The largest stable nucleus has a nucleon number of 209, and the smallest has a nucleon number of 1. If each nucleus is assumed to be a sphere, what is the ratio (largest/smallest) of the surface areas of these spheres? (Area209/Area1)

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sphere volume ... 4/3 π r^3

surface area ... 4 π r^2

s.a. = k v^(2/3)

Area209 / Area1 = k[209^(2/3)] / k[1^(2/3)] = 209^(2/3)

To find the ratio of the surface areas of the two spheres, we need to calculate the areas and then divide them.

The formula for the surface area of a sphere is:

Surface Area = 4 * π * r^2

Where r is the radius of the sphere. In this case, we don't have the radius directly, but we have the nucleon number, which is proportional to the mass number (A), given by:

A = number of protons (Z) + number of neutrons (N)

Let's calculate the radii (r1 and r209) for the two spheres:

For the smallest nucleus with a nucleon number of 1, we have A1 = 1.
We assume that the nucleus is a hydrogen atom, which has one proton and no neutrons.
So, Z1 = 1, and N1 = 0.
A1 = Z1 + N1 = 1.
Therefore, r1 = A1^(1/3) = 1^(1/3) = 1.

For the largest nucleus with a nucleon number of 209, we have A209 = 209.
We assume that the nucleus is lead-209, which has 82 protons and 127 neutrons.
So, Z209 = 82, and N209 = 127.
A209 = Z209 + N209 = 82 + 127 = 209.
Therefore, r209 = A209^(1/3) = 209^(1/3) ≈ 5.96.

Now, let's calculate the surface areas of the spheres using the formula:

Surface Area1 = 4 * π * r1^2 = 4 * π * 1^2 = 4 * π.
Surface Area209 = 4 * π * r209^2 = 4 * π * (5.96)^2 ≈ 4 * 3.14 * 35.52 ≈ 445.03 * π.

Finally, we can calculate the ratio of the surface areas:

Ratio = Surface Area209 / Surface Area1 = (445.03 * π) / (4 * π) = 445.03 / 4 ≈ 111.26.

So, the ratio of the surface areas of the spheres is approximately 111.26.