Centrifuge-based uranium enrichment: Natural uranium is composed of two isotopes

Question

Centrifuge-based uranium enrichment: Natural uranium is composed of two isotopes:^238 U and^235 U, with percentages of 99.27% and 0.72%, respectively. If uranium hexafluoride gas UF_6 is injected into a rapidly spinning hollow metal cylinder with inner radius R, the equilibrium pressure of the gas is largest at the inner radius and isotopic concentration difference between the axis and the inner radius allow enrichment the concentration of^235 U. Write down the Lagrangian Lscr(|q_k, q^dot_k|) for particles of mass m moving in a cylindrical coordinate system rotating at angular velocity infinity and use a Legendre transformation Hscr({q_k, p_k}) = sigma_k p_k q_k - Lscr, to show that the one-particle Hamiltonian Hscr in that cylindrical coordinate system is Hscr(r, theta, z, p_theta, p_theta, p_z) = p^2_r/2m + (p^2_theta - mr^2 omega)^2/2mr^2 + p^2_z/2m. Ignore the internal degrees of freedom of the molecules since they will not affect the density as a function of position. Show that the one-particle partition function shown here can be written Q_1(V, T) = 1/h^3 integral^infinity_-infinity dp_r integral^infinity_-infinity dp_theta integral^infinity_-infinity dp_z integral^R_0 dr integral^2 pi_0 d theta integral^H_0 dz exp(-beta Hscr), by constructing the Jacobian of transformation between the cartesian and the cylindrical coordinates for the phase space integral. Evaluate the partition function Q_1 in a closed form and determine the Helmholtz free energy of this system.

Explanation / Answer

Solution 6.7

We know that doppler shift of wavelength = 0 Sqrt( (1+v/c)/(1-v/c) )

Here we have to assume that v/c << 1

We also know that

E = mc^2 and E = hc/

From the above relation it is clear that the relative intensity is directly proportional to the exp (-mc^2 (-0))

Here -ve sgn represents reverse direction

And we also know that Intensity is inversly proportional to 0^2T

So relative intensity I = 2Kb0^2T

Hence combining both abive eqquations, we get:-

I() directly propotional to exp (-mc^2 (-0)/2Kb0^2T)

Hence proved.

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---