Question

Energy of \(\mathrm{H}\) -atom in the ground state is \(-13.6 \mathrm{eV}\), hence energy in the second excited state is: (a) \(-6.8 \mathrm{eV}\) (b) \(-3.4 \mathrm{eV}\) (c) \(-1.51 \mathrm{eV}\) (d) \(-4.53 \mathrm{eV}\)

Short answer

Energy in the second excited state is \(-1.51 \mathrm{eV}\), option (c).

Step-by-step solution

Understand the Energy Levels of Hydrogen

The energy levels for a hydrogen atom are given by the formula \( E_n = \frac{-13.6}{n^2} \, \mathrm{eV} \), where \( n \) is the principal quantum number corresponding to the energy level. The ground state is \( n=1 \).

Identify the Principal Quantum Number for the Second Excited State

The ground state corresponds to \( n=1 \), the first excited state corresponds to \( n=2 \), and the second excited state corresponds to \( n=3 \). In this problem, we need to find the energy when \( n=3 \).

Calculate the Energy of the Second Excited State

Plug \( n=3 \) into the energy level formula: \( E_3 = \frac{-13.6}{3^2} = \frac{-13.6}{9} \, \mathrm{eV} \). Calculate \( E_3 \).

Simplify the Expression

Perform the division: \( \frac{-13.6}{9} = -1.51 \, \mathrm{eV} \).

Match the Calculated Energy with the Options

The calculated energy for the second excited state, \( -1.51 \, \mathrm{eV} \), corresponds to option (c).

Key concepts

Energy Levels

For a hydrogen atom, energy levels refer to the distinct levels of energy that the electron within the atom can occupy. These energy levels are quantized, meaning that only certain discrete energy levels are allowed, rather than a continuous range of values. The concept of quantized energy levels was introduced through the Bohr Model, a significant breakthrough in atomic physics.

When looking at the formula for the energy levels of hydrogen, given by \[ E_n = \frac{-13.6}{n^2} \, \text{eV} \]we see that the principal quantum number \( n \) plays a pivotal role. This number determines the energy level, where \( n=1 \) represents the ground state, the lowest energy level, and increasing \( n \) values correspond to higher energy states.

• Ground State (\( n=1 \)): Lowest energy level with energy of \(-13.6 \text{ eV}\)
• First Excited State (\( n=2 \))
• Second Excited State (\( n=3 \)) and so on.

As \( n \) increases, the energy becomes less negative, indicating that the electron is further from the nucleus. Calculating specific energy states involves substituting different \( n \) values into the formula to find their corresponding energy. For example, in the problem above, the energy for \( n=3 \) calculated to \(-1.51 \text{ eV}\).

Hydrogen Atom

The hydrogen atom is the simplest atom in the universe and serves as a fundamental building block for understanding atomic physics. A hydrogen atom consists of one electron and one proton, making its internal structure straightforward. This simplicity allowed the hydrogen atom to be a primary model for early atomic studies, shedding light on the behavior of more complex atoms.

Due to its single electron, the hydrogen atom was key in the development of quantum mechanics. Researchers could precisely calculate the allowed energy levels, which are critical in understanding atomic behavior. In the Bohr model, electrons travel in distinct orbits around the nucleus, with specific energy levels associated with each orbit.

• Electron orbits are defined by specific quantum numbers.
• The model is pivotal in understanding atomic emissions.
• Quantum transitions between energy levels result in spectral lines.

The Bohr Model offers great insight into electron transitions that occur when electrons absorb or emit energy, moving from one orbit to another. This understanding is crucial in various applications, such as spectroscopy, where studying the light emitted by atoms helps determine their properties.

Quantum Numbers

Quantum numbers are fundamental values used to describe the quantum state of particles, such as electrons, within an atom. Essentially, they serve as an address system that details an electron's probable location and energy level within an atom. For the hydrogen atom, understanding quantum numbers is essential for describing its energy levels and electron orbits.

The principal quantum number \( n \) is one of the four quantum numbers that provide the key to defining energy states in the hydrogen atom. It directly correlates to the atom's energy level and the size of the electron's orbit. In the given exercise, the switch from \( n=1 \) to \( n=3 \) represents a change from the ground state to the second excited state, giving rise to a change in energy.

• Principal Quantum Number (\( n \)): Specifies energy level and size of the orbital. It ranges from 1 to infinity.
• Other quantum numbers include azimuthal, magnetic, and spin, which describe electron aspects like shape and orientation.
• All quantum numbers follow the Pauli exclusion principle, implying no two electrons in an atom can have the same set of quantum numbers.

Each energy state made possible by these quantum numbers has a corresponding physical meaning, and transitions between these states are marked by the atom's absorption or emission of energy.