Plutonium- 239 is used as the energy source for heart pacemakers and space probes. It decays by alpha emission. (a) Calculate \(\Delta m\) in grams when one mole of Pu-239 decays. (b) How much energy (in kilojoules) is given off by the decay of \(2.00 \mathrm{mg}\) of \(\mathrm{Pu}-239 ?\)

To find the mass difference, we first need the atomic masses of Plutonium-239 (Pu-239) and the resulting element after the decay, which is Uranium-235 (U-235). Additionally, we also need the atomic mass of the alpha particle. Atomic masses (in amu): Plutonium-239 (Pu-239) = 239.052 amu, Uranium-235 (U-235) = 235.044 amu, and an alpha particle (He-4) = 4.002 amu.

When one mole of Pu-239 decays, it decreases its atomic mass and releases an alpha particle. To find the mass difference, we can subtract the sum of the atomic masses of U-235 and alpha particle from the atomic mass of Pu-239: \(\Delta m = m_{\text{Pu-239}} - (m_{\text{U-235}} + m_{\text{He-4}}) = 239.052 - (235.044 + 4.002) = 0.006\,\text{amu}\) Now, let's convert the mass difference from atomic mass units (amu) to grams: \(\Delta m \, (\text{grams}) = \Delta m \, (\text{amu}) \times \frac{1\, \text{mole}}{6.022 \times 10^{23}\, \text{atoms}} \times \frac{1.661 \times 10^{-24}\, \text{grams}}{1\, \text{amu}} = 0.006 \times \frac{1.661 \times 10^{-24}}{6.022 \times 10^{23}} = 1.66 \times 10^{-11}\, \text{grams}\) So, the mass difference when one mole of Pu-239 decays is \(1.66 \times 10^{-11}\) grams.

To find the energy released by the decay of 2.00 mg of Pu-239, we first need to determine the number of moles in 2.00 mg of Pu-239: Number of moles of Pu-239 = \(\frac{2.00 \times 10^{-3}\, \text{grams}}{239.052\, \text{grams/mol}} = 8.37 \times 10^{-6}\, \text{moles}\) Now, we can use the mass-energy equivalence formula, \(E = mc^2\), where \(m\) is the mass difference and \(c\) is the speed of light (\(2.998 \times 10^8\, \text{m/s}\)), to calculate the energy released: \(E = mc^2 = (1.66 \times 10^{-11}\, \text{grams}) \times (2.998 \times 10^8\, \text{m/s})^2 \times \frac{1\, \text{Joule}}{1\, \text{gram} \cdot (\text{m/s})^2}\) Energy per mole = \(1.49 \times 10^9\, \text{Joules}\) Finally, find the total energy released from 2.00 mg of Pu-239: Total energy = (Energy per mole) \(\times\) (Number of moles of Pu-239) = \((1.49 \times 10^9\, \text{Joules}) \times (8.37 \times 10^{-6}\, \text{moles}) = 1.25 \times 10^4\, \text{Joules}\) Now, let's convert the energy to kilojoules: Total energy (in kilojoules) = \(\frac{1.25 \times 10^4}{10^3} = 12.5\, \text{kilojoules}\) So, the energy released by the decay of 2.00 mg of Pu-239 is 12.5 kilojoules.