Question

Consider the fusion of B-10 with an alpha particle. The products of the fusion are \(\mathrm{C}-13\) and a proton. (a) Write a nuclear reaction for this process. (b) How much energy is released when \(1.00 \mathrm{~g}\) of \(\mathrm{B}-10\) is fused with an \(\alpha\) -particle?

Short answer

Answer: The nuclear reaction for the fusion of B-10 with an alpha particle is: \(_{5}^{10}\mathrm{B} + _{2}^{4}\mathrm{He} \rightarrow _{6}^{13}\mathrm{C} + _{1}^{1}\mathrm{H}\) When 1.00 g of B-10 is fused with an alpha particle, \(2.75 \times 10^{11} \mathrm{J}\) of energy is released.

Step-by-step solution

Write the nuclear reaction equation

To write the nuclear reaction for this process, we must represent the reactants and products using their symbols and atomic numbers. Reactants: \(\mathrm{B-10}\) = \(_{5}^{10}\mathrm{B}\) (Boron-10) \(\alpha\)-particle = \(_{2}^{4}\mathrm{He}\) (Helium-4) Products: \(\mathrm{C-13}\) = \(_{6}^{13}\mathrm{C}\) (Carbon-13) proton = \(_{1}^{1}\mathrm{H}\) (Hydrogen-1) Now, let's combine these reactants and products into a single equation: \(_{5}^{10}\mathrm{B} + _{2}^{4}\mathrm{He} \rightarrow _{6}^{13}\mathrm{C} + _{1}^{1}\mathrm{H}\)

Find the mass difference between the reactants and products

We need to find the mass difference between the reactants and products to calculate the energy released. Masses (in atomic mass units): \(_{5}^{10}\mathrm{B}\): 10.0129370 u \(_{2}^{4}\mathrm{He}\): 4.0015060 u \(_{6}^{13}\mathrm{C}\): 13.0033548 u \(_{1}^{1}\mathrm{H}\): 1.0078250 u Mass difference = (mass of reactants) - (mass of products) Mass difference = (10.0129370 + 4.0015060) - (13.0033548 + 1.0078250) = 0.0032632 u

Find the energy released per reaction

Using the mass-energy equivalence formula, we can determine the energy released per reaction. The formula is: Energy = mass difference × \(c^2\) Here, c is the speed of light (\(3.00 \times 10^8 \mathrm{m/s}\)), and we need to convert the mass difference in atomic mass units (u) to kilograms. 1 u = \(1.6605 \times 10^{-27} \mathrm{kg}\) Mass difference = 0.0032632 u × \(1.6605 \times 10^{-27} \mathrm{kg/u}\) = \(5.42 \times 10^{-30} \mathrm{kg}\) Energy = \((5.42 \times 10^{-30} \mathrm{kg})\times (3.00 \times 10^8 \mathrm{m/s})^2\) = \(4.58 \times 10^{-12} \mathrm{J}\)

Determine the number of reactions for 1.00 g of B-10

To determine the number of reactions that occur when 1.00 g of B-10 is fused with an alpha particle, we need to convert the mass of B-10 into moles and then find the number of atoms involved. Moles of B-10 = \(\frac{1.00 \mathrm{g}}{10.0129370 \mathrm{g/mol}}\) = 0.09988 mol number of atoms = moles × Avogadro's number number of atoms = \(0.09988 \mathrm{mol} \times 6.022 \times 10^{23} \mathrm{atoms/mol}\) = \(6.01 \times 10^{22} \mathrm{atoms}\)

Calculate the total energy released

Now that we know the energy released per reaction and the number of reactions, we can calculate the total energy released. Total energy = Energy per reaction × number of reactions Total energy = \((4.58 \times 10^{-12} \mathrm{J})\times (6.01 \times 10^{22} \mathrm{atoms})\) = \(2.75 \times 10^{11} \mathrm{J}\) So, when 1.00 g of B-10 is fused with an alpha particle, \(2.75 \times 10^{11} \mathrm{J}\) of energy is released.