Problem 148 - A Complete Resource Book in Chemistry for JEE Main

Question

Hydrogen atoms in the ground state are excited by means of monochromatic radiation of wavelength \(970.6 \AA\). How many different lines are possible in the resulting emission spectrum?

Step-by-Step Solution

Answer

Eight different lines are possible in the emission spectrum.

Step 1: Understand the Excitation of Hydrogen Atoms

When hydrogen atoms in the ground state (where electrons are in the lowest energy level, n=1) absorb a photon, they become excited to higher energy levels. The given wavelength is 970.6 Å (1 Ångström = 10^-10 meters), which means energy is absorbed to excite the electrons.

Step 2: Calculate the Energy of the Photon

Calculate the energy of a photon using the equation: \[ E = \frac{hc}{\lambda} \]where \( h = 6.626 \times 10^{-34} \) J·s is Planck's constant, \( c = 3 \times 10^8 \) m/s is the speed of light, and \( \lambda = 970.6 \times 10^{-10} \) m is the wavelength.Calculate the energy:

Step 3: Convert Energy to Electron Volts

Convert the energy from Joules to electron volts (eV), knowing that 1 eV = 1.602 x 10^-19 J. Once calculated, this energy determines the final energy level 'n' to which electrons are excited based on the energy difference between levels.

Step 4: Determine the Excited State

Use the Bohr model formula for energy levels:\[ E_n = -13.6 \frac{Z^2}{n^2} \text{ eV} \]For hydrogen, \( Z = 1 \). Compare the calculated energy with potential energy levels to find the highest excited state 'n'.

Step 5: Calculate the Possible Emission Transitions

Determine the number of possible emission lines. Electrons can transition from any excited state 'n > 1' to lower states (n=1 to n=2, n=3 to n=1, etc.). For each initial level 'n', possible transitions include all lower levels. Use the formula for the number of lines: \[ \frac{n(n-1)}{2} \]where 'n' is the highest reached energy level.

Step 6: Final Calculation

Based on the highest reached energy level (calculated from the photon energy), count all the transitions. Each transition corresponds to a distinct emission line.

Key Concepts

Photon Energy

When we talk about photons in the context of the hydrogen emission spectrum, we are referring to packets of energy. Each photon has an energy directly related to its wavelength or frequency. The relationship between the energy of a photon and its wavelength is given by the equation:\[ E = \frac{hc}{\lambda} \]Here, \( E \) represents the energy of the photon, \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ J·s})\), \( c \) is the speed of light \((3 \times 10^8 \text{ m/s})\), and \( \lambda \) is the wavelength of the photon.

• Shorter wavelengths have higher energy photons.
• Longer wavelengths have lower energy photons.

To excite a hydrogen atom, photons with sufficient energy are required. This is determined by calculating the photon's energy using the above formula. Once you compute the energy, you can relate it to the possible energy levels in the atom through the Bohr Model, which predicts how electrons will respond when they absorb this energy.

Bohr Model

The Bohr Model is a foundational concept in quantum mechanics that explains how electrons reside in specific energy levels around the nucleus in an atom. According to this model:

• Electrons orbit the nucleus in defined circular paths called energy levels or "shells." These are denoted by 'n', where \( n = 1, 2, 3, \ldots \).
• Each energy level corresponds to a certain energy value, and an electron can only inhabit those discrete energy levels.

The Bohr Model helps in understanding why electrons absorb or emit energy. When a photon of sufficient energy interacts with an electron, it can jump to a higher level, and when it comes back down, it emits a photon of energy. This model is essential for calculating the electron transitions as it provides the formula for energy levels:\[ E_n = -13.6 \frac{Z^2}{n^2} \text{ eV} \]In the case of hydrogen, the atomic number \( Z = 1 \), simplifying the calculations. The model accurately predicts the wavelengths of the series of spectral emission lines.

Energy Levels

Energy levels in an atom determine how much energy an electron can have when it moves from one orbit to another. For hydrogen:

• The first energy level \((n=1)\) is the ground state, where the electron usually resides.
• Higher energy levels \((n=2, 3, 4, \ldots)\) are excited states.
• The difference in energy between two levels affects the amount of energy absorbed or released.

For hydrogen atoms in particular, the energy levels are neatly calculated using the Bohr model, and these levels are indexable with numbers \( n \).When an electron absorbs a specific amount of energy, it moves to a higher energy state. For example, if a photon with enough energy hits a hydrogen electron at \( n=1 \), it can leap to a higher state like \( n=2 \) or more, depending on the photon's energy.

Transitions

Transitions occur when electrons in an atom move from one energy level to another. There are two main types of transitions:

• **Absorption:** When an electron absorbs energy and moves to a higher energy level (higher \( n \)).
• **Emission:** When an electron falls back to a lower energy level, releasing energy in the form of light (a photon).

Each possible transition results in a photon being emitted or absorbed, contributing to what we call the emission spectrum. This spectrum consists of various lines, each representing a different transition. Analytically, the number of possible emission lines from an electron in state \( n \) can be calculated with the formula:\[ \frac{n(n-1)}{2} \]This formula accounts for all possible paths an electron can take between energy levels before reaching the ground state. The hydrogen emission spectrum is thus composed of lines associated with these transitions, forming distinct series like the Lyman, Balmer, and Paschen series, depending on the final level the electron falls into.