Question

We separate U- 235 from U- 238 by fluorinating a sample of uranium to form UF \(_{6}\) (which is a gas) and then taking advantage of the different rates of effusion and diffusion for compounds containing the two isotopes. Calculate the ratio of effusion rates for \({ }^{238} \mathrm{UF}_{6}\) and \({ }^{235} \mathrm{UF}_{6} .\) The atomic mass of \(\mathrm{U}-235\) is \(235.054 \mathrm{amu}\) and that of U-238 is 238.051 amu.

Short answer

\( \frac{Rate_{235}}{Rate_{238}} = 1.0043 \)

Step-by-step solution

Understand Graham's Law of Effusion

Graham's Law states that the rate of effusion for a gas is inversely proportional to the square root of its molar mass. This can be represented by the formula \( \frac{Rate_{1}}{Rate_{2}} = \sqrt{\frac{M_{2}}{M_{1}}} \) where \( Rate_{1} \) and \( Rate_{2} \) are the effusion rates of two gases, and \( M_{1} \) and \( M_{2} \) are their respective molar masses.

Calculate the Molar Mass of \( \mathrm{UF}_{6} \) Isotopes

For \( ^{238}\mathrm{UF}_{6} \) the molar mass is the sum of the mass of uranium-238 and six fluorine atoms (each fluorine atom approximately 19 amu): \( M_{238} = 238.051 + 6(19) \). Similarly, for \( ^{235}\mathrm{UF}_{6} \) it is: \( M_{235} = 235.054 + 6(19) \) amu.

Substitute the Molar Masses into Graham's Law

Now substitute the molar masses into Graham's Law: \( \frac{Rate_{235}}{Rate_{238}} = \sqrt{\frac{M_{238}}{M_{235}}} \)

Calculate the Ratio of Effusion Rates

Substitute the calculated molar masses into the equation: \( \frac{Rate_{235}}{Rate_{238}} = \sqrt{\frac{238.051 + 6(19)}{235.054 + 6(19)}} \) and simplify to find the ratio.

Simplify and Solve

Performing the calculation provides the ratio: \( \frac{Rate_{235}}{Rate_{238}} = \sqrt{\frac{252.051}{249.054}} \) which can be further simplified to get the final ratio.

Key concepts

Effusion and Diffusion

When it comes to the movement of gases, effusion and diffusion are two processes that often get mentioned together, yet they refer to different phenomena. Effusion is a process where gas molecules pass through a tiny opening into a vacuum or an area with lower pressure. Think of it as a tiny balloon with a pinhole; the air that escapes out is a result of effusion. It's a bit pickier about its surroundings and usually involves a barrier that the gas needs to overcome.

Diffusion, on the other hand, is when gas molecules spread out from an area of higher concentration to an area of lower concentration until they are evenly distributed. Imagine walking into a room where someone has sprayed a perfume; the way the scent spreads throughout the room is diffusion at work. It's driven mostly by the random, constant motion of the molecules, rather than by a pressure difference like effusion.

Molar Mass Calculation

Molar mass calculation is a fundamental concept in chemistry that involves finding the mass of one mole of a substance. A mole is a unit that represents Avogadro's number (\(6.022 \times 10^{23}\) particles) of atoms or molecules. The molar mass is expressed in grams per mole (g/mol). Calculating molar mass is straightforward when we know the atomic masses of each element in the compound from the periodic table.

For a molecule like \text{UF}_6, you'd need to find the atomic mass of uranium (which can vary based on its isotope, as evidenced by U-235 and U-238) and add it to the mass of six fluorine atoms. The sum would give you the molar mass of that particular UF6 isotope, which is crucial for determining other properties such as effusion rates, as indicated by Graham's Law.

Isotope Separation

Isotope separation is a technique used in nuclear chemistry and physics to separate isotopes of the same element, which differ in the number of neutrons and thus in mass. Uranium isotope separation is particularly important in the context of nuclear fuel preparation and for nuclear weapons. The slight differences in mass between the isotopes make separation possible but also challenging.

For uranium isotope separation, gas centrifuge and gaseous diffusion methods are commonly used. Both exploit the mass difference by converting the uranium into a gaseous compound like UF6 and employing methods that allow the heavier U-238 to be separated from the lighter U-235. The process is critical as U-235 is needed for its fission properties in nuclear reactors and bombs, while natural uranium contains only about 0.7% of this isotope.

Gas Effusion Rates

The concept of gas effusion rates revolves around how quickly a gas can escape through a small hole, and it is well described by Graham's Law of Effusion. According to the law, the rate of effusion of a gas is inversely proportional to the square root of its molar mass. This means that a lighter gas will effuse faster than a heavier one due to its smaller molar mass.

To calculate the effusion rate ratio of two gases, as in the step-by-step problem involving UF6 isotopes, you'd use the formula: \[ \frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}} \]This equation is a valuable tool not only for academic exercises but also for practical applications in fields such as environmental science, chemical engineering, and the nuclear industry, where understanding gas behavior is crucial.