Question

One way of separating oxygen isotopes is by gaseous diffusion of carbon monoxide. The gaseous diffusion process behaves like an effusion process. Calculate the relative rates of effusion of \(^{12} \mathrm{C}^{16} \mathrm{O},^{12} \mathrm{C}^{17} \mathrm{O},\) and \(^{12} \mathrm{C}^{18} \mathrm{O} .\) Name some advantages and disadvantages of separating oxygen isotopes by gaseous diffusion of carbon dioxide instead of carbon monoxide.

Short answer

The relative rates of effusion for the given isotopes are: \( \frac{Rate_{(^{12}C^{16}O)}}{Rate_{(^{12}C^{17}O)}} = \sqrt{\frac{29}{28}}\) and \( \frac{Rate_{(^{12}C^{16}O)}}{Rate_{(^{12}C^{18}O)}} = \sqrt{\frac{30}{28}}\) Advantages of using Carbon Dioxide (CO2) instead of Carbon Monoxide (CO) include safer handling due to CO2 being non-toxic and non-flammable. Disadvantages include slower diffusion rates due to higher molar mass and the potential for solid hydrate formation at low temperatures, complicating the process.

Step-by-step solution

Understand 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. Mathematically, it can be written as: \[ \frac{Rate_1}{Rate_2} = \sqrt{\frac{Molar \ Mass_2}{Molar \ Mass_1}}\] Where, Rate_1 and Rate_2 are the rates of effusion of the two gases, and Molar Mass_1 and Molar Mass_2 are their respective molar masses.

Calculate the molar masses of the given isotopes

Given isotopes of Carbon Monoxide: \(^{12}C^{16}O\), \(^{12}C^{17}O\), and \(^{12}C^{18}O\) Molar mass of \(^{12}C^{16}O\): 12 (Carbon) + 16 (Oxygen) = 28g/mol Molar mass of \(^{12}C^{17}O\): 12 (Carbon) + 17 (Oxygen) = 29g/mol Molar mass of \(^{12}C^{18}O\): 12 (Carbon) + 18 (Oxygen) = 30g/mol

Calculate the relative rates of effusion

Using Graham's law, we will find the relative rates of effusion for the given isotopes. Relative rate of \(^{12}C^{16}O\) and \(^{12}C^{17}O\): \[ \frac{Rate_{(^{12}C^{16}O)}}{Rate_{(^{12}C^{17}O)}} = \sqrt{\frac{29}{28}}\] Relative rate of \(^{12}C^{16}O\) and \(^{12}C^{18}O\): \[ \frac{Rate_{(^{12}C^{16}O)}}{Rate_{(^{12}C^{18}O)}} = \sqrt{\frac{30}{28}}\]

Advantages and Disadvantages of using Carbon Dioxide

Let's discuss some advantages and disadvantages of using Carbon Dioxide instead of Carbon Monoxide for the gaseous diffusion process. Advantages: 1. Carbon Dioxide (CO2) is a non-toxic and non-flammable gas, unlike Carbon Monoxide (CO) which is highly toxic and flammable. Hence, handling CO2 is safer than handling CO. Disadvantages: 1. The molar mass of CO2 is higher than that of CO, which could lead to slower diffusion rates. 2. Carbon Dioxide can form solid hydrates at low temperatures, which might complicate the gaseous diffusion process.

Key concepts

Graham's Law of Effusion

Graham's law of effusion provides a simple yet powerful way to understand how gases move through small holes or apertures. It's especially useful when comparing two different gases. According to this law, the rate at which a gas effuses—which means how fast it can escape through a small hole—is inversely proportional to the square root of its molar mass.

This can be expressed with the formula:

• \( \frac{Rate_1}{Rate_2} = \sqrt{\frac{Molar \ Mass_2}{Molar \ Mass_1}} \)

The takeaway here is that a lighter gas will effuse faster than a heavier gas.

For example, when comparing isotopes of carbon monoxide like \(^{12}C^{16}O\) and \(^{12}C^{17}O\), you can calculate their relative effusion rates using their molar masses: 28 g/mol for \(^{12}C^{16}O\) and 29 g/mol for \(^{12}C^{17}O\). This calculation helps in separating these isotopes effectively.

Isotope Separation

Isotope separation is a fascinating and complex topic in chemistry. It involves sorting isotopes—or atoms with the same number of protons but different numbers of neutrons—of a specific element.

For example, in the case of oxygen isotopes such as \(^{16}O\), \(^{17}O\), and \(^{18}O\), separating these isotopes requires precision and careful technique. One effective method of achieving this separation is by gaseous diffusion.

Using carbon monoxide as a carrier gas, the isotopes can be separated based on their slight differences in effusion rates, as dictated by Graham's law of effusion. The method leverages the fact that lighter isotopes will effuse faster than heavier ones, which can be crucial in applications requiring specific isotopes.

However, the choice of gas, such as carbon monoxide versus carbon dioxide, also impacts the process based on factors like the toxicity of the gas and the efficiency of the diffusion process.

Carbon Monoxide

Carbon monoxide (CO) is a simple molecule composed of one carbon atom and one oxygen atom. Despite its simplicity, it's known for being highly toxic to humans and animals because it binds to hemoglobin more effectively than oxygen does, which can lead to serious health issues.

However, in scientific processes like isotopic separation, carbon monoxide can be useful due to its relatively low molar mass, which facilitates faster diffusion rates.

• CO's low flammability, compared to its gaseous relative carbon dioxide (CO2), makes it a better candidate for certain diffusion processes.
• It's worth noting that its toxicity does pose significant handling risks, necessitating strict safety protocols and controlled environments.

When used in the gaseous diffusion process, such as in separating oxygen isotopes, its properties lend themselves to high efficiency. It’s an interesting example of how a chemical’s properties make it suitable for specific industrial and scientific advancements.