Question

Consider a small virus having a diameter of 10 nm. The atoms of the intracellular fluid are confined within the virus. Suppose we model the virus as a 10-nm-long “box.” What is the ground state energy (in eV) of a sodium ion confined in this box?

Short answer

The ground state energy (in eV) of a sodium ion is${\mathrm{E}}_1=9\times {10}^{-8}\mathrm{eV}$

Step-by-step solution

Given information

Consider a tiny virus with a 10 nm diameter. Within the virus, the atoms of the internal fluid are contained.

Determine ground state energy  of a sodium ion 

We can begin by expressing the energy of a particle in a box as follows:

$$\begin{gathered} {\mathrm{E}}_{\mathrm{n}}=\frac{{\mathrm{n}}^2{\mathrm{h}}^2}{8{\mathrm{mL}}^2} \\ \mathrm{n}=1\text{(ground state energy)} \\ {\mathrm{E}}_1=\frac{1^2\times {\left(6.63\times {10}^{-34}\right)}^2}{\left(8\times 3\times 82\times {10}^{-26}\right)\times {\left(10\times {10}^{-9}\right)}^2}\text{(substitute)} \\ {\mathrm{E}}_1=1.44\times {10}^{-26}\mathrm{J}=9\times {10}^{-8}\mathrm{eV} \end{gathered}$$