Question

How many orbitals in an atom can have the designation \(5 p,\) \(3 d_{z}, 4 d, n=5, n=4 ?\)

Short answer

The numbers of orbitals for each designation are as follows: - \(5 p\) : 3 orbitals - \(3 d_{z}\) : 1 orbital - \(4 d\) : 5 orbitals - \(n = 5\) : 25 orbitals - \(n = 4\) : 16 orbitals

Step-by-step solution

Recall quantum number rules

The rules that must be followed for selecting quantum numbers are: 1. \(n\): any positive integer (1, 2, 3,...) 2. \(l\): any integer from 0 to \((n - 1)\) 3. \(m_l\): any integer from \(-l\) to \(+l\) With these rules in mind, we can find the allowed orbitals for each designation.

Find orbitals for \(5 p\)

For the \(5 p\) orbitals: - \(n = 5\) - \(l = 1\) (since p orbitals correspond to \(l = 1\)) Now, we have to find the possible values of \(m_l\). If \(l = 1\), then \(m_l\) can be \(-1\), \(0\), or \(1\). So there are 3 orbitals in the \(5 p\) designation.

Find orbitals for \(3 d_{z}\)

For the \(3 d_{z}\) orbital: - \(n = 3\) - \(l = 2\) (since d orbitals correspond to \(l = 2\)) Now, we have to find the possible values of \(m_l\) that correspond to the \(d_{z}\) designation. \(d_{z}\) is just one of the \(5\) possible \(d\) orbitals. So there is only 1 orbital for the \(3 d_{z}\) designation.

Find orbitals for \(4 d\)

For the \(4 d\) orbitals: - \(n = 4\) - \(l = 2\) (since d orbitals correspond to \(l = 2\)) Now, we have to find the possible values of \(m_l\). If \(l = 2\), then \(m_l\) can be \(-2\), \(-1\), \(0\), \(1\), or \(2\). So there are 5 orbitals in the \(4 d\) designation.

Find orbitals for \(n = 5\) and \(n = 4\)

For the orbitals with \(n = 5\) and \(n = 4\), we have to find all the possible \(l\) and \(m_l\) values for each. For \(n = 5\): - \(l\) can be \(0,1,2,3\), or \(4\) For each value of \(l\), we will find all the possible \(m_l\) values and hence, count the orbitals for each \(l\) value. \(l\) Values | Allowed \(m_l\) Values | Number of Orbitals ----------- | -------------------- | ----------------- 0 | 0 | 1 1 | -1, 0, 1 | 3 2 | -2,-1, 0, 1, 2 | 5 3 | -3,-2,-1, 0, 1, 2, 3 | 7 4 | -4,-3,-2,-1,0,1,2,3,4 | 9 In this case, there are a total of 1+3+5+7+9 = 25 orbitals for \(n = 5\). For \(n = 4\): - \(l\) can be \(0,1,2\), or \(3\) For each value of \(l\), we will find all the possible \(m_l\) values and hence, count the orbitals for each \(l\) value. \(l\) Values | Allowed \(m_l\) Values | Number of Orbitals ----------- | -------------------- | ----------------- 0 | 0 | 1 1 | -1, 0, 1 | 3 2 | -2,-1, 0, 1, 2 | 5 3 | -3,-2,-1, 0, 1, 2, 3 | 7 In this case, there are a total of 1+3+5+7 = 16 orbitals for \(n = 4\). To sum up, the numbers of orbitals for each designation are as follows: \(5 p\) : 3 orbitals \(3 d_{z}\) : 1 orbital \(4 d\) : 5 orbitals \(n = 5\) : 25 orbitals \(n = 4\) : 16 orbitals

Key concepts

Atomic Orbitals

When studying the structure of atoms, atomic orbitals are a fundamental concept. These orbitals represent the regions in space where the probability of finding an electron is highest. According to the quantum mechanical model, orbitals are defined by quantum numbers.

In the given exercise, the designations such as '5p' correspond to specific orbitals. The number '5' represents the principal quantum number (), which indicates the energy level and size of the orbital. 'p' indicates the azimuthal quantum number (), also known as the angular momentum quantum number, which determines the shape of the orbital. In this case, for 'p' orbitals, is equal to 1. This value of gives rise to three possible magnetic quantum numbers (), corresponding to the three different orientations that p orbitals can have in space. Therefore, for a '5p' orbital, there are 3 possible orientations, which essentially means 3 different orbitals.

Similarly, with '4d' and '3dz' designations, the d is implied to equaling 2, which yields d orbitals with more complex shapes. The 'dz' suggests a specific orientation along the z-axis, but for counting purposes, it's crucial to understand that there are 5 different 'd' orbitals within a given energy level because can take on values from -2 to +2.

Quantum Mechanical Model

The quantum mechanical model of the atom presents a detailed understanding of how electrons are arranged and how they behave. This model characterizes electron positions in terms of probability distributions rather than fixed orbits. The concept of quantum numbers is the backbone of this model, as each electron in an atom is described by a unique set of quantum numbers that dictate their energy level, angular momentum, orientation, and spin.

The principal quantum number (), as discussed, defines the energy level. The azimuthal () or angular momentum quantum number dictates the shape of the orbital, and it ranges from 0 to ( - 1). For a given value of , the magnetic quantum number () takes values from - to + and describes the orientation of the orbital in three-dimensional space. The application of this rule enables us to systematically understand the presence of multiple orbitals within a subshell (like 'p' or 'd'), as seen in the exercise with '5p' or '4d' orbitals.

Electron Configuration

Electron configuration is the arrangement of electrons in an atom's orbitals. It follows a set of principles and patterns known as Aufbau principle, Hund's rule, and the Pauli exclusion principle. The configuration is usually noted in terms of the energy levels and types of orbitals the electrons occupy.

For instance, the notation '5p' seen in the exercise indicates the electron is in the fifth energy level (n=5) and in one of the 'p' orbitals. The numbers of orbitals for different values of and reflect that these orbitals accommodate different number of electrons, which is essential in predicting chemical behavior. For each principal energy level (), the number of possible orbitals increases. The exercise shows that not only are there individual orbitals for specific designations like '5p' or '3dz', but entire sets of orbitals corresponding to a particular principal quantum number (like all orbitals with = 5).

Understanding the possible values of , , and in the quantum mechanical model allows us to deduce the electron capacity within each energy level. With the exercise's guidance on allowed orbitals in an atom for various designations, it becomes clear how the organization of electrons contributes to the understanding of atomic structure and chemistry.