Physical Chemistry-Ii

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Paper No and Title 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No and 13: Molar heat capacities and Electronic Partition Title function Module Tag CHE_P6_M13

Chemistry Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No. 13: Molar heat capacities and Electronic Partition function

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TABLE OF CONTENTS

1. Learning Outcomes 2. Introduction 3. Molar Heat Capacities – Solids 3.1 Einstein Theory of Heat Capacity of Solids 4. Partition function – Internal Structure of Atom 4.1 Electronic Partition Function 5. Summary

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1. Learning Outcomes

After studying this module, you shall be able to

 Learn about molar heat capacities of solids  Learn about Einstein theory of heat capacities of solids  Derive electronic partition function

2. Introduction

In our previous module, we did talk about molar heat capacity of gases. Molar heat capacities provide a testing ground for verifying the theoretical statistical mechanical results as heat capacities are experimentally determinable quantities and are characteristic of a given system or substance. Before proceeding with our discussion, let us recapitulate some important definitions and concepts: Heat capacity (퐶) is defined as the heat required to raise the temperature of a substance by 1°C.

Specific heat capacity (퐶𝑔) is defined as the heat required to raise the temperature of a 1g of substance by 1°C.

Molar Heat capacity (퐶푚) is defined as the heat required to raise the temperature of a 1mole of substance by 1°C. There is a general notion that molar heat capacity of any substance is independent of temperature. But experimentally it is found that the value of molar heat capacity is different Chemistry Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No. 13: Molar heat capacities and Electronic Partition function

______at different temperatures. And we have seen that this temperature dependence comes from the vibrational degree of freedom associated with the system for gases in our previous module. In this module, we will talk about molar heat capacities of solids and will see how temperature variations affect molar heat capacities followed by discussion on electronic and nuclear partition function.

3. Molar Heat Capacities - Solids

In solids, we mainly consider elements, i.e., the atomic crystals of different elements like C, Si, Cu etc. In our earlier module, we categorized a system of distinguishable and localized particles as solid or rather now one can say a crystalline “atomic” solid. Dulong – Petit law proposed in 1819 states that the specific heat capacity of a solid element 3푅 is equal to where R is the universal gas constant (in J/K.mol) and M is the molar mass 푀 (in kg/mol). 3푅 퐶 = ...(1) 푠 푀 Or, 푐푎푙표푟𝑖푒 퐶 . 푀 = 3푅 ~ 6 ~ 25 퐽/퐾 …(2) 푠 푘푒푙푣𝑖푛

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퐶 퐶 . 푀 = ( ) 푀 = 3푅 푠 푚 Where C is the total heat capacity, m is the mass and M is the atomic weight. 푚 = 푛 (푛표. 표푓 푚표푙푒푠) 푀 퐶 퐶 퐶 . 푀 = ( ) 푀 = = 3푅 푠 푚 푛 Therefore, heat capacity of most solid crystalline substances is 3R per mole of substance.

Dulong and Petit did not state their law in terms of the gas constant R as it was not known then. Instead, they measured the values of specific heat capacities of substances and found that when multiplied by their atomic weights, the values were found to be nearly constant,

and equal to a value which was later recognized to be 3R.

This result can be explained by the principle of equipartition of energy (classical approach) by treating every atom of the solid as a linear harmonic oscillator with six degrees of 1 freedom each contributing an energy of 푘 푇. 2 퐵

Then, 퐶푉푚 becomes, 푅 퐶 = (퐶 ) + (퐶 ) + (퐶 ) = 6 ( ) = 3푅 …(3) 푉푚 푇 푉 푅 푉 푉 푉 2 휕퐸 Where 퐶푉푚 = ( ) (퐴푡 푐표푛푠푡푎푛푡 푣표푙푢푚푒) …(4) 휕푇 푉 Where E refers to the energy of the system.

And (퐶푇)푉 is the translational contribution, (퐶푅)푉 is the rotational contribution and (퐶푉)푉 is the vibrational contribution to heat capacity. It is important to note that nowhere in the above discussion on the thermodynamic rule “Dulong – Petit law”, we saw temperature dependence of heat capacity. In fact, specific heat capacity has been assumed to be independent of temperature. However, many

______exceptions are known to this law and it was found that specific heat capacity depends on temperature. Also, the value is not 6 푐푎푙표푟𝑖푒 for all the elements.

Figure 1: Heat Capacity versus Temperature plot for solids. Although heat capacity for many elements is close to 3R at room temperature, low temperature measurements found strong temperature dependence of heat capacity.

Einstein proposed a theory to explain the temperature dependence of heat capacity of solids which is popularly referred to as “Einstein theory of heat capacity of solids” which forms a part of discussion of our next section.

3.1 Einstein Theory of Heat Capacity of Solids Einstein used statistical mechanics to account for temperature dependence of molar heat capacities. He modelled a non-linear solid, referred to as “Einstein Solid” and supposed

that it had 푁푎 number of atoms.

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If there are 푁푎 number of atoms in a particle (individual entity of a system), there are 3푁푎 degrees of freedom

Linear molecule Non-Linear molecule 3 Translational 3 Translational 2 Rotational 3 Rotational 3푁 - 5 Vibrational 3푁 - 6 Vibrational 푎 푎

So, total contribution to heat capacity will take into account translational, rotational and vibrational degrees of freedom.

퐶푉푚 = (퐶푇)푉 + (퐶푅)푉 + (퐶푉)푉

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Einstein Solid – It is a model of a solid based on two assumptions:

1. Each atom / particle in the system is an independent quantum harmonic oscillator –

model for vibrational energy (the atoms vibrating about their individual fixed positions are assumed to be loosely coupled, that is, to be independent of each other in the sense that the energy of each one at any instant is independent of that of its neighbors).

2. All atoms oscillate independently with the same frequency in the sample (the frequency depends on the strength of the restoring force acting on the atom, i.e. the strength of the chemical bonds within the solid). Note that this assumption is an approximation, since in reality there are forces of attraction between the atoms and a continual exchange of energy among them. This assumption combined with the quantum mechanical concepts, suggests that the allowed energies for a one dimensional harmonic oscillator have the form, 1 퐸 = ℎ휈 (푉 + ) 푉 2 where V = 0, 1, 2, 3, ...... (Vibrational quantum number) and  = Frequency of linear harmonic oscillator. ℎ휈 We find that for 푉 = 0, vibrational energy has the value 퐸 = and is not zero . We 푉 2 refer this energy as zero-point energy of the system which means the system is not at rest even in the lowest vibrational energy state.

It is further assumed that the overall vibrations are effectively three dimensional with no one direction favoured over any other, i.e. isotropic. So, a sample of 푁 will 푎 have 3푁푎 oscillations associated with it.

______1 Instead of classical law of equipartition, that is a contribution of 푘 푇 for each degree of 2 퐵 freedom, Einstein used the mean energy of Planck's oscillator for each degree of freedom (vibrational contribution). Let us see the mathematical formulation of this now,

For the non-linear solid,

3 3 퐶 = 푘 + 푘 + (3푁 − 6)푘 푓 …(5) 푉푚 2 퐵 2 퐵 푎 퐵

휃 2 푉 휃푉 푒 푇 ℎ휈 Where 푓 = ( ) 2 휃푉 = …(6) 푇 휃푉 푘퐵 (푒 푇 −1)

푓 is referred to as Einstein function and its value ranges from 0 to 1. The expression (6) is obtained by substituting vibrational energy in equation (5) which has the form,

푅휃 푅휃 퐸 = ( 푉 + 푉 ) 푉 2 휃푉 (푒 푇 −1) 푉

For two solids at same temperature, f will be different and 휃푉 will be different. Equation (5) can be re-written as,

퐶푉푚 = 3푘퐵 + 3푁푎푘퐵푓 − 6푘퐵푓 …(7)

퐶푉푚 = 3푁푎푘퐵푓 + (3 − 6푓)푘퐵 …(8)

If the number of atoms in the system is very large, the second term in the above expression becomes insignificant and can be neglected.

퐶푉푚~3푁푎푘퐵푓 …(9)

′ If 푁푎 = 푁퐴 (퐴푣표푔푎푑푟표 푠 푛푢푚푏푒푟) 푡ℎ푒푛 푁퐴푘퐵 = 푅

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퐶푉푚 ~ 3푅푓 …(10)

From the above expression, we find that the value of heat capacity can vary from 0 to 1 depending on the value of f.

Figure 2: Einstein Plot (this plot is independent of the nature of the element)

So now let us see, how the Einstein function makes heat capacity temperature dependent.

Case I: At very high temperatures, 푇 ≫ 휃푉 휃 푉 ~푣푒푟푦 푠푚푎푙푙 푇

휃 Since, 푉 (= 푥)~푣푒푟푦 푠푚푎푙푙, 푇

푥2 푒푥 = 1 + 푥 + ~1 + 푥 (푛푒푔푙푒푐푡𝑖푛푔 ℎ𝑖푔ℎ푒푟 푡푒푟푚푠) …(11) 2!

______Using equation (11) in equation (6), we get

휃푉 2 휃푉 휃푉 푒 푇 푓 = ( ) 2 = 푒 푇 …(12) 푇 휃푉 (1+푒 푇 −1) At very very high temperature,

휃푉 푇 → ∞ ; 푒 푇 ~푒0~1 …(13)

푓 = 1 ….(14)

Substituting equation (14) in equation (10) gives,

퐶푉푚 = 3푅 …(15)

Case II: At very low temperatures, 푇 ≪ 휃 푉 휃 푉 ~푣푒푟푦 푙푎푟푔푒 푇

휃푉 푒 푇 ≫ 1 휃 2 푉 2 휃푉 푒 푇 휃푉 1 푓 = ( ) 2 = ( ) 휃 …(16) 푇 휃푉 푇 푉 (푒 푇 ) 푒 푇 Expanding the exponential in the denominator gives,

2 2 휃푉 1 휃푉 1 푓 = ( ) 휃 = ( ) 2 …(17) 푇 푉 푇 휃푉 휃푉 1 푇 (1+ +( ) +⋯….) 푒 푇 푇 2! Re-writing the above equation gives,

1 푓 = 2 …(18) 푇 푇 1 휃 1 (( ) + + + 푉 …….) 휃푉 휃푉 2! 푇 3!

푇 ~ 푣푒푟푦 푣푒푟푦 푠푚푎푙푙 휃푉

______푓~0 …(19)

Substituting equation (19) in equation (10) gives,

퐶푉푚 = 0 …(20)

So, in all have heat capacity varying from 0 to 3R depending on the temperature conditions of the system.

푪푽 = ퟑ푹풇

Low temperature High temperature 풇 = ퟎ 풇 = ퟏ

푪푽풎 = ퟎ 푪푽풎 = ퟑ푹

However, this theory too had drawbacks: 1. Einstein considered vibrational degree of freedom of one particle only and extended it

to (3푁푎 − 6) degrees of freedom.

2. All the (3푁푎 − 6) vibrations were assumed to be having same frequency and the solid was assumed to be isotropic. For example, in 푂 = 퐶 = 푂, all types of vibrations are possible – symmetric, anti-symmetric, bending and all have different energies. 3. Vibrations were assumed to be simple harmonic which is very ideal and anharmonicity was not taken into account.

These shortcomings were latter improved in Debye Model.

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In all, Einstein model was able to prove 1. Molar heat capacity varies from 0 to 3R. 2. Temperature dependence of molar heat capacity And again we found that the temperature dependence of heat capacity is verified from the vibrational contribution term.

4. Partition function – Internal Structure of Atom

In this discussion on partition functions of bulk systems comprising of monoatomic and polyatomic particles, we considered only the translational, rotational and vibrational contributions. We have so far ignored the internal structure of the particle i.e. the electrons and the nucleus and their contribution to partition function.

In this section, we will talk about electronic partition function (푞푒) and we will take nuclear

partition function (푞푛) in detail in our next module.

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4.1 Electronic Partition function The general expression for electronic partition function can be by the expression, 휀 − 푒 푘 푇 푞푒 = ∑𝑖 (푔푒푒 퐵 ) …(21) 𝑖

휀 휀 휀 − 0 − 01 − 02 푘 푇 푘 푇 푘 푇 푞푒 = 푔0푒 퐵 + 푔1푒 퐵 + 푔2푒 퐵 + ⋯ …(22)

In general, there is no relation for electronic energy levels. Electronic energy has no fixed value. This is because such relations are can only be obtained from solution of Schrödinger equation and Schrödinger equation cannot be solved exactly for a system with more than one electron. Though in principle, it is possible to solve Schrödinger equation for the electronic states and energies of an atom or a molecule, it is more convenient to obtain data from spectroscopic studies. And it has been found that for most molecules, the excited electronic Chemistry Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No. 13: Molar heat capacities and Electronic Partition function

______energy levels lie so far above the ground state compared with 푘퐵푇 that the molecules may be considered to occupy the ground state as ordinary temperatures. (Electronic energy levels are very far apart and therefore, we need to supply very large amount of energy for transition from one level to another). And thus, contributions to electronic partition function from excited electronic states can be neglected.      ...... 01 02 03 휀 − 0 푘 푇 푞푒 = 푔0푒 퐵 …(23)

Bringing all the energy levels (translational, rotational, vibrational …) to the same scale that means assuming the ground electronic energy level is the zero point level i.e. energy of ground state is assumed to be zero, we get electronic partition function as,

0 − 푘 푇 푞푒 = 푔0푒 퐵 = 푔0 …(24) So, the electronic partition function becomes equal to the degeneracy of the ground electronic state.

In the Russell – Saunders coupling scheme, the degeneracy of an atomic electronic state is given be the expression, 푔0 = 2퐽 + 1 푤ℎ푒푟푒 퐽 = 푡표푡푎푙 푎푛푔푢푙푎푟 푚표푚푒푛푡푢푚 = 퐿 + 푆 (for atoms Z≤ 30) The ground electronic states of free atoms are generally degenerate. For example – Hydrogen 1 1 atom with electronic configuration 1푠1; 푆 = 푎푛푑 퐿 = 0; 퐽 = 푎푛푑 2 2 1 푔 = 2 × + 1 = 2 (Doublet) 0 2 The ground states of most molecules and stable ions are invariably non-degenerate, i.e. are singlets (푔0 = 1) And for atoms like chlorine and molecules like NO, the difference between ground state and the first excited state is small and therefore, the contribution from excited state must be taken into account.

______Once, the electronic partition function is known, one can calculate the bulk thermodynamic properties. We did derive the expressions for energy and entropy in terms of partition function following Maxwell Boltzmann Statistics.

Energy

푁 2 휕 ln 푞푒 퐸푒 = 푘퐵푇 ( ) …(25) 휕푇 푉 The above expression can also be written as,

2 휕 ln 푞푒 퐸푒 = 푁푘퐵푇 ( ) …(26) 휕푇 푉

Substituting 푞푒 from equation (24) in the above expression gives,

2 휕 ln 𝑔0 퐸푒 = 푁푘퐵푇 ( ) = 0 …(27) 휕푇 푉

Entropy

(퐸 ) 푆 = [푘 ln((푞 )푁) + 푒 ] …(28) 푒 퐵 푒 푇

Substituting 푞푒 from equation (24) and 퐸푒 from equation (27) in the above expression gives,

(0) 푆 = [푁푘 ln(푞 ) + ] = 푁푘 ln(푞 ) …(29) 푒 퐵 푒 푇 퐵 푒

Heat Capacity

휕퐸푒 퐶푉푚 = ( ) …(30) 푒 휕푇 푉

Substituting equation (27) in the above equation gives,

휕퐸푒 퐶푉푚 = ( ) = 0 …(31) 푒 휕푇 푉 Chemistry Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No. 13: Molar heat capacities and Electronic Partition function

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Similarly, other thermodynamic properties like G, H, A, etc. may be calculated. For most molecules, there is no contribution of electronic energy to the thermodynamic properties at ordinary temperatures. Moreover, the electronic partition function is likely to be nearly independent of temperature and therefore, the electronic contributions to

퐸, 퐻, 퐶푉 푎푛푑 퐶푃 are all zero.

In the next module, we will talk about nuclear partition function in detail.

5. Summary

 Heat capacities are experimentally determinable quantities and are characteristic of a given system or substance. Molar heat capacities provide a testing ground for verifying the theoretical statistical mechanical results.  There is a general notion that molar heat capacity of any substance is independent of temperature. But experimentally it is found that the value of molar heat capacity is different at different temperatures. And this temperature dependence is verified from the vibrational contribution term.  Einstein theory of heat capacity of solids explained the temperature dependence of heat capacity of solids.  In statistical mechanics, electronic partition function is the part of partition function which results from the electronic part of the system under consideration.

푞푒 = 푔0

퐸푒 = 0

푆푒 = 푁푘퐵 ln(푞푒)

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퐶푉푚푒 = 0