Question

Three electrons are trapped in three different one-dimensional infinite potential wells of widths (a) 50pm (b)200pm, and (c)100pm . Rank the electrons according to their ground-state energies, greatest first.

Short answer

(a) The ground state energy of an electron that trapped in the potential well of widths 50pm is $2.41\times {10}^{-17}J$.

(b) The ground state energy of an electron that trapped in the potential well of widths 200pm is $1.51\times {10}^{-18}J$.

(c) The ground state energy of an electron that trapped in the potential well of widths 100pm is $6.03\times {10}^{-18}J$.

The electron that trapped in the potential well of width 50pm has highest ground state energy, followed by the electron that trapped in the potential well of width 100pm and the electron that trapped in the potential well of width 200pm has lowest ground state energy.

Step-by-step solution

Identification of the given data

The given data can be listed below as,

The width of first potential well is,$L_1=50pm$ .

The width of the second potential well is,$L_2=200pm$ .

The width of the third potential well is, $L_3=100pm$.

The quantum number for ground state is, n = 1 .

Significance of electron in an infinite potential well

An electron confined to an infinite potential well can exist in only specific states. The relation between the electron's energy and the potential well's width is an inverse one.

(a) Determination the ground state energy of an electron that trapped in the potential well of widths

The expression to calculate the ground state energy of an electron that trapped in the potential well of widths 50pm is expressed as,

$E_n=\left(\frac{h^2}{8m{L}_1^2}\right)n^2$

Here,$E_n$ is the energy of an electron that trapped in the potential well of width 50pm , is the Plank’s constant whose value is$6.63\times {10}^{-34}J.s$ , is the mass an electron whose value is$9.11\times {10}^{-31}kg$ .

Substitute all the known values in the above equation.

role="math" localid="1661762987913" $$\begin{gathered} E_1=\left[\frac{{\left(6.63\times {10}^{-34}J.s\right)}^2}{8\left(9.11\times {10}^{-31}kg\right)\left(50pm\times {\displaystyle \frac{{10}^{-12}m}{1pm}}\right)}\right]{\left(1\right)}^2 \\ \approx 2.41\times {10}^{-17}{J}^2.s^2/kg.m^2 \\ \approx \left(2.41\times {10}^{-17}{J}^2.s^2/kg.m^2\right)\left(\frac{1J}{1J^2.s^2/kg.m^2}\right) \\ \approx 2.41\times {10}^{-17}J \end{gathered}$$

Thus, the ground state energy of an electron that trapped in the potential well of widths 50pm is$2.41\times {10}^{-17}J$ .

(b) Determination the ground state energy of an electron that trapped in the potential well of widths 200pm

The expression to calculate the ground state energy of an electron that trapped in the potential well of widths 200pm is expressed as,

$\left(E_n\right)\text{'}=\left(\frac{h^2}{8m{L}_2^2}\right)n^2$

Here,$\left(E_n\right)\text{'}$ is the energy of an electron that trapped in the potential well of width 200pm .

Substitute all the known values in the above equation.

$$\begin{gathered} \left(E_n\right)\text{'}\text{'}=\left[\frac{{\left(6.63\times {10}^{-34}J.s\right)}^2}{8\left(9.11\times {10}^{-31}kg\right){\left(50pm\times {\displaystyle \frac{{10}^{-12}m}{1pm}}\right)}^2}\right]{\left(1\right)}^2 \\ \approx 2.41\times {10}^{-17}{J}^2.s^2/kg.m^2 \\ \approx \left(2.41\times {10}^{-17}{J}^2.s^2/kg.m^2\right)\left(\frac{1J}{1J^2.s^2/kg.m^2}\right) \\ \approx 2.41\times {10}^{-17}J \end{gathered}$$

Thus, the ground state energy of an electron that trapped in the potential well of widths is$2.41\times {10}^{-17}J$ .

(c) Determination the ground state energy of an electron that trapped in the potential well of widths

The expression to calculate the ground state energy of an electron that trapped in the potential well of widths 100pm is expressed as,

$\left(E_n\right)\text{'}\text{'}=\left(\frac{h^2}{8m{L}_3^2}\right)n^2$

Here,$\left(E_n\right)\text{'}\text{'}$ is the energy of an electron that trapped in the potential well of width.

Substitute all the known values in the above equation.

$$\begin{gathered} \left(E_n\right)\text{'}\text{'}=\left[\frac{{\left(6.63\times {10}^{-34}J.s\right)}^2}{8\left(9.11\times {10}^{-31}kg\right){\left(100pm\times {\displaystyle \frac{{10}^{-12}m}{1pm}}\right)}^2}\right]{\left(1\right)}^2 \\ \approx 6.03\times {10}^{-18}{J}^2.s^2/kg.m^2 \\ \approx \left(6.03\times {10}^{-18}{J}^2.s^2/kg.m^2\right)\left(\frac{1J}{1J^2.s^2/kg.m^2}\right) \\ \approx 6.03\times {10}^{-18}J \end{gathered}$$

Thus, the ground state energy of an electron that trapped in the potential well of widths 100pm is$6.03\times {10}^{-18}J$ .

From the above calculation, one can observe that the electron that trapped in the potential well of width 50pm has highest ground state energy, followed by the electron that trapped in the potential well of width 100pm and the electron that trapped in the potential well of width 200pm has lowest ground state energy.