How many times larger is the volume of the nucleus ²⁰⁸Pb Z = 82) compared to the nucleus ⁴⁰Ca Z = 20)?

Question

Quizplus · Accepted Answer

To find the volume, we have to use the formula V = (4/3)πr^3, where r is the radius of the nucleus. For the nucleus ²⁰⁸Pb, Z = 82, so it has 82 protons. The number of neutrons can be found by subtracting the atomic number from the mass number. From the symbol given, we know that the mass number is 208, so the number of neutrons is 208 - 82 = 126. The radius of the nucleus can be estimated using the empirical formula: r = r0 * A^(1/3), where r0 is a constant equal to about 1.2 x 10^-15 m, and A is the mass number. Using this formula, we can estimate the radius of the Pb nucleus to be: r(Pb) = 1.2 x 10^-15 m * 208^(1/3) = 7.27 x 10^-15 m Using the same method, we can estimate the radius of the Ca nucleus: r(Ca) = 1.2 x 10^-15 m * 40^(1/3) = 4.14 x 10^-15 m Now we can find the ratio of their volumes: V(Pb) / V(Ca) = (4/3)πr(Pb)^3 / (4/3)πr(Ca)^3 V(Pb) / V(Ca) = (r(Pb) / r(Ca))^3 V(Pb) / V(Ca) = (7.27 x 10^-15 m / 4.14 x 10^-15 m)^3 V(Pb) / V(Ca) = 4.12^3 V(Pb) / V(Ca) = 71.5 Therefore, the volume of the Pb nucleus is 71.5 times larger than the volume of the Ca nucleus. But since the question asks for the ratio in terms of multiples (i.e., "how many times larger"), we need to subtract 1: V(Pb) / V(Ca) = 71.5 V(Pb) = 71.5 * V(Ca) V(Pb) = 71.5 * 1 = 71.5 So the volume of the Pb nucleus is 71.5 times larger than the volume of the Ca nucleus. Therefore, the answer is D.

How Many Times Larger Is the Volume of the Nucleus