There are two important isotopes of uranium, ^235 U and ^238 U ; these isotopes are nearly ident

step 1 In this question, it is about the enrichment of uranium. So using a technique based on the diffe

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. Separating the isotopes is called uranium enrichment (and is often in the news as of this writing, because of concerns that some countries are enriching uranium with the goal of making nuclear weapons.) One of the techniques for enrichment, gas diffusion, is based on the different molecular speeds of uranium hexafluoride gas, UF $_{6}$. (a) The molar masses of $^{235} \mathrm{U}$ and $^{238} \mathrm{UF}_{6}$ are $349.0 \mathrm{g} / \mathrm{mol}$ and $352.0 \mathrm{g} / \mathrm{mol}$ respectively. What is the ratio of their typical speeds $v_{\mathrm{rms}} ?$ (b) At what temperature would their typical speeds differ by 1.00 m/s? (c) Do your answers in this problem imply that this technique may be difficult?

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. Separating the isotopes is called uranium enrichment (and is often in the news as of this writing, because of concerns that some countries are enriching uranium with the goal of making nuclear weapons.) One of the techniques for enrichment, gas diffusion, is based on the different molecular speeds of uranium hexafluoride gas, UF $_{6}$. (a) The molar masses of
$^{235} \mathrm{U}$ and $^{238} \mathrm{UF}_{6}$ are $349.0 \mathrm{g} / \mathrm{mol}$ and $352.0 \mathrm{g} / \mathrm{mol}$
respectively. What is the ratio of their typical speeds $v_{\mathrm{rms}} ?$
(b) At what temperature would their typical speeds differ by 1.00 m/s? (c) Do your answers in this problem imply that this technique may be difficult?

Key Concepts

Temperature Dependence in Molecular Speeds

Temperature plays a vital role in molecular behavior since it influences the average kinetic energy of the molecules. The relationship between temperature, molecular mass, and RMS speed means that even minor differences in molar mass will result in very small differences in speed, which can become a significant challenge when trying to separate isotopes using diffusion techniques.

Gas Diffusion and Isotope Separation

Gas diffusion is a technique used to separate isotopes based on their slight differences in molecular speeds due to their different molar masses. In processes like uranium enrichment, the small differences in RMS speeds of uranium hexafluoride molecules containing different uranium isotopes are exploited to gradually increase the concentration of the desired fissile isotope.

Isotopes

Isotopes are variants of elements that have the same number of protons but differ in the number of neutrons. This variation in neutron count leads to different atomic masses, which, while not significantly affecting chemical properties, can result in distinctly different physical behaviors such as in nuclear reactions or separation methods.

Molar Mass

Molar mass is the mass of one mole of a given substance and is a critical parameter when comparing isotopes. The slight difference in molar masses between isotopes is central to understanding and exploiting differences in behavior, such as their molecular speeds in a gas.

Kinetic Molecular Theory

The kinetic molecular theory explains the motion of molecules in a gas and relates their speeds to temperature and molecular mass. According to this theory, the root-mean-square (RMS) speed of a molecule is given by the equation v????? = ?(3RT/M), which shows that molecular speed is inversely proportional to the square root of the molar mass for a fixed temperature.

Transcript

00:01 In this question, it is about the enrichment of uranium.

00:05 So using a technique based on the different molecular speeds of uranium hexafluoride gas.

00:12 Okay, so there are three parts in this question.

00:15 In part a, we want to find a ratio of their typical root mean square speeds.

00:21 So, okay.

00:27 So you'll be looking at the formula, the rooming square speed formula, vrm s equals to square root trit over molar mass okay so from here we can see that the rooming square speed is inversely proportional to the square root of the molar mass so i'm going to call one to be uh two three five uf6 and then i'm going to call uh two to be two three eight uf6 so that m1, so molams 1 is 349 grams per more and then m2 is 352 grams per more.

01:24 Then our ratio, in the first question is the final ratio of the typical vrms.

01:33 So we have vrms 1 divided by vrms 2 is equal to m2 divided by m1 and square root.

01:47 So you get 352 divided by 349 square root and you get 1 .004.

01:57 Okay, so this is the answer for part a.

02:02 Okay, so for the same temperature, the lighter molecules will travel faster, which is what is being shown in this ratio over here.

02:13 Okay, one is the lighter molecule, okay? two is the heavier molecule.

02:19 Okay, so in part b, we want to find the temperature so that there are typical speeds differ by one meters per second.

02:28 So that means we need vrms1 minus vrms2 to be 1 meter per second...

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