A nuclear reaction of the kind \({ }_{2}^{3} \mathrm{He}+{ }_{6}^{12} \mathrm{C} \rightarrow \mathrm{X}+\alpha\) is called a pick-up nuclear reaction. a) Why is it called a pick-up reaction, that is, what is picked up, what picked it up, and where did it come from? b) What is the resulting nucleus X? c) What is the \(\mathrm{Q}\) -value of this reaction? d) Is this reaction endothermic or exothermic?

In a pick-up nuclear reaction, a projectile nucleus picks up a nucleon (or nucleons) from a target nucleus to form a new nucleus. In this case, the projectile nucleus is \({}_{2}^{3} \mathrm{He}\) (helium), the target nucleus is \({}_{6}^{12}\mathrm{C}\) (carbon), and the resulting nucleus X is the product of helium picking up a nucleon (or nucleons) from carbon.

Given the nuclear reaction: \({}_{2}^{3} \mathrm{He}+{ }_{6}^{12}\mathrm{C} \rightarrow \mathrm{X}+\alpha\) , we can find X by conserving nucleon number and proton number. The initial nucleon number is: \(3+12=15\) The initial proton number is: \(2+6=8\) As the alpha particle has a nucleon number of 4 and a proton number of 2 (alpha particle is \({}_{2}^4\mathrm{He}\)), subtract these numbers from initial nucleon and proton numbers to determine X: Nucleon number of X: \(15 - 4 = 11\) Proton number of X: \(8 - 2 = 6\) Therefore, the resulting nucleus X is \({}_{6}^{11}\mathrm{C}\) (carbon with a mass number 11).

The Q-value of the reaction, denoted as \(\mathrm{Q}\), represents the energy released (if positive) or absorbed (if negative) during the reaction. It can be calculated as the difference in the masses of the reactants and the masses of the products, multiplied by the speed of light c^2: \(\mathrm{Q} = (\Delta m) c^2\) Here, \(\Delta m\) represents the mass difference between the reactants and the products. Mass of \({}_{2}^{3}\mathrm{He}\) = 3.01603 u Mass of \({}_{6}^{12}\mathrm{C}\) = 12.00000 u Add both for mass of reactants Mass of \({}_{6}^{11}\mathrm{C}\) = 11.01143 u Mass of alpha particle ( \({}_{2}^4\mathrm{He}\)) = 4.00151 u Add both for mass of products Next, find the mass difference, \(\Delta m\): \(\Delta m = \textrm{mass of reactants} - \textrm{mass of products}\) \(\Delta m = (3.01603 + 12.00000) - (11.01143 + 4.00151)\) \(\Delta m = 0.00309 \textrm{ u}\) Finally, calculate Q using the energy conversion constant for atomic mass units to MeV, which is 931.5 MeV/c²: \(\mathrm{Q} = (\Delta m)(c^2) = (0.00309 \textrm{ u})(931.5 \textrm{ MeV/c}^2)\) \(\mathrm{Q} \approx 2.88 \textrm{ MeV}\)

Since the Q-value of the reaction is positive, it indicates that the reaction releases energy, and therefore, it is an exothermic reaction.