Obtain molar specific heat solid by using law of equipartition of energy .

Explanation

The law of equipartition of energy can use to predict molar specific heat capacities of solids.
Consider a solid of N atoms each vibrating about its mean position.
An oscillator in one dimension has average energy
`=2xx1/2k_B T =k_BT` where , `k_B`=Boltzmann constant.
In three dimensions , the average energy = `3k_B T`
`therefore` For a mole of a solid , the total energy = Average energy x No. of atoms in one mole
`therefore U=3k_B T xx N_A`
`therefore` U=3RT [`because k_B N_A=R`]
`therefore (dU)/(dT)=3R` ...(1)
Specific heat capacity of mole of solid,
`C=(DeltaQ)/(DeltaT)`....(2) [`because mu=1`]
Now, from first law of thermodynamics,
`DeltaQ=DeltaU+DeltaW`
But, at constant pressure `DeltaW=PDeltaV`
`therefore DeltaQ=DeltaU+PDeltaV`
Now , since for a solid change in volume is negligible ,
`therefore P DeltaU=0`
`therefore DeltaQ=DeltaU`
`therefore` From equation (2) ,
`C=(DeltaU)/(DeltaT)` and From equation (1),
C=3R