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 disad- vantages of separating oxygen isotopes by gaseous diffusion of carbon dioxide instead of carbon monoxide.

First, we need to determine the molecular masses of the three carbon monoxide isotopes: \({ }^{12} \mathrm{C}^{16} \mathrm{O},{ }^{12} \mathrm{C}^{17} \mathrm{O}\), and \({ }^{12} \mathrm{C}^{18} \mathrm{O}\). \({ }^{12} \mathrm{C}^{16} \mathrm{O}\): Molecular mass = 12 (for carbon) + 16 (for oxygen) = 28 amu. \({ }^{12} \mathrm{C}^{17} \mathrm{O}\): Molecular mass = 12 (for carbon) + 17 (for oxygen) = 29 amu. \({ }^{12} \mathrm{C}^{18} \mathrm{O}\): Molecular mass = 12 (for carbon) + 18 (for oxygen) = 30 amu.

We can now use Graham's law of effusion formula to calculate the relative rates of effusion for the isotopes. 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: Rate of effusion = \(k \times \frac{1}{\sqrt{molar\ mass}}\) Here, k is a proportionality constant, which will be the same for all three isotopes in consideration.

We will calculate the relative rates of effusion for each isotope by choosing \({ }^{12} \mathrm{C}^{16} \mathrm{O}\) as a reference. Relative rate of effusion for \({ }^{12} \mathrm{C}^{16} \mathrm{O}\): \(\frac{Rate_{^{12}C^{16}O}}{Rate_{^{12}C^{16}O}} = \frac{\frac{k}{\sqrt{28}}}{\frac{k}{\sqrt{28}}}\)= 1 Relative rate of effusion for \({ }^{12} \mathrm{C}^{17} \mathrm{O}\): \(\frac{Rate_{^{12}C^{17}O}}{Rate_{^{12}C^{16}O}} = \frac{\frac{k}{\sqrt{29}}}{\frac{k}{\sqrt{28}}}\)= \(\frac{\sqrt{28}}{\sqrt{29}}\) Relative rate of effusion for \({ }^{12} \mathrm{C}^{18} \mathrm{O}\): \(\frac{Rate_{^{12}C^{18}O}}{Rate_{^{12}C^{16}O}} = \frac{\frac{k}{\sqrt{30}}}{\frac{k}{\sqrt{28}}}\)= \(\frac{\sqrt{28}}{\sqrt{30}}\)

Using carbon dioxide instead of carbon monoxide for oxygen isotope separation can have advantages and disadvantages. Advantages: 1. Carbon dioxide is a more stable and less toxic molecule compared to carbon monoxide, which makes the process safer. 2. Carbon dioxide is more abundant in nature and easier to obtain compared to carbon monoxide, which could result in a more cost-effective process. Disadvantages: 1. Carbon dioxide has a higher molecular mass than carbon monoxide (44 amu for CO₂ vs 28-30 amu for CO), which could lead to slower effusion and lower separation efficiency when it comes to separating isotopes. 2. Carbon dioxide can also form a variety of chemical species with different masses, which could complicate the separation process. In summary, the relative rates of effusion for \({ }^{12} \mathrm{C}^{16} \mathrm{O},{ }^{12} \mathrm{C}^{17} \mathrm{O}\), and \({ }^{12} \mathrm{C}^{18} \mathrm{O}\) are 1, \(\frac{\sqrt{28}}{\sqrt{29}}\), and \(\frac{\sqrt{28}}{\sqrt{30}}\), respectively. There are advantages and disadvantages to using carbon dioxide for oxygen isotope separation instead of carbon monoxide, which should be considered when determining the best method for the process.