Question

Question: Calculate the energy (in$\mathbf{kJ}\mathbf{/}\mathbf{mol}$) required to remove the electron in the ground state for each of the following one-electron species using the Bohr model.

a.$\mathbf{H}$

b.${\mathbf{He}}^{\mathbf{+}}$

c.${\mathbf{Li}}^{\mathbf{2}\mathbf{+}}$

d.${\mathbf{C}}^{\mathbf{5}\mathbf{+}}$

e.${\mathbf{Fe}}^{\mathbf{25}\mathbf{+}}$

Short answer

a) The energy for an$\mathrm{H}$ atom is$-1308797.9\mathrm{kJ}/\mathrm{mol}$ .

b) The energy for an ${\mathrm{He}}^{+}$atom is$-5247.2376\mathrm{kJ}/\mathrm{mol}$ .

c) The energy for an${\mathrm{Li}}^{2+}$atom is$-11806.2846\mathrm{kJ}/\mathrm{mol}$ .

d) The energy for an${\mathrm{C}}^{5+}$ atom is$-47225.1384\mathrm{kJ}/\mathrm{mol}$ .

e) The energy for an${\mathrm{Fe}}^{25+}$ atom is$-886783.1544\mathrm{kJ}/\mathrm{mol}$ .

Step-by-step solution

Concept used

Electronic transition is a phenomenon that occurs when electrons are emitted or absorbed from one energy level to another. If a transition from a lower to a higher energy level is noticed, absorption occurs. If you go from a greater to a lower energy level, you'll emit something.

Expression for energy of electrons in hydrogen atom is as follows:

$\mathrm{E}=-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\mathrm{z}}^2}{{\mathrm{n}}^2}\right)$ ……. (1)

Where,

E is the energy of electrons.

Z is the atomic number.

n is an integer.

Calculate the energy for H the atom

a)

For the H atom,

Value of Z is$1$ .

Value of n is$1$ .

When the values in equation (1) are substituted, we get

$$\begin{gathered} \mathrm{E}=-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\mathrm{z}}^2}{{\mathrm{n}}^2}\right) \\ =-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\left(1\right)}^2}{{\left(1\right)}^2}\right) \\ =-2.178\times {10}^{-18}\mathrm{J} \end{gathered}$$

This energy is to be converted into as follows:

$$\begin{gathered} \mathrm{E}=(-2.178\times {10}^{-18}\mathrm{J})\left(\frac{{10}^{-3}\mathrm{kJ}}{1\mathrm{kJ}}\right)\left(6.023\times {10}^{23}\right) \\ =-1308797.9\mathrm{kJ}/\mathrm{mol} \end{gathered}$$

Hence, the required energy is$-1308797.9\mathrm{kJ}/\mathrm{mol}$ .

Calculate the energy for He+ the atom.

b)

For ${\mathrm{He}}^{+}$atom,

Value of $\mathrm{z}$is $2$.

Value of$\mathrm{n}$ is$1$ .

When the values in equation (1) are substituted, we get

$$\begin{gathered} \mathrm{E}=-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\mathrm{z}}^2}{{\mathrm{n}}^2}\right) \\ =-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\left(2\right)}^2}{{\left(1\right)}^2}\right) \\ =-8.712\times {10}^{-18}\mathrm{J} \end{gathered}$$

This energy is to be converted into as follows:

$$\begin{gathered} \mathrm{E}=(-8.172\times {10}^{-18}\mathrm{J})\left(\frac{{10}^{-3}\mathrm{kJ}}{1\mathrm{kJ}}\right)\left(6.023\times {10}^{23}\right) \\ =-5247.2376\mathrm{kJ}/\mathrm{mol} \end{gathered}$$

Hence required energy is $-5247.2376\mathrm{kJ}/\mathrm{mol}$.

Calculate the energy for Li2+the atom

c)

For ${\mathrm{Li}}^{2+}$atom,

Value of$\mathrm{z}$ is$3$ .

Value of$\mathrm{n}$ is$1$ .

When the values in equation (1) are substituted, we get

$$\begin{gathered} \mathrm{E}=-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\mathrm{z}}^2}{{\mathrm{n}}^2}\right) \\ =-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\left(3\right)}^2}{{\left(1\right)}^2}\right) \\ =-19.602\times {10}^{-18}\mathrm{J} \end{gathered}$$

This energy is to be converted into : as

$$\begin{gathered} \mathrm{E}=(-19.602\times {10}^{-18}\mathrm{J})\left(\frac{{10}^{-3}\mathrm{kJ}}{1\mathrm{kJ}}\right)\left(6.023\times {10}^{23}\right) \\ =-11806.2846\mathrm{kJ}/\mathrm{mol} \end{gathered}$$

Hence, the required energy is $-11806.2846\mathrm{kJ}/\mathrm{mol}$.

Calculate the energy for C5+ the atom

d)

For${\mathrm{C}}^{5+}$ atom,

Value of$\mathrm{z}$ is$6$

Value of$\mathrm{n}$ is$1$

When the values in equation (1) are substituted, we get

$$\begin{gathered} \mathrm{E}=-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\mathrm{z}}^2}{{\mathrm{n}}^2}\right) \\ =-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\left(6\right)}^2}{{\left(1\right)}^2}\right) \\ =-78.408\times {10}^{-18}\mathrm{J} \end{gathered}$$

This energy is to be converted into as:

$$\begin{gathered} \mathrm{E}=(-78.408\times {10}^{-18}\mathrm{J})\left(\frac{{10}^{-3}\mathrm{kJ}}{1\mathrm{kJ}}\right)\left(6.023\times {10}^{23}\right) \\ =-47225.1384\mathrm{kJ}/\mathrm{mol} \end{gathered}$$

Hence, the required energy is$-47225.1384\mathrm{kJ}/\mathrm{mol}$ .

Calculate the energy for Fe25+the atom

e)

For ${\mathrm{Fe}}^{25+}$atom,

The value of Z is 6

The value of n is$1$

When the values in equation (1) are substituted, we get

$$\begin{gathered} \mathrm{E}=-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\mathrm{z}}^2}{{\mathrm{n}}^2}\right) \\ =-2.178\times {10}^{-18}\mathrm{J}\left(\frac{{\left(26\right)}^2}{{\left(1\right)}^2}\right) \\ =-1472.328\times {10}^{-18}\mathrm{J} \end{gathered}$$

This energy is to be converted into as:

$$\begin{gathered} \mathrm{E}=(-1472.328\times {10}^{-18}\mathrm{J})\left(\frac{{10}^{-3}\mathrm{kJ}}{1\mathrm{kJ}}\right)\left(6.023\times {10}^{23}\right) \\ =-886783.1544\mathrm{kJ}/\mathrm{mol} \end{gathered}$$

Hence, the required energy is$-886783.1544\mathrm{kJ}/\mathrm{mol}$ .