Problem 50 - Chemistry and Chemical Reactivity

Question

A A sample of uranium fluoride is found to effuse at the rate of \(17.7 \mathrm{mg} / \mathrm{h}\). Under comparable conditions, gaseous \(\mathrm{I}_{2}\) effuses at the rate of \(15.0 \mathrm{mg} / \mathrm{h} .\) What is the molar mass of the uranium fluoride? (Hint: Rates must be converted to units of moles per time.

Step-by-Step Solution

Answer

The molar mass of uranium fluoride is approximately 182.42 g/mol.

Step 1: Understand Graham's Law of Effusion

Graham's Law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Therefore, if two gases, A and B, have rates of effusion (Rate_A and Rate_B) and molar masses (M_A and M_B), the following relationship holds: \( \frac{\text{Rate}_A}{\text{Rate}_B} = \sqrt{\frac{M_B}{M_A}} \).

Step 2: Set Up the Equation Using Given Rates

We are given the rates of effusion for uranium fluoride (Rate_UF) and iodine \((\text{Rate}_{I_2})\). These are 17.7 mg/h and 15.0 mg/h, respectively. Substitute these into Graham's Law: \( \frac{17.7}{15.0} = \sqrt{\frac{M_{I_2}}{M_{UF}}} \).

Step 3: Calculate the Molar Mass of Iodine

To proceed further, first determine the molar mass of iodine \((\text{I}_2)\). Since iodine has an atomic mass of approximately 127 u, the molar mass of \(\text{I}_2\) is \({2 \times 127 = 254 \, \text{g/mol}}\).

Step 4: Solve for Molar Mass of Uranium Fluoride

Insert the molar mass of iodine into the equation: \( \frac{17.7}{15.0} = \sqrt{\frac{254}{M_{UF}}} \). Square both sides and solve for \(M_{UF}\): \( \left(\frac{17.7}{15.0}\right)^2 = \frac{254}{M_{UF}} \). Simplifying yields \(M_{UF} = \frac{254}{\left(\frac{17.7}{15.0}\right)^2}\).

Step 5: Calculate Numerically

Perform the calculations: \( \left(\frac{17.7}{15.0}\right)^2 \approx 1.3921 \). Thus, \( M_{UF} \approx \frac{254}{1.3921} \approx 182.42 \, \text{g/mol} \).

Key Concepts

Molar Mass Calculation

Molar mass is a fundamental concept in chemistry that represents the mass of one mole of a given substance. It is usually expressed in grams per mole (g/mol). Calculating molar mass is crucial because it allows us to convert between the mass of a substance and the amount of substance in moles, which is essential for performing stoichiometric calculations in chemical reactions. To calculate the molar mass of a compound:

• Sum up the atomic masses of each element in the compound.
• Multiply the atomic masses by the number of each type of atom present in the formula.

For example, iodine (I) has an atomic mass of about 127 u. Thus, the molar mass of iodine gas (\(\text{I}_2\)) is calculated by \[\text{Molar Mass of \(\text{I}_2\)} = 2 \times 127 \, \text{g/mol} = 254 \, \text{g/mol}.\]Knowing how to determine molar mass is essential when using concepts like Graham's Law, which involves comparing different gases.

Rate of Effusion

Effusion is the process by which gas molecules escape through a tiny hole into a vacuum. Graham's Law of Effusion provides a way to compare the rates of effusion between two gases. It states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.In formula form, Graham's Law is represented as:\[\frac{\text{Rate}_A}{\text{Rate}_B} = \sqrt{\frac{M_B}{M_A}},\]where \(\text{Rate}_A\) and \(\text{Rate}_B\) are the rates of effusion for gases A and B, and \(M_A\) and \(M_B\) are their respective molar masses.Graham's Law allows us to predict how fast a gas will effuse compared to another. For instance, in the exercise given, uranium fluoride and iodine are compared. By rearranging the law's formula, you can solve for unknown molar masses or rates once the others are known.

Educational Chemistry Problem

Solving chemistry problems effectively requires a good grasp of fundamental principles and the ability to apply them. The exercise involving uranium fluoride and iodine is a perfect example. This type of educational chemistry problem challenges students to apply their understanding of Graham's Law of Effusion, alongside the skill of molar mass calculation. To solve the uranium fluoride problem, students need to:

• Identify the relationship between two gases given by Graham's Law.
• Convert provided data, like effusion rates, to the necessary units if required.
• Perform calculations accurately by carefully substituting values into the equations.

Problem-solving in chemistry not only enhances critical thinking but also deepens comprehension of how principles interconnect, like the relation between molar mass and effusion rate. By following systematic steps and understanding each part of the process, students gain confidence in tackling diverse chemistry challenges.