Problem 178 - A Complete Resource Book in Chemistry for JEE Main

Question

Energy of an electron is given by, \(\mathrm{E}=-2.178 \times 10^{-18}\left(\frac{\mathrm{Z}^{2}}{\mathrm{n}^{2}}\right)\) Wavelength of light required to excite an electron in an hydrogen atom from level \(\mathrm{n}=1\) to \(\mathrm{n}=2\) will be \(\left(\mathrm{h}=6.62 \times 10^{-34} \mathrm{Js}\right.\) and \(\left.\mathrm{c}=3.0 \times 10^{\mathrm{s}} \mathrm{ms}^{-1}\right)\) (a) \(6.500 \times 10^{-7} \mathrm{~m}\) (b) \(8.500 \times 10^{-7} \mathrm{~m}\) (c) \(1.214 \times 10^{-7} \mathrm{~m}\) (d) \(2.816 \times 10^{-7} \mathrm{~m}\)

Step-by-Step Solution

Answer

The wavelength is \(1.214 \times 10^{-7} \text{ m}\), option (c).

Step 1: Understand the Formula for Energy Levels

The energy of an electron in a hydrogen atom is given by the formula \( \mathrm{E} = -2.178 \times 10^{-18} \left( \frac{\mathrm{Z}^{2}}{\mathrm{n}^{2}} \right) \). Here, \( \mathrm{Z} \) is the atomic number, which is 1 for hydrogen, and \( \mathrm{n} \) is the principal quantum number of the electron's orbit.

Step 2: Calculate Energy for n=1

Substitute \( \mathrm{Z} = 1 \) and \( \mathrm{n} = 1 \) into the energy formula: \[\mathrm{E}_1 = -2.178 \times 10^{-18} \left( \frac{1^2}{1^2} \right) = -2.178 \times 10^{-18} \text{ J}.\]

Step 3: Calculate Energy for n=2

Substitute \( \mathrm{Z} = 1 \) and \( \mathrm{n} = 2 \) into the formula: \[\mathrm{E}_2 = -2.178 \times 10^{-18} \left( \frac{1^2}{2^2} \right) = -2.178 \times 10^{-18} \times 0.25 = -5.445 \times 10^{-19} \text{ J}.\]

Step 4: Find the Energy Required for Excitation

Calculate the energy difference required to excite the electron from \( n = 1 \) to \( n = 2 \): \[\Delta E = \mathrm{E}_2 - \mathrm{E}_1 = -5.445 \times 10^{-19} - (-2.178 \times 10^{-18}) = 1.6335 \times 10^{-18} \text{ J}.\]

Step 5: Use the Energy-Wavelength Relation

The relation between energy and wavelength is given by \( \Delta E = \frac{hc}{\lambda} \). Rearrange to solve for wavelength \( \lambda \):\[\lambda = \frac{hc}{\Delta E}.\]

Step 6: Calculate the Wavelength

Substitute the known values \( h = 6.62 \times 10^{-34} \text{ Js} \), \( c = 3.0 \times 10^8 \text{ m/s} \), and \( \Delta E = 1.6335 \times 10^{-18} \text{ J} \) into the formula:\[\lambda = \frac{6.62 \times 10^{-34} \times 3.0 \times 10^8}{1.6335 \times 10^{-18}} \approx 1.214 \times 10^{-7} \text{ m}.\]

Step 7: Choose the Correct Option

Compare the calculated wavelength with the options given. The closest match is option (c) \(1.214 \times 10^{-7} \text{ m}\).

Key Concepts

Energy Levels in Hydrogen

In the Bohr Model of the Hydrogen Atom, energy levels are crucial for understanding electron behavior. These levels are specific regions where electrons can reside, and they are quantized, meaning electrons are only allowed at certain distances from the nucleus. This quantization occurs because of the nature of the electron's wave-like behavior in the atom.

The energy of an electron in a hydrogen atom is calculated using a significant formula: \[ E = -2.178 \times 10^{-18} \left( \frac{Z^2}{n^2} \right) \]

• Z is the atomic number, which is 1 for hydrogen.
• n is the principal quantum number, representing the energy level.

This formula shows that energy decreases as the principal quantum number, n, increases. Electrons at lower energy levels (smaller n values) have higher energy magnitudes (more negative values), meaning they are more tightly bound to the nucleus. Conversely, increasing n means higher energy levels where the electron is less tightly bound.

Excitation Wavelength Calculation

To understand excitation, think of moving an electron from one energy level to a higher one. This transition requires energy, which can be supplied by a photon of light. The amount of energy absorbed is exactly equal to the difference in energy between the two levels.

• Initially, calculate the energy difference (\( \Delta E \)) between the two levels using the energy formula.
• For hydrogen, going from n=1 to n=2:\[ \Delta E = E_2 - E_1 = -5.445 \times 10^{-19} - (-2.178 \times 10^{-18}) = 1.6335 \times 10^{-18} \text{ J}. \]

Next, use the energy-wavelength relation:\[ \Delta E = \frac{hc}{\lambda} \]

• h is Planck's constant \(6.62 \times 10^{-34}\) Js.
• c is the speed of light \(3.0 \times 10^8\) m/s.
• \(\lambda\) is the wavelength of light required for the excitation.

Solving for \(\lambda\), gives:\[ \lambda = \frac{hc}{\Delta E} = \frac{6.62 \times 10^{-34} \times 3.0 \times 10^8}{1.6335 \times 10^{-18}} \approx 1.214 \times 10^{-7} \text{ m}. \]This calculated wavelength helps predict the color of the light that can induce transitions between energy levels in hydrogen.

Quantum Mechanics

Quantum mechanics provides a rich framework for understanding atomic and subatomic particles' behaviors, including electrons in atoms. Fundamentally, it suggests that electrons behave both as particles and waves, a duality reflected in their quantization into energy levels.

One of the core aspects of quantum mechanics is the Schrödinger equation, a mathematical formula that describes how the quantum state of a physical system changes over time. It also predicts the probability of finding an electron at a certain place, differing from the Bohr model which provided fixed orbits.

• Heisenberg's Uncertainty Principle, another key concept, states that one cannot simultaneously know the exact position and momentum of an electron.
• Wave functions, introduced in Schrödinger's work, define the probability density of an electron’s location at any given point in space.

These principles explain why atomic spectra are discrete and why only certain wavelength transitions are allowed. When studying the hydrogen atom via the Bohr model and quantum mechanics, we gain insights into how energy levels form and allow for such precise wavelength measurements as seen in excitation calculations.