−1

Solution: The dissociation reaction is:

𝐻2𝑂 (𝑔) → 𝐻2(𝑔) + 1

2𝑂2(𝑔) ∆𝐻°(291 𝐾) = 241750 𝐽

∆𝐶𝑃 = 𝐶𝑃(𝐻2) +1

2𝐶𝑃(𝑂2) − 𝐶𝑃(𝐻2𝑂)

= (28.83 +1

2∗ 29.12 − 33.56) 𝐽 𝐾−1𝑚𝑜𝑙−1

= 9.83 𝐽 𝐾−1𝑚𝑜𝑙−1

Therefore,

∆𝐶𝑃 ∗ ∆𝑇 = (9.83𝐽 𝐾−1𝑚𝑜𝑙−1 ) ∗ (50 𝐾)

= 491.5 𝐽 𝑚𝑜𝑙−1

∆𝐻°(341 𝐾) = ∆𝐻°(291 𝐾) + ∆𝐶𝑃 ∗ ∆𝑇

= 241750 + 491.5 = 242241.5 𝐽 𝑚𝑜𝑙−1

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12. Summary

The branch of chemistry which deals with the energy changes involved in

chemical reaction is called thermochemistry

The relation between enthalpy at constant volume (qv) and enthalpy at constant

pressure (qp) is given by:

p v gq q n RT

Enthalpies of reactions at 25ºC can be determined if ΔHºf values of the

reactants and products involved in the reactions are known as

ΔHº = ΣΔHºf (products) - ΣΔHºf (reactants)

Kirchhoff equation is given by

P, C P, D P, A P, B PP

( H)cC dC aC bC C

T

Maximum flame temperature is defined as the maximum temperature

attained by the flame zone (containing the resultant gases) due to the heat evolved

by the combustion of the fuel under adiabatic conditions at constant pressure

If the combustion is carried out under adiabatic conditions at constant volume,

the maximum temperature attained is called maximum explosion temperature

Hess’s law was established by the Russian chemist German H. Hess in 1840

Hess’s law states that the amount of heat evolved and absorbed in a process,

including a chemical change, is the same whether the process takes place in one or

several steps, i.e., total change in enthalpy do not change during the course of the

reaction

Hess’s law states that changes in enthalpy are additive. Thus for a single

reaction change in enthalpy ΔH is given by:

𝛥𝐻𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛º = ∑ 𝛥𝐻𝑓(𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠)

º − ∑ 𝛥𝐻𝑓(𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠)º

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Subject Chemistry Paper No and Title 10: Physical