Question

When electromagnetic radiation at wavelength $300\mathrm{nm}$ falls on the surface of sodium, electrons are emitted with a kinetic energy of $1.68\times {10}^5{\mathrm{Jmol}}^{-1}$. What is the minimum energy needed to remove an electron from sodium?

Answer 1

The minimum energy required to remove an electron from sodium

The minimum energy required to remove an electron can be determined by subtracting the kinetic energy from the energy given by the electromagnetic radiation.

Step 1: Given data

$$\begin{gathered} k.e.=1.68\times {10}^5{\mathrm{Jmol}}^{-1} \\ \mathrm{\lambda }=300\mathrm{nm} \end{gathered}$$

Here, $k.e.$ is the kinetic energy, and $\lambda$ is the wavelength.

Step 2: Calculation of electromagnetic radiation

The equation for electromagnetic radiation can be written as follows:

$E_1=\frac{hc}{\lambda }$

Here, $E_1$ is the energy given by the electromagnetic radiation, $h$ is the Plank's constant, and $c$ is the speed of light.

$$\begin{gathered} E_1=\frac{6.626\times {10}^{-34}\mathrm{Js}\times 3\times {10}^8\mathrm{m}/\mathrm{s}}{300\times {10}^{-9}\mathrm{m}} \\ {E}_1=6.626\times {10}^{-19}\mathrm{J} \end{gathered}$$

Step 3: Calculation of the minimum energy required to remove an electron from sodium

The equation for the minimum energy required can be written as follows:

$E_{\min }=E_1-k.e.$

Here, $E_{\min }$ is the minimum energy required.

$$\begin{gathered} E_{\min }=6.626\times {10}^{-19}\mathrm{J}-\frac{1.68\times {10}^5\mathrm{J}/\mathrm{mol}}{6.023\times {10}^{23}{\mathrm{mol}}^{-1}} \\ {E}_{\min }=6.626\times {10}^{-19}\mathrm{J}-2.79\times {10}^{-19}\mathrm{J} \\ {\mathrm{E}}_{\min }=3.836\times {10}^{-19}\mathrm{J} \end{gathered}$$

Hence, the minimum energy required is $3.836\times {10}^{-19}\mathrm{J}$.