QUARTER FINAL STAGE 2024
YILO KROBO
KOFORIDUA SENIOR HIGH TECH
KNUST SHS
ROUND 2 - SPEED RACE
QUESTION
Find the energy of the photon emitted in the 4p to 1s transition of the hydrogen atom leaving your answer in electron volts to 2 significant figures.
ANSWER: 13 ELECTRON VOLTS
SOLUTION 1:
$E_n = -\dfrac{13.6}{n^2}\text{ eV}$
$E_1 = -\dfrac{13.6}{1^2}$
$E_1 = -13.6\text{ eV}$
$E_4 = -\dfrac{13.6}{4^2}$
$E_4 = -0.85\text{ eV}$
$\Delta E = E_4 - E_1$
$\Delta E = -0.85 - (-13.6)$
$\Delta E = 12.75\text{ eV}$
$\Delta E \approx 13\text{ eV}$
SOLUTION 2:
Precompute: the photon energies of the common hydrogen transitions ($13.6$ times the fixed fraction)
$2 \to 1$: $13.6 \times \dfrac{3}{4} = 10.2\text{ eV}$
$3 \to 1$: $13.6 \times \dfrac{8}{9} \approx 12.1\text{ eV}$
$4 \to 1$: $13.6 \times \dfrac{15}{16} = 12.75\text{ eV}$
$3 \to 2$: $13.6 \times \dfrac{5}{36} \approx 1.89\text{ eV}$
$4 \to 2$: $13.6 \times \dfrac{3}{16} = 2.55\text{ eV}$
Memorise these five values.
4p to 1s is the $4 \to 1$ transition.
$\Delta E = 12.75\text{ eV}$
$\Delta E \approx 13\text{ eV}$
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PRACTICE QUESTIONS
1. QUESTION: Find the energy of the photon emitted in the 2p to 1s transition of the hydrogen atom leaving your answer in electron volts to three significant figures.
ANSWER: 10.2 eV or $1.02\times10^{1}$ eV
SOLUTION:
$\Delta E = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\text{ eV}$
$\Delta E = 13.6\left(\dfrac{1}{1^2} - \dfrac{1}{2^2}\right)$
$\Delta E = 13.6 \times \dfrac{3}{4}$
$\Delta E = \dfrac{136\times10^{-1} \times 3}{4}$
$\Delta E = 34 \times 3 \times10^{-1}$
$\Delta E = 10.2\text{ eV}$
2. QUESTION: Find the energy of the photon emitted in the 4d to 2p transition of the hydrogen atom leaving your answer in electron volts to three significant figures.
ANSWER: 2.55 eV
SOLUTION:
$\Delta E = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\text{ eV}$
$\Delta E = 13.6\left(\dfrac{1}{2^2} - \dfrac{1}{4^2}\right)$
$\Delta E = 13.6 \times \dfrac{3}{16}$
$\Delta E = \dfrac{136\times10^{-1} \times 3}{16}$
$\Delta E = \dfrac{17 \times 3}{2} \times10^{-1}$
$\Delta E = 2.55\text{ eV}$
3. QUESTION: Find the energy of the photon emitted in the 5f to 4d transition of the hydrogen atom leaving your answer in electron volts to two significant figures.
ANSWER: 0.31 eV or $3.1\times10^{-1}$ eV
SOLUTION:
$\Delta E = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\text{ eV}$
$\Delta E = 13.6\left(\dfrac{1}{4^2} - \dfrac{1}{5^2}\right)$
$\Delta E = 13.6 \times \dfrac{9}{400}$
$\Delta E = \dfrac{136\times10^{-1} \times 9}{400}$
$\Delta E = \dfrac{17 \times 9}{50} \times10^{-1}$
$\Delta E \approx 0.31\text{ eV}$
4. QUESTION: Find the energy of the photon emitted in the 10p to 5s transition of the hydrogen atom leaving your answer in electron volts to two significant figures.
ANSWER: 0.41 eV or $4.1\times10^{-1}$ eV
SOLUTION:
$\Delta E = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\text{ eV}$
$\Delta E = 13.6\left(\dfrac{1}{5^2} - \dfrac{1}{10^2}\right)$
$\Delta E = 13.6 \times \dfrac{3}{100}$
$\Delta E = \dfrac{136\times10^{-1} \times 3}{100}$
$\Delta E = \dfrac{34 \times 3}{25} \times10^{-1}$
$\Delta E \approx 0.41\text{ eV}$
5. QUESTION: Find the energy of the photon emitted in the 20d to 5p transition of the hydrogen atom leaving your answer in electron volts to two significant figures.
ANSWER: 0.51 eV or $5.1\times10^{-1}$ eV
SOLUTION:
$\Delta E = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\text{ eV}$
$\Delta E = 13.6\left(\dfrac{1}{5^2} - \dfrac{1}{20^2}\right)$
$\Delta E = 13.6 \times \dfrac{3}{80}$
$\Delta E = \dfrac{136\times10^{-1} \times 3}{80}$
$\Delta E = \dfrac{17 \times 3}{10} \times10^{-1}$
$\Delta E = 0.51\text{ eV}$
6. QUESTION: Find the energy of the photon emitted in the 20p to 4s transition of the hydrogen atom leaving your answer in electron volts to two significant figures.
ANSWER: 0.82 eV or $8.2\times10^{-1}$ eV
SOLUTION:
$\Delta E = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\text{ eV}$
$\Delta E = 13.6\left(\dfrac{1}{4^2} - \dfrac{1}{20^2}\right)$
$\Delta E = 13.6 \times \dfrac{3}{50}$
$\Delta E = \dfrac{136\times10^{-1} \times 3}{50}$
$\Delta E = \dfrac{68 \times 3}{25} \times10^{-1}$
$\Delta E \approx 0.82\text{ eV}$
7. QUESTION: Find the energy of the photon emitted in the 10d to 8p transition of the hydrogen atom leaving your answer in electron volts to three significant figures.
ANSWER: 0.0765 eV or $7.65\times10^{-2}$ eV
SOLUTION:
$\Delta E = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\text{ eV}$
$\Delta E = 13.6\left(\dfrac{1}{8^2} - \dfrac{1}{10^2}\right)$
$\Delta E = 13.6 \times \dfrac{9}{1600}$
$\Delta E = \dfrac{136\times10^{-1} \times 9}{1600}$
$\Delta E = \dfrac{17 \times 9}{200} \times10^{-1}$
$\Delta E = 0.0765\text{ eV}$
8. QUESTION: Find the energy of the photon emitted in the 20f to 10d transition of the hydrogen atom leaving your answer in electron volts to two significant figures.
ANSWER: 0.10 eV or $1.0\times10^{-1}$ eV
SOLUTION:
$\Delta E = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\text{ eV}$
$\Delta E = 13.6\left(\dfrac{1}{10^2} - \dfrac{1}{20^2}\right)$
$\Delta E = 13.6 \times \dfrac{3}{400}$
$\Delta E = \dfrac{136\times10^{-1} \times 3}{400}$
$\Delta E = \dfrac{17 \times 3}{50} \times10^{-1}$
$\Delta E \approx 0.10\text{ eV}$
9. QUESTION: Find the energy of the photon emitted in the 15d to 12p transition of the hydrogen atom leaving your answer in electron volts to two significant figures.
ANSWER: 0.034 eV or $3.4\times10^{-2}$ eV
SOLUTION:
$\Delta E = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\text{ eV}$
$\Delta E = 13.6\left(\dfrac{1}{12^2} - \dfrac{1}{15^2}\right)$
$\Delta E = 13.6 \times \dfrac{1}{400}$
$\Delta E = \dfrac{136\times10^{-1}}{400}$
$\Delta E = \dfrac{17}{50} \times10^{-1}$
$\Delta E = 0.034\text{ eV}$