Question

The electrons, identified by quantum numbers \(\mathrm{n}\) and \(l\) (i) \(\mathrm{n}=4 . l=1\) (ii) \(\mathrm{n}=4, l=0\) (iii) \(\mathrm{n}=3, l=2\) (iv) \(\mathrm{n}\) \(=3, l=1\) can be placed in order of increasing energy from the lowest to highest as: (a) (iv) \(<(\mathrm{ii})<(\mathrm{iii})<(\mathrm{i})\) (b) \((\mathrm{ii})<(\mathrm{iv})<(\mathrm{i})<(\mathrm{iii})\) (c) (i) \(<(\mathrm{iii})<(\mathrm{ii})<\) (iv) (d) (iii) \(<(\mathrm{i})<(\mathrm{iv})<(\mathrm{ii})\)

Short answer

The correct order is (a): (iv) < (ii) < (iii) < (i).

Step-by-step solution

Determine Energy Level of Subshells

The energy of an electron in an atom is primarily determined by its principal quantum number \(n\) and angular momentum quantum number \(l\). The subshells are arranged in order of increasing energy primarily by their \((n+l)\) value. If two subshells have the same \((n+l)\) value, the one with the lower \(n\) value is lower in energy.

Calculate \((n+l)\) Values for Each Option

We need to calculate \((n+l)\) for each set of quantum numbers:- For (i), \((n=4, l=1)\), \((n+l) = 4+1 = 5\).- For (ii), \((n=4, l=0)\), \((n+l) = 4+0 = 4\).- For (iii), \((n=3, l=2)\), \((n+l) = 3+2 = 5\).- For (iv), \((n=3, l=1)\), \((n+l) = 3+1 = 4\).

Arrange Subshells by Energy Levels

Now, arrange the options based on their \((n+l)\) values from lowest to highest:1. (ii) and (iv) both have \((n+l) = 4\), but (iv) has a lower \(n\) and is therefore lower in energy: \((n=3, l=1)\) before \((n=4, l=0)\).2. (i) and (iii) both have \((n+l) = 5\), but (iii) has a lower \(n\): (iii) \((n=3, l=2)\) before (i) \((n=4, l=1)\).Order: (iv) < (ii) < (iii) < (i).

Match with Given Options

Compare our order (iv) < (ii) < (iii) < (i) with the provided answer choices: - Option (a): (iv) < (ii) < (iii) < (i) perfectly matches our calculated order.

Key concepts

Energy Levels

In the exciting world of quantum physics, the concept of energy levels is fundamental in understanding how electrons are organized within an atom. Electrons occupy specific energy levels, which are often depicted as concentric shells around the nucleus. These energy levels determine the stability and configuration of an atom. The closer the electron is to the nucleus, the lower the energy level it occupies.

• Energy levels correspond to the principal quantum number, which we will explore further in the next section.
• The energy level of electron subshells is also influenced by the angular momentum quantum number, or the shape of the orbital.

Each distinct arrangement of electrons is called a subshell, and understanding their order is crucial to predicting atomic behavior in chemical reactions.

Principal Quantum Number

The principal quantum number, denoted as \(n\), plays a significant role in determining the energy levels of electrons. It is a positive integer ranging from 1 to any larger number, theoretically up to infinity. This number tells us the number of shells in which the electrons reside, and it defines the overall size and energy level of the electron shell.

• A higher \(n\) value indicates an electron is further away from the nucleus and has higher energy.
• This number determines the number of possible subshells - for each \(n\), there are \(n\) subshells.

For example, if \(n = 4\), this means the atom has four main energy levels, with the fourth level being larger and energetically higher than levels with smaller \(n\) values. Thus, understanding \(n\) is crucial in identifying the electron configuration in atoms.

Angular Momentum Quantum Number

The angular momentum quantum number, symbolized as \(l\), represents the shape of the orbital and helps classify subshells. It ranges from 0 to \(n-1\) for each principal quantum number \(n\). This quantum number defines the subshell occupied by the electron within a main energy level.

• If \(l = 0\), the orbital is spherical, known as the s subshell.
• If \(l = 1\), the orbital has a dumbbell shape, known as the p subshell.
• If \(l = 2\), the orbital is more complex, known as the d subshell, and so on.

The value of \(l\) also plays a part in influencing the subshell energy, as described in the section discussing energy levels and subshell energy order.

Subshell Energy Order

The subshell energy order is determined by the sum of the principal quantum number \(n\) and the angular momentum quantum number \(l\). This combination is often abbreviated as \((n+l)\). Subshells with a lower \((n+l)\) are lower in energy.

• A helpful rule: If two subshells have the same \((n+l)\) value, the one with the lower \(n\) value is lower in energy.
• To determine the energy ranking of subshells, calculate \((n+l)\) for each.

For example, comparing subshells 3p and 4s, both have \((n+l)\) values of 4, but because n for the 3p subshell is less than the 4s, 3p has a lower energy. Hence, understanding subshell energy order helps reveal the electronic structure of atoms, guiding how they interact and bond in reactions.