Atoms And Nuclei

To find the frequency of the photon emitted when an electron transitions from the Bohr orbit to the ground state in a hydrogen atom, we start with the following equations:Coulomb's Lawfor the centripetal force: Bohr's Quantization Condition: When we divide these equations, we derive the speed of the electron, , as: Given in the problem: Thus: This allows us to express the speed of the electron in the orbit as: The normalized speed is given by: Solving for : To find the change in energy when the electron transitions from the orbit to the ground state: For , we calculate the frequency : Finally, calculating using a calculator gives:

Step 1: Calculate the total energy in joules equivalent to 20,000 units of TNT.$ $ units of TNT is:$ 8.368 \times 10^{13} \mathrm{~J} \left(6.022 \times 10^{23}\right) (239 \mathrm{~g} / \mathrm{mole}) $Thus, the closest estimate of the amount of Plutonium required is approximately 1 kg .So, the closest estimate of the amount of Plutonium required for a 20,000 units of TNT explosion is around 1 kg .

In Bohr's atomic model, the radius of an electron orbit is given by the formula: This can be simplified to show that: where: is the principal quantum number, is the atomic number (indicating the number of protons in the nucleus, or the effective charge).For this problem, the values of and are crucial since other factors remain constant irrespective of the orbital angular momentum. The question states that the nucleus has a charge , which implies that the atomic number for the second nucleus is double that of the first one, i.e., . The principal quantum number remains unchanged, hence .Thus, we have: Substituting the values: Therefore, the radius of the orbit for an electron with the same principal quantum number around a nucleus of charge is .

Calculate Mass Defect for Each Isotope:For :Number of protons : 8Number of neutrons : Mass of 8 protons: Mass of 8 neutrons: Total mass of nucleons: Mass defect : For :Number of protons : 8Number of neutrons : Mass of 8 protons: Mass of 10 neutrons: Total mass of nucleons: Mass defect : Calculate Binding Energies in :Binding energy for : Binding energy for : Difference in Binding Energies:Calculate the difference: Therefore, the difference between the binding energies of the and nuclei is .