Q.A 3310 Å photon liberates an electron from a material with energy $3 \times 10^{-19}$ J, while another 5000 Å photon ejects an electron with energy $0.972 \times 10^{-19}$ J from the same material. Determine the value of Planck's constant and the threshold wavelength of the material.

Step 1. Write Einstein's equation for each photon: $\dfrac{hc}{\lambda_1}-\phi_0=K_1$ and $\dfrac{hc}{\lambda_2}-\phi_0=K_2$, with $\lambda_1=3310$ Å, $K_1=3\times10^{-19}$ J and $\lambda_2=5000$ Å, $K_2=0.972\times10^{-19}$ J.

Step 2. Subtracting eliminates $\phi_0$: $hc\left(\dfrac{1}{\lambda_1}-\dfrac{1}{\lambda_2}\right)=K_1-K_2=2.028\times10^{-19}\ \text{J}$.

Step 3. Computing $\dfrac{1}{\lambda_1}-\dfrac{1}{\lambda_2}=\dfrac{1}{3310\times10^{-10}}-\dfrac{1}{5000\times10^{-10}}\approx1.021\times10^{6}\ \text{m}^{-1}$, so $hc\approx\dfrac{2.028\times10^{-19}}{1.021\times10^{6}}\approx1.986\times10^{-25}$, giving $h=\dfrac{1.986\times10^{-25}}{3\times10^{8}}\approx6.62\times10^{-34}\ \text{Js}$. …

Write Einstein's equation for both given photon/electron-energy pairs and subtract to eliminate the unknown work function, solving first fo …

• Forgetting to convert both wavelengths to the same unit (metres) before taking the reciprocal difference. …