InterviewSolution
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InterviewSolution
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Derive the formula expressing molar heat capacity of a unidimensional crystal, a chain of identical atoms, as a function of temperature `T` if the Debye temperature of the chain is equal ot `Theta`. Simplyfie the octained expression for the case `Tgt Theta` |
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Answer» In the Debye approximation the number of modes per unit frequency interval is given by `dN= (1)/(pi v)d omega 0 le omega le(k Theta)/(ħ)` But `(k Theta)/(ħ)= pin_(0)v` Thus `dN=(l)/(pi v)d omega, 0 le omega le pin_(0)v` The enrgy per mode is `lt E ge(1)/(2)ħ omega+(ħ omega)/(e^(ħ omega//kT)-1)` The total interval energy of the chain is `U=(l)/(pi v) int_(0)^(pin_(0)v) (1)/(2)ħ omega d omega` `+(l)/(piv)int_(0)^(pi n_(0)v)( ħ omega)/(e^( ħ omega//kT)-1) domega=(l ħ)/(4pi v)(pi n_(0)v)^(2)+(l ħ)/(4 pi v)(pi n_(0)v)^(2)+(l)/(pi v ħ)(kT)^(2) int_(0)^(Theta//T)(xdx)/(e^(x)-1)= ln_(0)k.( ħ)/(k)(pi n_(0)v).(1)/(4)` `+ln_(0)k(T^(2))/((pi n_(0)vħ//k)) int_(0)^(Theta//T)(x dx)/(e^(x)-1)` We put `ln_(0)k=R` for `1` mole of the chain Then `U=R Theta{(1)/(4)+((I)/(Theta))^(2) int_(0)^(Theta//T)(x dx)/(e^(x)-1)}` Hence the molar heat capacity is by defferentiation `C_(v)=((delU)/(delT))=R[2((T)/(Theta))int_(0)^(Theta//T)(xdx)/(e^(x)-1)-(Theta//T)/(e^(e//T)-1)]` when `T gt gt Theta, C_(v)~~R` |
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