Question

We obtain uranium- 235 from \(U-238\) by fluorinating the 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 amu, and that of \(\mathrm{U}-238\) is 238.051 amu.

Short answer

The ratio of the effusion rates for 238UF6 to 235UF6 is sqrt((235.054 + 6*19) / (238.051 + 6*19)).

Step-by-step solution

Understanding Graham's Law of Effusion

Graham's Law of Effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. This can be written mathematically as Rate1 / Rate2 = sqrt(M2 / M1), where 'Rate1' and 'Rate2' are the effusion rates of two different gases, and 'M1' and 'M2' are their respective molar masses.

Calculate the Molar Mass of Each Isotope

First, calculate the molar mass of each gas molecule. Since the fluorine atoms do not change, only the mass of the uranium isotope will affect the molar mass of each UF6 molecule. The molar mass of each molecule is the sum of the mass of the uranium isotope and the mass of the 6 fluorine atoms, which have an atomic mass of about 19 amu each. Thus, Molar Mass of 238UF6 = 238.051 amu + 6(19 amu), and Molar Mass of 235UF6 = 235.054 amu + 6(19 amu).

Substitute the Values into Graham's Law

Next, substitute the calculated molar masses into Graham's Law to find the ratio of the effusion rates. Using the equation Rate1 / Rate2 = sqrt(M2 / M1), identify Rate1 as the rate of effusion for 238UF6 and Rate2 as the rate of effusion for 235UF6. Then substitute M1 = Molar Mass of 238UF6 and M2 = Molar Mass of 235UF6.

Simplify and Calculate the Ratio

Calculate the ratio by taking the square root of the ratio of the molar masses. The ratio of the effusion rates (Rate238UF6 / Rate235UF6) is equal to sqrt((235.054 + 6*19) / (238.051 + 6*19)). Simplify the numerical values and compute the square root to find the ratio of the effusion rates.

Key concepts

Effusion Rates

Effusion is a process where gas molecules escape through a tiny hole into a vacuum without colliding with other molecules. Graham's Law of Effusion is a fundamental concept used in chemistry to compare the effusion rates of two different gases. According to this law, the rate at which a gas effuses is inversely proportional to the square root of its molar mass. Mathematically, it is expressed as:

\( \frac{Rate1}{Rate2} = \sqrt{\frac{M2}{M1}} \)

where Rate1 and Rate2 are the effusion rates of gas 1 and gas 2, respectively, and M1 and M2 are their respective molar masses. This relationship shows that lighter gas molecules effuse faster than heavier ones, which is a principle utilized in methods such as isotope separation.

Isotope Separation

Isotope separation is essential in many fields, including nuclear power and medical diagnostics. One of the techniques for isotope separation involves exploiting effusion rates governed by Graham's Law. Different isotopes of the same element have slightly different molar masses. By converting these isotopes into a gaseous form, such as uranium hexafluoride (UF6), we can separate them based on their rate of effusion through a porous barrier. This method is particulary prominent in the enrichment of uranium, where the desired uranium-235 is separated from the more abundant uranium-238. Despite their similar chemical properties, their slight mass difference allows for separation due to the variance in effusion rates.

Molar Mass Calculation

In order to apply Graham's Law of Effusion, we must first calculate the molar mass of the gases involved. The molar mass is the mass of one mole of a substance and is usually expressed in grams per mole (g/mol). For a compound like UF6, this calculation involves summing the atomic masses of all atoms within a molecule — one uranium atom and six fluorine atoms. Since we're dealing with isotopes, it's crucial to use the specific atomic mass of the isotope in question for an accurate determination. The molar mass directly influences the effusion rate and thus it's a key factor in calculating the separation efficiency for different isotopes.

Uranium Hexafluoride

Uranium hexafluoride (UF6) is a compound of high importance in nuclear technology. It's the material of choice for uranium enrichment due to its unique property of being a gas at relatively low temperatures. This characteristic makes it suitable for gas centrifuge and diffusion processes, which are used to separate isotopes of uranium based on their mass. The compound consists of one uranium atom combined with six fluorine atoms.

The molar mass of UF6 will vary slightly depending on the isotope of uranium present: uranium-235 or uranium-238. These differences, although minimal, are significant enough to affect the effusion rates and thus enable the use of Graham's Law for isotope separation in nuclear fuel preparation.