____________________________________________________________________________________________________ Subject CHEMISTRY 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 ____________________________________________________________________________________________________ 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 Chemistry Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No. 13: Molar heat capacities and Electronic Partition function ____________________________________________________________________________________________________ 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 (퐶) 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, 푐푎푙표푟푒 퐶 . 푀 = 3푅 ~ 6 ~ 25 퐽/퐾 …(2) 푠 푘푒푙푣푛 Chemistry Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No. 13: Molar heat capacities and Electronic Partition function ____________________________________________________________________________________________________ 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 Chemistry Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No. 13: Molar heat capacities and Electronic Partition function ____________________________________________________________________________________________________ exceptions are known to this law and it was found that specific heat capacity depends on temperature. Also, the value is not 6 푐푎푙표푟푒 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. Chemistry Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No. 13: Molar heat capacities and Electronic Partition function ____________________________________________________________________________________________________ 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 Chemistry Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No. 13: Molar heat capacities and Electronic Partition function ____________________________________________________________________________________________________ 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 퐸푉 = ℎ휈 (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 푁푎 will have 3푁푎 oscillations associated with it. Chemistry Paper No. 6: PHYSICAL CHEMISTRY-II (Statistical Thermodynamics, Chemical Dynamics, Electrochemistry and Macromolecules) Module No. 13: Molar heat capacities and Electronic Partition function ____________________________________________________________________________________________________ 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