sackur tetrode equation

Associative of partition function

Z=αeβEα=ijeβ(Ei(a)+Ej(b)) \mathcal Z = \sum_\alpha e^{-\beta E_\alpha} = \sum_i\sum_j e^{-\beta(E^{(a)}_i + E^{(b)}_j)}

=(eβE1(a)+eβE2(a)+)(eβE1(b)+eβE2(b)+)=ZaZb= (e^{-\beta E^{(a)}_1} + e^{-\beta E^{(a)}_2} + \cdots)(e^{-\beta E^{(b)}_1} + e^{-\beta E^{(b)}_2} + \cdots) = \mathcal Z_a \cdot \mathcal Z_b

Partion function of ideal gas

E=12mv2E = \frac{1}{2} m v^2


Z=αeβ12mvα20eβfrac12mv2g(v)dv\mathcal Z = \sum_\alpha e^{-\beta\frac{1}{2}mv^2_\alpha}\;\to\; \int^\infty_0 e^{-\beta frac{1}{2}mv^2}g(v)\,dv

g(v)g(v) is density of states

Quantumn magic

Infinte square wall

ψn=2Lsin(kx),(k=nπL)\psi_n = \sqrt{\frac{2}{L}} \sin(kx), \quad (k=\frac{n\pi}{L})


ψ(x,y,z)=ψn(x)ψn(y)ψn(z)=2Lsin(kxx)2Lsin(kyy)2Lsin(kzz)\psi(x, y, z) = \psi_n(x)\psi_n(y)\psi_n(z) = \sqrt{\frac{2}{L}} \sin(k_x x)\sqrt{\frac{2}{L}} \sin(k_y y)\sqrt{\frac{2}{L}} \sin(k_z z)

=2L3/2sin(kxx)sin(kyy)sin(kzz)=\frac{2}{L}^{3/2} \sin(k_x x) \sin(k_y y)\sin(k_z z)

because expectation energy is

E=22mn2π2L2\langle E\rangle= \frac{\hbar^2}{2m}\frac{n^2\pi^2}{L^2}

in case of 3-D space,

E=22m(nx2π2Lx2+ny2π2Ly2+nz2π2Lz2)\langle E\rangle= \frac{\hbar^2}{2m}(\frac{n_x^2\pi^2}{L_x^2}+\frac{n_y^2\pi^2}{L_y^2}+\frac{n_z^2\pi^2}{L_z^2})

=22m(kx2+ky2+kz2)=22mk2=\frac{\hbar^2}{2m}(k^2_x+k^2_y+k^2_z) = \frac{\hbar^2}{2m}\vec k^2


ki=niπLi,i=x,y,zk_i = \frac{n_i\pi}{L_i}, \quad i = x, y, z

k2=2mE2\vec k^2 = 2m \frac{\langle E \rangle}{\hbar^2}

Z=αeβE=eβ22mk2\mathcal Z = \sum_\alpha e^{-\beta E} = \sum e^{-\beta \frac{\hbar^2}{2m}\vec k^2}

Density of state

surface

surface: 18(4πk2)\frac{1}{8}(4\pi k^2)

volume: surface ×dk\times dk

density of state:volume in k-space of one of a shell volume k-space occupied per allowed state\frac{\text{volume in k-space of one of a shell }}{\text{volume k-space occupied per allowed state}}

=volume(πL)3=k2L32π2dk=k2V2π2dk=g(k),number of state in (k,k+dk)= \frac{\text{volume}}{(\frac{\pi}{L})^3} = \frac{k^2L^3}{2\pi^2}dk = \frac{k^2V}{2\pi^2}dk = g(k),\quad \text{number of state in }(k,\, k+dk)

Thermal wavelength

Z=0eβ22mk2k2V2π2dk=V8π2(8m3πβ36)12\mathcal Z = \int_0^\infty e^{-\beta \frac{\hbar^2}{2m}\vec k^2}\frac{k^2V}{2\pi^2}dk = \frac{V}{8\pi^2}(\frac{8m^3\pi}{\beta^3\hbar^6})^{\frac{1}{2}}

=V3(mkT2π)32,β=1kT= \frac{V}{\hbar^3}(\frac{mkT}{2\pi})^{\frac{3}{2}}, \quad \beta=\frac{1}{kT}

=VnQ,(nQ1/3=2πmkT,[m])=V \cdot n_Q,\quad (n_Q^{-1/3} = \frac{\hbar}{\sqrt{2\pi mkT}},\quad [m])

nQn_Q is quantum concentration.

the de-brogil wavelenth is λ=p\lambda = \frac{\hbar}{p}, we set 2πmkT\sqrt{2\pi mkT} be average thermal momentum.

λth=2πmkT,thermal wave length\lambda_{th} = \frac{\hbar}{\sqrt{2\pi mkT}},\quad \text {thermal wave length}

Indistinguishable

U(internal energy), F(helmholzt), S(entropy) Partition function

U=ddβln(Zn)U = -\frac{d}{d\beta}\ln(\mathcal Z_n), F=kbTln(Z)F=-k_bT\ln(\mathcal Z), S=UFTS=\frac{U-F}{T}

Gibb's paradox

fig

The partition function of N free particles is

Zn=Zn=(V3(m2πβ)32)N==(VnQ)NZ_n = Z^n = (\frac{V}{\hbar^3}(\frac{m}{2\pi \beta})^{\frac{3}{2}})^N==(V\cdot n_Q)^N


U=Nddβln(V3(m2πβ)32)=N32dln(2πβ)dβ=N3kT2U = -N\frac{d}{d\beta} \ln(\frac{V}{\hbar^3}(\frac{m}{2\pi \beta})^{\frac{3}{2}})=N\frac{3}{2}\frac{d\ln(2\pi \beta)}{d\beta}=N\frac{3kT}{2}


F=kTNln(VnQ)F=-kTNln(V\cdot n_Q)


S=N3k2+kNln(V)S=N\frac{3k}{2} + kNln(V)


Sbefore=2(N23k2+kN2ln(V2))S_{before} = 2(\frac{N}{2}\frac{3k}{2} + k\frac{N}{2}ln(\frac{V}{2}))

Safter=N3k2+kNln(V))S_{after} = N\frac{3k}{2} + kNln(V))

SafterSbefore=kNln(V)kNln(V2)0S_{after} - S_{before} = kNln(V) - kNln(\frac{V}{2}) \neq 0

Indistinguishability of the particles

Zn=Z1nN!Z_n = \frac{Z_1^n}{N!}

Zn=ZnN!=1N!(V3(m2πβ)32)N=(VnQ)NN!Z_n = \frac{Z^n}{N!} = \frac{1}{N!}(\frac{V}{\hbar^3}(\frac{m}{2\pi \beta})^{\frac{3}{2}})^N=\frac{(V\cdot n_Q)^N}{N!}


U=Nddβln(1N!V3(m2πβ)32)=N32dln(2πβ)dβ=N3kT2U = -N\frac{d}{d\beta} \ln(\frac{1}{N!}\frac{V}{\hbar^3}(\frac{m}{2\pi \beta})^{\frac{3}{2}})=N\frac{3}{2}\frac{d\ln(2\pi \beta)}{d\beta}=N\frac{3kT}{2}


F=kT(Nln(VnQ)ln(N!))=kT(Nln(VnQ)Nln(N)+N)=kTNln(VnQeN)F=-kT(Nln(V\cdot n_Q)-\ln(N!))=-kT(Nln(V\cdot n_Q)-N\ln(N) + N)=-kTN\ln(\frac{V\cdot n_Q\cdot e}{N})


S=kN(32+ln(VnQeN))=kNln(VnQe5/2N),Sackur-tetrode equationS=kN(\frac{3}{2} + \ln(\frac{V\cdot n_Q\cdot e}{N}))=kN\ln(\frac{V\cdot n_Q\cdot e^{5/2}}{N}),\quad \text{Sackur-tetrode equation}

Sbefore=2(kN2ln(VnQe5/2N))S_{before} = 2(k\frac{N}{2}\ln(\frac{V\cdot n_Q\cdot e^{5/2}}{N}))

Safter=kNln(VnQe5/2N))S_{after} = kN\ln(\frac{V\cdot n_Q\cdot e^{5/2}}{N}))

SafterSbefore=0S_{after} - S_{before} = 0