Question

|You learned in Chapter 37 that, except for hydrogen, the mass of a nucleus with atomic number Z is larger than the mass of the Z protons. The additional mass was ultimately discovered to be due to neutrons, but prior to the discovery of the neutron it was suggested that a nucleus with mass number A might contain A protons and (A-Z) electrons. Such a nucleus would have the mase of A protone, but ite net charge would be only Z o.

a. We know that the diameter of a nuclens is approximately 10 fmm. Model the nucleus as a one-dimensional box and find the minimum range of speeds that an electron would have in such a box.

b. What does your answer imply about the possibility that the nucleus contains electrons? Explain.

Short answer

Thus, the minimum range of speeds that electron have is from $0\mathrm{m}/\mathrm{s}$ to $1.82\times {10}^{10}\mathrm{m}/\mathrm{s}$

The existence of electron inside the nucleus is not possible.

Step-by-step solution

part (a) step 1: Given information

Diameter of the nucleus $\Delta x$

$$\begin{gathered} =10\mathrm{fm}=10\mathrm{fm}\left(\frac{{10}^{-15}\mathrm{m}}{1.0\mathrm{fm}}\right) \\ \Delta x=10\times {10}^{-15}\mathrm{m} \end{gathered}$$

Formula to be used,

According to Heisenberg uncertainty principle, the velocity is given by

$\Delta vx=\frac{h}{2m\Delta x}\dots ..\dots \dots \left(1\right)$

Where,$\Delta \mathbf{vx}$is uncertainty velocity, $\Delta \mathrm{x}$ is the uncertainity in position, $\mathrm{m}$is the mass, h is palnk's constant.

part(b) step 2: solution

$\text{Applying h}=6.63\times {10}^{-34}\mathrm{JS},\mathrm{m}=9.11\times {10}^{-15}\mathrm{m}\text{in eq}\left(1\right)$

$$\begin{gathered} \Delta vx=\frac{6.63\times {10}^{-3/}\mathrm{SS}}{2\left(9.11\times {10}^{-31}\mathrm{kg}\right)\left(10\times {10}^{-33}\mathrm{m}\right)} \\ \Delta vx=3.64\times {10}^{10}\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore the possible range of velocities is from $-1.82\times {10}^{10}\mathrm{m}/\mathrm{s}$to $1.85\times {10}^{10}\mathrm{m}/\mathrm{s}.$The speed cannot have negative values,

Therefore the possible range of speed is from 0 to $1.82\times {10}^{10}\mathrm{m}/\mathrm{s}$

part(b) step 3: Given information 

The electron has a speed of $1.82\times {10}^{10}\mathrm{m}/\mathrm{s}$inside the nucleus is greater than the speed of light(which is not possible). Therefore, the existence of electron inside the nucleus is not possible.