Question

An electron that will have the highest energy in the set is (1) \(3,2,1, \frac{1}{2}\) (2) \(4,2,-1, \frac{1}{2}\) (3) \(4,1,0,-\frac{1}{2}\) (4) \(5,0,0, \frac{1}{2}\)

Short answer

Option (4) with quantum numbers ewline (5,0,0, \frac{1}{2}ewline) will have the highest energy.

Step-by-step solution

Understand Quantum Numbers

Each electron in an atom is described by a set of four quantum numbers: principal quantum number (ewline newline), azimuthal quantum number (ewline lewline), magnetic quantum number (ewline m_lewline), and spin quantum number (ewline m_sewline). The energy of an electron depends primarily on the principal quantum number (ewline newline) and, to a lesser extent, on the azimuthal quantum number (ewline lewline). Generally, a higher ewline newline means higher energy.

Compare Principal Quantum Numbers

Analyze the principal quantum numbers (ewline newline) given in the options:ewline (1) ewline newline = 3, (2) ewline newline = 4, (3) ewline newline = 4, (4) ewline newline = 5. The set with the highest ewline newline will have the highest energy. Clearly, option (4) has the highest ewline newline value (5).

Conclusion

Since the principal quantum number primarily determines the energy level and option (4) has the highest ewline newline (5), the electron in option (4) will have the highest energy.

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \(n\), is a crucial parameter in quantum mechanics. It describes the energy level of an electron in an atom, which can take positive integer values (1, 2, 3,...). A larger \(n\) value corresponds to a higher energy electron and a larger orbit in which the electron is likely to be found. Thus, electrons in the shell with \(n = 5\) are generally higher in energy compared to those in shells with lower \(n\) values. This principal quantum number primarily determines the electron's energy and its distance from the nucleus.

Azimuthal Quantum Number

The azimuthal quantum number, represented as \(l\), further defines the shape of the electron's orbital within a given principal quantum number. It can take values from 0 to \(n-1\). For instance, if \(n = 4\), \(l\) can be 0, 1, 2, or 3. The different values of \(l\) correspond to different orbital shapes:

• \(l = 0\): 's' orbital (spherical shape)
• \(l = 1\): 'p' orbital (dumbbell shape)
• \(l = 2\): 'd' orbital (clover shape)
• \(l = 3\): 'f' orbital (complex shapes)

Along with the principal quantum number, the azimuthal number influences electron energy, where higher \(l\) generally means slightly higher energy within the same principal level.

Electron Energy Levels

Electron energy levels in an atom are defined by the quantum numbers, primarily the principal quantum number \(n\). Higher \(n\) and \(l\) values mean higher energy. The sequence of energy levels grows with each added electron shell: K (n=1), L (n=2), M (n=3), N (n=4), and so on. Electrons in these levels occupy different sublevels (s, p, d, f), influencing how they interact and bond with other atoms. For simplicity, electron configurations are often expressed by the Aufbau principle, which states that electrons fill lower-energy orbitals before moving to higher ones.

Magnetic Quantum Number

The magnetic quantum number, denoted \(m_l\), specifies the orientation of an electron's orbital around the nucleus. Its value ranges from \(-l\) to \(+l\), including zero. For example, if \(l = 2\) (a 'd' orbital), \(m_l\) can be -2, -1, 0, 1, or 2. Each \(m_l\) value represents a possible orientation of the orbital in space, impacting how the electrons in an atom are arranged and, consequently, how they contribute to the atom's overall magnetic properties.

Spin Quantum Number

The spin quantum number, \(m_s\), describes the intrinsic spin of an electron within an orbital. Spin can be either +1/2 or -1/2, often denoted as 'spin up' or 'spin down.' This quantum number is essential in determining the magnetic moment of an electron, influencing the atom's magnetic properties. Paired electrons in an orbital have opposite spins, which is a principle outlined by the Pauli exclusion principle, asserting that no two electrons in an atom can have the same set of all four quantum numbers.