Question

The ratio of the areas within the electron orbits for the first excited state to the ground state for the hydrogen atom is (a) \(2: 1\) (b) \(4: 1\) (c) \(8: 1\) (d) \(16: 1\)

Short answer

The ratio of the areas within the electron orbits for the first excited state to the ground state for the hydrogen atom is 16:1.

Step-by-step solution

Understand the Bohr Model for Hydrogen Atom

According to the Bohr model, the energy levels of an electron in a hydrogen atom are quantized. The area enclosed by the electron orbit (which can be approximated to a circular path for simplicity in Bohr's model) is directly proportional to the square of the principal quantum number (n) of the orbit. The ground state corresponds to n=1, while the first excited state corresponds to n=2.

Relationship Between the Areas of Orbits

To find the ratio of the areas within the electron orbits for different states, use the formula for the area of a circle, which is proportional to the square of its radius. In Bohr's model, the radius of the orbit is proportional to n^2. Therefore, the areas of the orbits are proportional to n^4, since area is proportional to the square of the radius.

Calculate the Ratio of the Areas

For the ground state (n=1) the area is proportional to 1^4, which is 1. For the first excited state (n=2), the area is proportional to 2^4, which is 16. The ratio of the areas is therefore 16:1.

Key concepts

Quantum Numbers

At the heart of the Bohr model and quantum mechanics, quantum numbers play a pivotal role in describing the properties of electrons within atoms. Each electron in an atom is identified by a unique set of quantum numbers, analogous to how a person might be identified by a distinct combination of name, birth date, and social security number.

There are four quantum numbers: the principal quantum number (), the angular momentum quantum number (), the magnetic quantum number (), and the spin quantum number (). The principal quantum number, , indicates the energy level and size of the electron orbit, with higher values of corresponding to larger and higher energy orbits. Bohr's model specifically uses to define electron orbits in hydrogen atoms. The other quantum numbers further detail the shape of the orbital () and the orientation and spin of electrons within these orbitals ( and ).

Knowing these quantum numbers is crucial because they determine how electrons are arranged in an atom and how they interact with energy, such as light. This foundational concept is key to solving problems like our textbook exercise that requires a comparison between different energy levels, dictated by the principal quantum number.

Electron Orbits

Electron orbits, often visualized as circular paths around the nucleus of an atom, are a major feature of the Bohr model. While modern quantum mechanics has evolved to describe electron behavior in terms of probability clouds or orbitals, the Bohr model simplifies this concept, instead representing each energy level as a distinct path traveled by the electron.

In Bohr's model, these orbits correspond to specific energy levels, and electrons can only occupy these defined levels. An orbit is not just an arbitrary path but is quantized; this means the electron's energy and the orbit's size are precisely dictated by the principal quantum number (). Notably, when performing calculations involving electron orbits, such as comparing areas within these paths, the Bohr model gives us a platform to understand how the radii and subsequent areas of these orbits grow with .

Students often visualize these orbits as concentric circles around the nucleus, with each subsequent orbit being larger and containing more energy. That's why, in our exercise, the concept of electron orbits is crucial for understanding why the area of an orbit in an excited state is significantly larger than that of an orbit in the ground state.

Excited State

An excited state in an atom occurs when an electron has absorbed energy and has moved from a lower energy level, or 'ground state', to a higher energy level. In the context of the Bohr model for the hydrogen atom, the excited state refers to when the electron occupies any orbit with a principal quantum number () greater than 1.

The energy difference between these states is quantized, meaning the electron must absorb or release a precise amount of energy to transition from one state to another. This phenomenon is visually represented in emission or absorption spectra, which are unique to each element and critical for understanding atomic structures.

When discussing the first excited state, we refer to the electron's position at =2. The transition between the ground state (=1) and the first excited state represents a fundamental change in the atom’s energy. This leap in the principal quantum number substantially increases the area of the electron's orbit, aligning with our exercise’s inquiry into the relative sizes of these orbits. Specifically, since the area is proportional to the square of the radius, which in turn is proportional to squared, the first excited state's orbit area will be exponentially larger than that of the ground state.