There are two important isotopes of uranium- 235 U and ^238 U ; these isotopes are nearly identi

There are two important isotopes of uranium- 235 U and ^238...

There are two important isotopes of uranium- $235 \mathrm{U}$ and $^{238} \mathrm{U} ;$ these isotopes are nearly identical chemically but have different atomic masses. Only $^{235} \mathrm{U}$ is very useful in nuclear reactors. One of the techniques for separating them (gas diffusion) is based on the different average velocities $v_{\mathrm{rms}}$ of uranium hexafluoride gas, UF $_{6} .$ (a) The molecular masses for $^{235} \mathrm{U}$ UF $_{6}$ and $^{238} \mathrm{U}$ UF $_{6}$ are $349.0 \mathrm{g} / \mathrm{mol}$,and $352.0 \mathrm{g} / \mathrm{mol}$, respectively. What is the ratio of their average velocities? (b) At what temperature would their average velocities differ by $1.00 \mathrm{m} / \mathrm{s} ?(\mathrm{c})$ Do your answers in this problem imply that this technique may be difficult?

Key Concepts

Isotopic Separation

Isotopic separation involves enriching one isotope over another based on slight differences in their physical properties rather than chemical properties. Since isotopes of the same element share chemical behavior, separation techniques exploit differences in mass, such as those observed in their kinetic energies or diffusion rates. This is particularly important in nuclear applications where specific isotopes, despite their similar chemistry, serve different functions.

Root?Mean?Square (rms) Velocity

The root?mean?square velocity is a statistical measure of the speed of particles in a gas, defined by the equation v?rms? = ?(3RT/M), where R is the gas constant, T is the absolute temperature, and M is the molecular mass. This concept is fundamental in understanding how differences in mass, even if slight, lead to measurable differences in the speed (and hence kinetic energy) of molecules at a given temperature.

Gas Diffusion

Gas diffusion is a separation method that relies on the fact that lighter molecules move faster than heavier ones. In a porous barrier or membrane, these differences in average velocities allow for a gradual separation of isotopes. Even small differences in molecular masses can result in different diffusion rates, though the effectiveness of the separation process depends on the relative scale of these differences.

Temperature Dependence in Kinetic Theory

Temperature directly influences the kinetic energy of gas molecules and therefore their velocities. In the context of separating isotopes, the temperature setting can determine how significant the difference in rms velocities will be. Adjusting temperature affects both the kinetic energy distribution and the extent to which small mass differences can produce a meaningful separation in speeds.

Transcript

00:01 So we can say for part a, the ratio of the root mean squared speed of the two uranium atoms, we can say that v .s, the root mean squared speed for uranium 235, divided by the root mean squared speed for uranium 238, would be equalling the square root of 3 times k, the boltsman constant, multiplied by the temperature divided by the mass of uranium 235.

00:37 This would be divided by square root of three times boltzmann's constant times the temperature, divided by the mass of uranium 238.

00:48 And so this would be equalling the square root of the mass of uranium 238 divided by the mass of uranium 235.

00:58 And so we can then say that the root, the root, mean squared speed are v rms 235 divided by vs of rms 238 would be equalling essentially square root of 352 grams per mole divided by avagos number this would be divided by the square root of 349 divided by abogado's number and this is equaling 1 .004.

01:42 This would be the required ratio for part a.

01:46 For part b, then, we can say that the root mean squared speeds of the uranium hexafluoride consisting of uranium 235, we can say that vrms for 235 would be equaling 1 .004 times the rms.

02:15 Vrms for 238 and so we can then say that this is going to be equaling vrms times 235 multiplied by vrms for 238 for 238 plus 0 .004 times vrms for 238 and so we can then substitute, let's subtract and say vrms 235 minus vrms 238, equaling 0 .004, vrms 238.

03:13 And we're going to substitute one meter per second for this difference here.

03:19 And then we can substitute in the formula for the root means squared speed.

03:26 And so we can say that then one meter per second is going to be equal to 0 .004 times the square root of 3 times boltzman constant, bultzman's constant times t divided by the mass of uranium 238.

03:44 And so we can say solving for the temperature, this would be equal to 1 meter per.

03:52 Per second divided by .004 quantity squared multiplied by the mass of uranium 238 divided by three times boltzmann's constant and so we can find for part b the temperature t this would be equaling again 1 .00 meters per second divided by point zero zero four quantity squared multiplied by 352 .0 times 10 to the negative third kilograms per mole...

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