Chapter 11: Problem 3

A mixture of two common isotopes of oxygen, oxygen-16 and oxygen-18, is put in a chamber that is then spun around at a very high speed. It is found that one isotope is more concentrated near the axis of rotation of the chamber and the other is more concentrated near the outer part of the chamber. Why is that, and which isotope is where?

Short Answer

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Explain why. Answer: Oxygen-16 is more concentrated near the axis of rotation, and oxygen-18 is more concentrated near the outer part of the chamber. This is because the centrifugal force acting on oxygen-16 is weaker than that on oxygen-18, causing oxygen-16 to stay closer to the axis, while the stronger centrifugal force pushes oxygen-18 towards the outer part of the chamber.

Step by step solution

Understand the principle of a centrifuge

A centrifuge is a device that spins at high speeds to separate substances based on their mass or density. The centrifugal force is stronger towards the outer part of the spinning chamber, which helps separate substances with different densities (or mass-to-volume ratios). When a mixture of isotopes is placed in the centrifuge, the isotopes experience different centrifugal forces due to their differing masses.

Calculate the centrifugal force exerted on each isotope

The centrifugal force can be calculated using the following formula: Centrifugal Force (F) = m * r * ω^2 where m is the mass of the isotope, r is the distance from the axis of rotation, and ω is the angular velocity of the spinning chamber. In this case, we have two isotopes of oxygen: oxygen-16 and oxygen-18 with masses of 16 and 18 atomic mass units (amu), respectively. The angular velocity (ω) will be the same for both isotopes as they are in the same chamber. We only need to find the ratio of the centrifugal forces acting on each isotope. Let's assume that the r and ω values are the same for each isotope to simplify calculations. F_16/F_18 = (16 * r * ω^2)/(18 * r * ω^2)

Find the ratio of the centrifugal forces acting on both isotopes

From the previous step, we can now determine the ratio of the centrifugal forces acting on both isotopes: F_16/F_18 = (16* r * ω^2) /(18 * r * ω^2) Canceling the common terms, we get: F_16/F_18 = 16/18 = 8/9 Since 8/9 < 1, the centrifugal force acting on oxygen-16 is less than that on oxygen-18.

Determine the location of each isotope in the chamber

As the centrifugal force acting on oxygen-16 is less than that on oxygen-18, oxygen-16 will be more concentrated near the axis of rotation (where the centrifugal force is weaker). In contrast, oxygen-18, which experiences a greater centrifugal force, will be pushed more towards the outer part of the chamber. In conclusion, oxygen-16 is more concentrated near the axis of rotation, and oxygen-18 is more concentrated near the outer part of the chamber.

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Short Answer

Step by step solution

Understand the principle of a centrifuge

Calculate the centrifugal force exerted on each isotope

Find the ratio of the centrifugal forces acting on both isotopes

Determine the location of each isotope in the chamber

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