Question

The binding energy Per nucleon of deuteron $\left({ }^{2}{ }_{1} \mathrm{H}\right)\( and Lielium nucleus \){ }_{2}{ }^{4}{ }_{2} \mathrm{He}$ ) is \(1.1 \mathrm{MeV}\) and \(7.0 \mathrm{MeV}\). respectively. If two deuteron react to form a single helium nucleus, the energy released is (A) \(23.6 \mathrm{MeV}\) (B) \(26.9 \mathrm{MeV}\) (C) \(13.9 \mathrm{MeV}\) (D) \(19.2 \mathrm{MeV}\)

Short answer

The energy released in the reaction when two deuterons react to form a single helium nucleus is \(23.6\,\text{MeV}\).

Step-by-step solution

Calculate the total binding energy of two deuterons

First, we need to find the total binding energy of two deuterons: Binding energy per nucleon of a deuteron: \(1.1\,\text{MeV}\) Each deuteron contains 2 nucleons, so the binding energy for a single deuteron is: \[\text{Binding energy of a deuteron} = 1.1\,\text{MeV/nucleon} \times 2\,\text{nucleons} = 2.2\,\text{MeV}\] Since we have two deuterons, the total binding energy for two deuterons is: \[\text{Total binding energy of two deuterons} = 2.2\,\text{MeV/deuteron} \times 2\,\text{deuterons} = 4.4\,\text{MeV}\]

Calculate the binding energy of a helium nucleus

Now, we need to find the total binding energy of the helium nucleus: Binding energy per nucleon of a helium nucleus: \(7.0\,\text{MeV}\) The helium nucleus contains 4 nucleons, so the binding energy for a single helium nucleus is: \[\text{Binding energy of a helium nucleus} = 7.0\,\text{MeV/nucleon} \times 4\,\text{nucleons} = 28.0\,\text{MeV}\]

Calculate the energy released in the reaction

The energy released in the reaction is the difference between the total binding energy of the initial system (two deuterons) and the final system (one helium nucleus): \[\text{Energy released} = \text{Binding energy of the helium nucleus} - \text{Total binding energy of two deuterons}\] \[\text{Energy released} = 28.0\,\text{MeV} - 4.4\,\text{MeV} = 23.6\,\text{MeV}\] The energy released in the reaction is \(23.6\,\text{MeV}\). Therefore, the correct answer is (A) \(23.6\,\text{MeV}\).