An atom in a gas discharge tube has a ground state energy level of -10.4\text{ eV} and an … — JAMB UTME

An atom in a gas discharge tube has a ground state energy level of $-10.4\text{ eV}$ and an excited energy level of $-3.8\text{ eV}$ . An electron absorbs a single photon to undergo a transition directly from the ground state to this excited level. If the frequency of the absorbed photon is expressed as $x \times 10^{15}\text{ Hz}$ , what is the numerical value of $x$ ? (Take Planck's constant $h = 6.6 \times 10^{-34}\text{ J}\cdot\text{s}$ and $1\text{ eV} = 1.6 \times 10^{-19}\text{ J}$ )

Answer: 1.6

Answer

The numerical value of

x

is 1.6.

The energy of the absorbed photon is equal to the difference between the excited state and ground state energies:

\Delta E = -3.8\text{ eV} - (-10.4\text{ eV}) = 6.6\text{ eV}

. Converting this to Joules yields

6.6 \times 1.6 \times 10^{-19}\text{ J} = 1.056 \times 10^{-18}\text{ J}

. Using Einstein's photon relation

E = hf

, the frequency is

f = \frac{1.056 \times 10^{-18}}{6.6 \times 10^{-34}} = 1.6 \times 10^{15}\text{ Hz}

, which gives

x = 1.6

Step-by-Step Solution

Determine the energy absorbed during transition

\Delta E = 6.6\text{ eV}

The energy of the absorbed photon equals the energy difference between the initial and final states:

\Delta E = -3.8\text{ eV} - (-10.4\text{ eV}) = 6.6\text{ eV}

Convert energy from eV to Joules

\Delta E = 1.056 \times 10^{-18}\text{ J}

Multiply by

1.6 \times 10^{-19}\text{ J/eV}

to convert energy to standard SI units.

Calculate photon frequency

f = 1.6 \times 10^{15}\text{ Hz}

Use Planck's energy equation

f = \frac{\Delta E}{h} = \frac{1.056 \times 10^{-18}\text{ J}}{6.6 \times 10^{-34}\text{ J}\cdot\text{s}} = 1.6 \times 10^{15}\text{ Hz}

Key Concept

Photon energy and atomic transition frequency

Estimated Time:2m 0s

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