Question

Because the neutron has no charge, its mass must be found in some way other than by using a mass spectrometer. When a neutron and a proton meet (assume both to be almost stationary), they combine and form a deuteron, emitting a gamma ray whose energy is 2.2233MeV. The masses of the proton and the deuteron are1.007276467uand 2.103553212u, respectively. Find the mass of the neutron from these data.

Short answer

The mass of the neutron from the data is 1.0087 u.

Step-by-step solution

The given data

1. Energy of the gamma ray,$E_{\gamma }=2.2233MeV$
2. Mass of the proton,$m_H=1.007276467u$
3. Mass of the deuteron,$m_d=2.013553212u$

Understanding the concept of mass and energy  

Considering the given statement of the problem, we can get an equation considering energy conservation that a proton and a neutron give rise to a deuteron with a gamma ray. Thus, the energies of proton and neutron on the left side balanced with the energy of deuteron and gamma-ray. Thus, one can get the required mass of the neutron by subtracting the total mass on the right side from the mass of the proton. Simply using the mass defect and binding energy relation, one can get the mass of the neutron.

The mass of the neutron can be calculated from the derived equation as follows:

$m_n=m_d-m_H+\frac{E_Y}{c^2}$ ….. (i)

Calculate the mass of neutron

It should be noted that when the problem statement says the “masses of the proton and the deuteron are….” they are actually referring to the corresponding atomic masses (given to very high precision). That is, the given masses include the “orbital” electrons. As in many computations in this context, this circumstance (of implicitly including electron masses in what should be a purely nuclear calculation) does not cause extra difficulty in the calculation. Thus, we can calculate the mass of the neutron using the given data in equation (i) as follows:

$\begin{array}{l}{\mathrm{m}}_{\mathrm{n}}=\left(2.013553212\right)-(1.007276467\mathrm{u})+\frac{(2.2233\mathrm{MeV})}{(931.5\mathrm{eV})}\\ =1.0062769+0.0023868\\ =1.0086637\mathrm{u}\\ \approx 1.0087\mathrm{u}\end{array}$

Here,$c^2=931.5MeV$

Hence, the value of the mass is 10087 u.