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 advan- tages and disadvantages of separating oxygen isotopes by gaseous diffusion of carbon dioxide instead of carbon monoxide.

Short answer

The relative rates of effusion of \({ }^{12} \mathrm{C}^{16}\mathrm{O}\), \({ }^{12} \mathrm{C}^{17}\mathrm{O}\), and \({ }^{12} \mathrm{C}^{18}\mathrm{O}\) are approximately 1:0.965:0.931. Using carbon dioxide instead of carbon monoxide for the separation process has some advantages, such as being less toxic and more stable, but might also lead to a less efficient separation due to smaller differences in relative effusion rates.

Step-by-step solution

Find the molar masses of the isotopes

Calculate the molar masses of the three isotopes of carbon monoxide by summing the atomic masses of the carbon and oxygen atoms. We will use the atomic mass unit (u) for the calculation: - \({ }^{12} \mathrm{C}^{16}\mathrm{O}\): \(12 u + 16 u = 28 u\) - \({ }^{12} \mathrm{C}^{17}\mathrm{O}\): \(12 u + 17 u = 29 u\) - \({ }^{12} \mathrm{C}^{18}\mathrm{O}\): \(12 u + 18 u = 30 u\)

Apply Graham's Law of Effusion

Graham's Law of Effusion states that the rate of effusion of two gases is inversely proportional to the square root of their molar masses: \(\frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}}\). We can use this formula to compare the rates of effusion of the three isotopes: Relative rate of effusion of \({ }^{12} \mathrm{C}^{16}\mathrm{O}\) to \({ }^{12} \mathrm{C}^{17}\mathrm{O}\): \(\frac{Rate_{16}}{Rate_{17}} = \sqrt{\frac{29u}{28u}} \approx 1.035\) Relative rate of effusion of \({ }^{12} \mathrm{C}^{16}\mathrm{O}\) to \({ }^{12} \mathrm{C}^{18}\mathrm{O}\): \(\frac{Rate_{16}}{Rate_{18}} = \sqrt{\frac{30u}{28u}} \approx 1.069\)

Advantages and disadvantages of using carbon dioxide

Separating oxygen isotopes by gaseous diffusion of carbon dioxide instead of carbon monoxide could have the following advantages and disadvantages: Advantages: 1. Carbon dioxide is less toxic, making the process safer for workers. 2. Carbon dioxide is a more stable molecule, which could lead to fewer complications or side reactions during the process. Disadvantages: 1. The molar masses of the isotopes would be higher in the case of carbon dioxide, which could lead to smaller differences in relative effusion rates, making the separation process more challenging or less efficient. 2. Carbon dioxide might be less reactive with certain materials, making it potentially more difficult to separate the isotopes after the diffusion process. In conclusion, the relative rates of effusion of \({ }^{12} \mathrm{C}^{16}\mathrm{O}\), \({ }^{12} \mathrm{C}^{17}\mathrm{O}\), and \({ }^{12} \mathrm{C}^{18}\mathrm{O}\) are approximately 1:0.965:0.931. While using carbon dioxide instead of carbon monoxide for the separation process has some advantages, it might also lead to a less efficient separation due to smaller differences in relative effusion rates.

Key concepts

Graham's Law

Graham's Law explains how gases effuse through small openings, a process highly related to their molar masses. This law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. If you have two different gases, you can compare their effusion rates using the formula:

• \( \frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}} \)

This means that a lighter gas (with a lower molar mass) effuses faster than a heavier one. In our context, the goal is to compare the rates of effusion for different isotopes of carbon monoxide. By understanding and applying Graham's Law, we can predict which isotope will pass through a barrier faster, a handy tool in isotope separation processes.

Oxygen Isotopes

Oxygen isotopes are forms of oxygen with different numbers of neutrons, giving them different atomic masses. The most common isotopes in our calculations are:

• \(^{16}\text{O}\)
• \(^{17}\text{O}\)
• \(^{18}\text{O}\)

Though they all contain eight protons (which defines them as oxygen), their neutron count makes them distinct. These distinctions are crucial in scientific studies, as they can affect chemical behavior and physical properties. In processes like gaseous diffusion, the tiny differences in mass due to isotopes are enough to separate them effectively. This subtle difference is what allows industries to exploit isotopic separations for various applications.

Molar Mass Calculation

Calculating the molar mass of any molecular compound is central to predicting its chemical behavior. For carbon monoxide isotopes like \(^{12}\text{C}^{16}\text{O}\), molar mass calculations involve adding the atomic masses of carbon and the specific oxygen isotope:

• \(^{12}\text{C}^{16}\text{O}: 12 \text{u} + 16 \text{u} = 28 \text{u}\)
• \(^{12}\text{C}^{17}\text{O}: 12 \text{u} + 17 \text{u} = 29 \text{u}\)
• \(^{12}\text{C}^{18}\text{O}: 12 \text{u} + 18 \text{u} = 30 \text{u}\)

Precise molar mass calculations are pivotal for applying Graham's Law, as they dictate how fast each isotope will effuse. A slight change in molar mass leads to noticeable differences in behavior during diffusion processes, allowing us to separate isotopes effectively.

Gaseous Diffusion

Gaseous diffusion is a technique used to separate gas molecules based on differences in their movement rate through a porous barrier. It mimics the natural process of effusion, where lighter molecules pass through barriers faster than heavier ones. This is especially important in separating isotopes, which only differ in mass.

• The process exploits Graham's Law to distinguish between competing isotopes/tiny mass differences.
• It is widely used in enriching uranium and separating isotopes in industries.

Despite its effectiveness, gaseous diffusion can be energy-intensive and is affected by the properties of the molecules involved such as size and shape. The diffusion efficiency depends on both the physical setup and the molecular characteristics of the gases considered.