Chapter 13: Problem 150
The number of orbitals having \((n+l)\) \(<5\) is (a) 9 (b) 8 (c) 4 (d) 10
Short Answer
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01
Understanding the quantum numbers
In quantum mechanics, every electron in an atom is described by four quantum numbers. The principal quantum number (), which describes the energy level or shell, and the azimuthal (also called the angular momentum) quantum number (), which defines the subshell or sublevel (s, p, d, f, etc.). The sum of and n is used to determine the energy order of subshells.
02
Enumerating the possibilities
We will list all combinations of and n values for which +n is less than 5. This will include all possible subshells within the given condition.
03
Listing the possible combinations
We start with 'n' at its lowest value, which is 1. Then we increment 'n' until adding the smallest possible 'l' (which is 0) would make (n+l) equal or larger than 5, thus exceeding the condition.For =1, n can be 0.For =2, n can be 0 or 1.For =3, n can only be 0 since 3 + 1 = 4 meets the condition but 3 + 2 = 5 does not.For =4, we cannot have any valid 'l' because the smallest value for 'l' (which is 0) will make +n = 4+0=4 which meets the condition.
04
Counting the number of orbitals
The number of orbitals for a given 'l' is +1. Therefore, for each possible value of 'l', we must count the corresponding number of orbitals and sum them up.For =1 and l=0, there is 1 orbital (the s orbital).For =2 and l=0, there is 1 orbital.For =2 and l=1, there are 3 orbitals (the p orbitals).For =3 and l=0, there is 1 orbital.Adding these up gives us 1 (for n=1, l=0) + 1 (for n=2, l=0) + 3 (for n=2, l=1) + 1 (for n=3, l=0), which equals 6 orbitals in total.
05
Arriving at the final answer
After enumerating all the possible orbitals and their count, we add up the orbitals: s orbitals (1+1+1) + p orbitals (3) = 6. Therefore, there are 6 orbitals with (n+l) < 5.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Orbital Quantum Mechanics
Quantum mechanics is the branch of physics dealing with the behavior of particles at the atomic and subatomic levels. When we talk about orbital quantum mechanics, we're diving into how electrons behave within atoms. Electrons aren't just randomly scattered around an atom; they exist in specific energy levels and regions called orbitals.
These orbitals are not definite paths, but rather probabilities of where an electron might be at any given time. They are defined by quantum numbers that provide a systematic nomenclature to specify the characteristics of electrons and their orbitals. Understanding the shape, size, and energy of these orbitals depends on these unique sets of numbers. In essence, they allow us to predict and explain the electronic structure and chemical behavior of atoms.
These orbitals are not definite paths, but rather probabilities of where an electron might be at any given time. They are defined by quantum numbers that provide a systematic nomenclature to specify the characteristics of electrons and their orbitals. Understanding the shape, size, and energy of these orbitals depends on these unique sets of numbers. In essence, they allow us to predict and explain the electronic structure and chemical behavior of atoms.
Principal Quantum Number
The principal quantum number, represented by the symbol \(n\), is a fundamental component in understanding orbital quantum mechanics. It denotes the energy level or shell in which the electron resides. Think of it as the 'address' indicating how far an electron is from the nucleus. The value of \(n\) starts at 1 and increases incrementally, where a higher \(n\) means the electron is in a higher energy level, and typically further away from the nucleus.
Role of the Principal Quantum Number
This number is crucial because it affects the energy that an electron possesses and therefore its reactivity. The principal quantum number also influences the size of the orbital. As \(n\) increases, orbitals become larger, accommodating more electrons. It's important not just for identifying the location of an electron but also for determining the maximum number of electrons that can exist within a particular energy level using the formula \(2n^2\).Azimuthal Quantum Number
Where the principal quantum number tells you the energy level, the azimuthal quantum number, denoted as \(l\), narrows down its 'neighborhood' or subshell. It can have values ranging from 0 to \(n-1\) for a given principal quantum number \(n\).
Each value of \(l\) correlates with a particular type of orbital. For instance, when \(l=0\), the orbital is an 's' orbital, which is spherical. When \(l=1\), it points to 'p' orbitals, which are dumbbell-shaped and come in three orientations. Higher values of \(l\) correspond to 'd' and 'f' orbitals, which are more complex in shape.
Each value of \(l\) correlates with a particular type of orbital. For instance, when \(l=0\), the orbital is an 's' orbital, which is spherical. When \(l=1\), it points to 'p' orbitals, which are dumbbell-shaped and come in three orientations. Higher values of \(l\) correspond to 'd' and 'f' orbitals, which are more complex in shape.