The answer to # of atoms is 0.033*10^24 cm^-3 The answer to linear attenuation i

Question

The answer to # of atoms is 0.033*10^24 cm^-3

The answer to linear attenuation is 0.46 cm^-1

The answer to half thickness is 1.5 cm

Please derive answer and show all work.

5.6 The element lead, M = 207.2, has a density of 11.35 g/cm^3. Find the number of atoms per cubic centimeter. If the total gamma ray cross section at 3 MeV is 14 barns, what an the linear attenuation coefficient and the half-thickness? The answer to # of atoms is 0.033*10^24 cm^-3 The answer to linear attenuation is 0.46 cm^-1 The answer to half thickness is 1.5 cm Please derive answer and show all work.

Explanation / Answer

molar mass of lead = 207.2 g

density = 11.35 g/cm^3

So,

number of atoms per cm^3 = ( density / molar mass) * Avogadro constant

number of atoms per cm^3 = (11.35 / 207.2) * 6.022 x 10^23 cm^-3

number of atoms per cm^3 = 0.33 x 10^23 cm^-3 = 0.033 x 10^24 cm^-3

Now,

14 barns = 14 * 1 x 10^-24 cm^2 = 14 x 10^-24 cm^2

Now linear attenuation coefficient = Number of atoms * 14 barns = 0.033 x 10^24 cm^-3 x 14 x 10^-24 cm^2

linear attenuation coefficient = 0.462 cm^-1

Now ,

For half thickness,

t(1/2) = - ln(0.5) / (mass attenuation coeff * density)

mass attenuation coefficent = linear attenuation coefficient / density

t(1/2) = - ln(0.5) / (linear attenuation coefficient)

t(1/2) = 0.6931 / 0.462 = 1.5 cm

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