Table of contents  
Degrees of Freedom  
Law of Equipartition of Energy  
Specific Heat Capacity  
Specific Heat Capacity of Solids  
Specific Heat Capacity of Water  
Mean Free Path  
Solved Question 
A single atom is free to move in space along the X, Y and Z axis. However, each of these movements requires energy. This is derived from the energy held by the atom. The Law of Equipartition of Energy defines the allocation of energy to each motion of the atom (translational, rotational and vibrational). Before we understand this law, let’s understand a concept called ‘Degrees of Freedom’.
Law of Equipartition of EnergyNow, let us look at some equations!
Q. ‘N′ moles of a diatomic gas in a cylinder are at a temperature ′T′. Heat is supplied to the cylinder such that the temperature remains constant but n moles of the diatomic gas get converted into monatomic gas. What is the change in the total kinetic energy of the gas?
(a) 5/2 nRT
(b) 1/2 nRT
(c) 0
(d) 3/2 nRT
Ans: (d)
Solution: Initial K.E. = (3/2) nRT . Number of moles in the final sample = 2n
Since the gas is changed to monoatomic gas, we have: K.E. of the final sample = (3/2) × 2nRT
Hence, the change in the K.E. = 3nRT – (3/2) nRT = 3/2 nRT.
Specific Heat Capacity is the amount of energy required by a single unit of a substance to change its temperature by one unit. When you supply energy to a solid, liquid or gas, its temperature changes. This change of temperature will be different for different substances like water, iron, oxygen gas, etc.
This energy is known as the Specific Heat Capacity of the substance and is denoted by ‘C’. Molar Specific Heat Capacity of a substance is C and is calculated for one mole of the substance. Mathematically we can write:
C = ΔQ/m
Further, when you supply energy to a substance, it may undergo a change in volume and/ or pressure, especially in gaseous substances. Hence, to determine the Specific Heat Capacity of gases, it is important to predetermine the pressure and volume under which you want to calculate C since it can have infinite values (depending on the values of pressure and volume). The Molar Specific Heat Capacity at constant volume is denoted by C_{v} and that at constant pressure is denoted by C_{p}.
According to the first law of thermodynamics ΔQ = ΔU + ΔW {change in heat of a system = change in internal energy + amount of work done}. Change in heat of a system (ΔQ) can also be calculated by multiplying Mass (m), Specific Heat Capacity (C) and change in Temperature (ΔT):
ΔQ = mCΔT Or,
mCΔT = ΔU + ΔW ————(1)
In monatomic gas, molecules have three translational degrees of freedom. At temperature ‘T’ the average energy of a monatomic molecule is (3/2)K_{B}T. Now, let’s look at one mole of such a gas at constant volume and calculate the internal energy (U):
U = (3/2) K_{B}T x N_{A} {where NA is Avogadro constant}
The total internal energy will by the internal energy of a single molecule multiplied by the number of molecules in one mole of the gas; which is Avogadro constant N_{A}. Now, Boltzmann’s constant (K_{B}) is the Gas constant (R) divided by N_{A}. Hence: U = (3/2)(R/N_{A})T × N_{A}
U = (3/2)RT ———————(2)
In equation (1), since the energy is supplied at constant volume: mC_{v}ΔT = ΔU + ΔW. For one mole of a gas, m = 1. Also, for calculating C_{v}, ΔT = 1. Since the volume is constant, ΔW = 0. Therefore, 1×C_{v}×1 = ΔU + 0
C_{v} = ΔU = (3/2)RT —————[refer (2)]
So, the molar specific heat capacity to change the temperature by 1 unit would be C_{v} = (3/2)R. For an ideal gas, C_{p} – C_{v} = R (Gas Constant). Therefore: C_{p} = R + C_{v} = R + (3/2)R = (5/2)R. The ratio of C_{p}:C_{v} (γ) is hence 5:3.
In case of diatomic gases, there are two possibilities:
U = (5/2)K_{B}T * N_{A} = (5/2)RT. Following the calculation used for monatomic gases:
C_{v} = (5/2)R
C_{p} = (7/2)R
γ = 7:5
U = [(5/2)K_{B}T + K_{B}T] * N_{A} = [(5/2)(R/N_{A})T + (R/N_{A})T] * N_{A} = (7/2)RT
Following the calculation used for monatomic gases: C_{v} = (7/2)R
C_{p} = (9/2)R and hence γ = 9:7
The degrees of freedom of polyatomic gases are:
Deploying the Law of Equipartition of Energy for calculation of internal energy, we get:
U = [(3/2)K_{B}T + (3/2)K_{B}T + fK_{B}T] * N_{A} = [(3/2)(R/N_{A})T + (3/2)(R/N_{A})T + f(R/N_{A})T] * N_{A}
U = (3 + f)RT
The molar specific heat capacities:
C_{v} = (3+f)R
C_{p} = (5+f)R
Using the Law of Equipartition of energy, the specific heat capacity of solids can be determined. Let us consider a mole of solid having N_{A} atoms. Each atom is oscillating along its mean position. Hence, the average energy in three dimensions of the atom would be:
3 * 2 * (1/2)K_{B}T = 3K_{B}T
For one mole of solid, the energy would be:
U = 3K_{B}T * N_{A} = 3(R/N_{A})T * N_{A} = 3RT —————–(3)
If the pressure is kept constant, then according to the laws of thermodynamics
ΔQ = ΔU + PΔV
In case of solids, the change in volume is ~0 if the energy supplied is not extremely high. Hence,
ΔQ = ΔU + P * 0 = ΔU
So, the molar specific heat capacity to change the temperature by 1 unit would be:
C = 3R ——————[refer (3)]
For the purpose of calculation of specific heat capacity, water is treated as a solid. A water molecule has three atoms (2 hydrogens and one oxygen). Hence, its internal energy would be:
U = (3 * 3K_{B}T)*N_{A} = 9K_{B}T*N_{A} = 9(R/N_{A})T * N_{A} = 9RT
And, following a similar calculation like solids: C = 9R
Q. One mole of an ideal monoatomic gas is mixed with 1 mole of an ideal diatomic gas. The molar specific heat of the mixture at constant volume is (in cal):
(a) 22 cal
(b) 4 cal
(c) 8 cal
(d) 12 cal
Ans: (b)
Solution: As we know C_{v} = (3/2) R for a monoatomic gas and C_{v} = (5/2) R for a diatomic case.
Thus for the mixture, average of both is = [(3/2) R + (5/2) R] /2 = 2R = 4 cal.
102 videos411 docs121 tests

1. What is the Law of Equipartition of Energy? 
2. How does the Law of Equipartition of Energy relate to the concept of degrees of freedom? 
3. What are examples of degrees of freedom in a molecule? 
4. How does the Law of Equipartition of Energy apply to gases? 
5. What are the implications of the Law of Equipartition of Energy in understanding the behavior of matter? 

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