 the power of equipartition of energy, law of equipartition, equally part of it, equally part of it, equally part of it, right. So we write down according to this law of a molecule equally divided among the all the possible modes, among all the possible modes of the of the energy. Among the all the possible modes of the energy, how do you take all the possible modes equally divided. Now what is number of modes means? Number of modes is basically telling you the degree of freedom, number of modes represents the degree of freedom. For monotomic, the helium atom is here, how many degrees of freedom it has, say 1, but that is not how degree of freedom is defined. The motion in x axis has nothing to do with motion in y axis, the independent they are 2, y and z. So for monotomic, third one is z. So if total energy that we have derived just now, 3 by 2 kT, how much? 1 by 2 kT, you should not say along x along y, it is like taking component of energy. It is not component of energy, it is like energy corresponding to this degree of freedom. So this much energy I will write like this, this much energy per degree of 2 kT. So energy depends only on temperature. Now let's try to find out the degree of freedom of the other molecules as well. By the way energy which you have written, nowhere it will be of the gas. It is simply written of an atom or a molecule. So you can find out degree of freedom for a gaseous atom or liquid atom or an atom corresponding to a solid. Now let's take monotomic, sorry diatomic gaseous atom. So this is covalent or ionic bonds or rigid bonds. Why such assumption is required? So that the bond is like a square. So since bonds typically covalent and ionic bonds are rigid, whereas hydrogen bonds can be treated as flexible. These hydrogen bonds can be treated as flexible bonds. So hydrogen bonds can behave like a spring. So hydration will happen along hydrogen bonds. Where solids, why the multiple molecules are stacked one up to the other and they are folding on to each other because there is no covalent bond. So in the solid, the bond between one atom and flexible. Oxygen, nitrogen could be anything. But this bond is rigid. So now tell me how many degrees of freedom. What kind of rotation? This is the molecule. One rotation is this, other is like that. What about this? This is negligible because the size of the atom is negligible. If there is a bit connected there, then you can't ignore this. So two degrees of freedom for the rotation and three for the translation. Total 5 degrees of freedom. So for diatomic, 5 degrees of freedom. Now can you tell me what is the energy of a diatomic gas molecule at a temperature T? How much it is? For a monatomic molecule. Now energy is 5 by 2 T. What it also says is that if there is a monatomic molecule and if I give 5 joules of energy, monatomic gas I give 5 joules of energy, 6 T, 2 KT, delta E, delta E, delta T1 and delta T2. Same energy. So delta T2 is equal to 3 by 5 times delta T1. So increase in temperature, it is specifically temperature. Because most of distribution of energy is higher, specifically. For a monatomic, it is not even diatomic. T and Cp for this. How much it is Cp? Cp will be 7 by 2 R. 7 by 5 R. Vibration mode. Because now we will be dealing with flexible bonds as well. Please write down vibration mode, generational degree of freedom. Flexible bond. This is a hydrogen bond or Vendor-Wall force. See Vendor-Wall we do not call bond. It is just a weak force or like a force. All right? Flexible bond acts like a spring. Even kinetic energy is very less, mass is very less. So it will have and because of that it can have potential energy as well as kinetic energy. So one vibration mode can have 2 degrees of freedom, kinetic and potential. Fine? So one flexible bond will contribute to 2 degrees of freedom. Ready? We will write down one flexible bond implies 2 degrees of freedom. Potential and kinetic. Please find out the specific heat of a gas having F vibration mode. F is the number of vibration mode. I mean F number of vibration mode. That is two times the specific heat, Cp and Cp quickly because it is very small. But if you call it on it, then you cannot ignore that. Just write it like that. Anyone? Cp and Cp. Total area of it has died into 6 plus 2F into half kT. This will be 3 plus kT. From here you can assume one mole multiplied by Cv delta T. So Cv is equal to 3 plus F into R. Procedure is same. This is Cv, Cp is 1. 4 plus F into R. So if I divide Cp and Cv I get 4 plus F divided by 3 plus F. So if F increases, the ratio of Cp and Cp will tend towards 1. If F is very large, both are almost fine. So Brahma will tend towards 1. If number of degree of freedom increases, any doubts till now? Anything? I mean theory which we have applied for the law of equipartition of energy tells you that per degree of freedom, one atom should have half kT energy. Now please take the case of solid. Is there a bond between the two atoms? What kind of bond? Is it a covalent bond? It is a band of all fours which is holding all the atom together. It is flexible. So how many degree of freedom one atom can have? Think and speak. What? It has to be vibration only. See one atom in a solid will not leave the solid and starts moving randomly here and there. It is not a gas. So the position of a solid and it cannot even rotate as well. 2 degree of freedom. So how many degree of freedom? 3 degree of freedom means 6. 3 vibration mode, 6 degree of freedom. So the energy of one molecule is how much? 6 into half kT. This is 3 kT. Now tell me what is the value of specific heat for a solid? Specific heat capacity for solid. For one molecule solid, change energy. So the value of Cp-Cv is R for gas. For solid, we ignore the expansion. So Cp and Cv does not exist. All the specific heats are same. So basically this 3R you get is the heat capacity per mole. What we were using? Heat capacity per kg. Yes or no? How much this is roughly? 3 into 8.31. This is 24.93. So this is joule per mole per kelvin. So in a way, according to Karate theory of gas, specific heat per mole. It does not matter whether it is a copper, steam, aluminum, whatever it is. It has same, is that true? It is true. Per mole specific heat is almost same, but per kg it is different because density is different. I will just tell you the few values. Please write down carbon, copper, lead, silver, tungsten. Take away, write down 0.4. Closer is for copper it is 24.4, 24.5. For lead it is 26.5. For silver it is 25.5. For tungsten it is this. They are same. Almost constant. Carbon is 6.1. Quantum physics. If you do not understand, give the logic. Quantum physics takes over here. Actually quantum physics takes over here. There is some quantum physics reasoning here why it is completely way off. But if you ignore carbon, most of these solids have the same specific heat per mole. So suppose I give, then you can calculate the specific heat of that solid.