: Includes Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics, ensembles, and kinetic theory.
ZN=1N!(Vλ3)Ncap Z sub cap N equals the fraction with numerator 1 and denominator cap N exclamation mark end-fraction open paren the fraction with numerator cap V and denominator lambda cubed end-fraction close paren to the cap N-th power 4. Calculate Helmholtz Free Energy ( Using Stirling’s approximation (
ZN=(Z1)N=[2cosh(βμB)]Ncap Z sub cap N equals open paren cap Z sub 1 close paren to the cap N-th power equals open bracket 2 hyperbolic cosine open paren beta mu cap B close paren close bracket to the cap N-th power Use the thermodynamic identity relating Observe which ensemble or thermodynamic law the author
⟨E⟩=−N(11+e−βϵ)(−ϵe−βϵ)=Nϵe−βϵ1+e−βϵ=Nϵeβϵ+1open angle bracket cap E close angle bracket equals negative cap N open paren the fraction with numerator 1 and denominator 1 plus e raised to the negative beta epsilon power end-fraction close paren open paren negative epsilon e raised to the negative beta epsilon power close paren equals the fraction with numerator cap N epsilon e raised to the negative beta epsilon power and denominator 1 plus e raised to the negative beta epsilon power end-fraction equals the fraction with numerator cap N epsilon and denominator e raised to the beta epsilon power plus 1 end-fraction Knowing that
W=∫V1V2PdVcap W equals integral from cap V sub 1 to cap V sub 2 of cap P space d cap V Since , we can substitute Find the canonical partition function ( ) and
If you get stuck, look only at the first two lines of the solution. Observe which ensemble or thermodynamic law the author chose to start with.
independent, distinguishable spin-1/2 particles in a magnetic field . Each spin can point up (energy −μBnegative mu cap B ) or down (energy +μBpositive mu cap B ). Find the canonical partition function ( ) and the average energy ( : Includes Maxwell-Boltzmann
EF=h22m(3N8πV)2/3cap E sub cap F equals the fraction with numerator h squared and denominator 2 m end-fraction open paren the fraction with numerator 3 cap N and denominator 8 pi cap V end-fraction close paren raised to the 2 / 3 power Advanced Study and Exam Strategies
must approach zero or a constant value based on the Third Law of Thermodynamics.
