Chapter 38: Problem 40

Find the atomic mass of iridium-193, whose binding energy is 7.94 MeV/nucleon.

Short Answer

Expert verified

The atomic mass of iridium-193 is \(29 u\).

Step by step solution

- Convert binding energy from MeV/nucleon to Joules/nucleon

1 MeV is approximately equal to \(1.60218 \times 10^{-13}\) Joules (by definition). So, the binding energy of iridium-193 in Joules/nucleon is \(7.94 MeV/nucleon \times 1.60218 \times 10^{-13} J/MeV = 1.27 \times 10^{-12} J/nucleon\).

- Find the total binding energy

As iridium-193 has 193 nucleons, the total binding energy is equal to \(1.27 \times 10^{-12} J/nucleon \times 193 nucleons = 2.45 \times 10^{-10} J\)

- Use Einstein’s mass-energy equivalence principle to find the atomic mass

Relating energy and mass through \(E = mc^2\) gives \(m = E/c^2\), where \(E\) is the total energy (or the total binding energy), \(m\) is the mass we want to find, and \(c\) is the speed of light (\(3 \times 10^8 m/s\)). Substituting the values gives \(m = 2.45 \times 10^{-10} J / (3 \times 10^8 m/s)^2 \approx 2.73 \times 10^{-27} kg\). Usually atomic masses are expressed in atomic mass units (u), where 1 u = \(1.660539 \times 10^{-27}\) kg. Converting the mass result to atomic mass units gives \(m = 2.73 \times 10^{-27} kg / 1.660539 \times 10^{-27} kg/u \approx 164 u\)

- Considering existence of binding energy

However, the above mass calculated was for the energy being present. The actual mass of the atom will be less than 193 u (as it is iridium-193) by the extent of the energy being present (in the form of reduced mass). So, the atomic mass of iridium-193 will be \(193 u - 164 u = 29 u\).

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Find the atomic mass of iridium-193, whose binding energy is 7.94 MeV/nucleon.

Short Answer

Step by step solution

- Convert binding energy from MeV/nucleon to Joules/nucleon

- Find the total binding energy

- Use Einstein’s mass-energy equivalence principle to find the atomic mass

- Considering existence of binding energy

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