Question

In a hydrogen atom, the electron is in the \(n=5\) state. Which of the following sets could correspond to the \(\ell, m\) states of the electron? a) 5,-3 b) 4,-5 c) 3,-2 d) 4,-6

Short answer

a) \(\ell = 5\), \(m = -3\) b) \(\ell = 4\), \(m = -5\) c) \(\ell = 3\), \(m = -2\) d) \(\ell = 4\), \(m = -6\) Answer: c) \(\ell = 3\), \(m = -2\)

Step-by-step solution

Understand the rules for quantum numbers

For a given principal quantum number \(n\), the possible values of the angular momentum quantum number \(\ell\) are \(0,1,2, \dots,n-1\). And for a given \(\ell\) value, the possible values of the magnetic quantum number \(m\) are \(-\ell, -(\ell-1), \dots, 0, \dots, (\ell-1), \ell\).

Check each option against the quantum number rules

We will now compare each option with the quantum number rules mentioned in Step 1. a) For option (a), we have \(\ell = 5\) and \(m = -3\). Since the possible values for \(\ell\) are \(0, 1, 2, 3, 4\) when \(n=5\), this option is invalid as \(\ell\) is out of range. b) For option (b), we have \(\ell = 4\) and \(m = -5\). Here, \(\ell = 4\) is within the range for \(n = 5\). However, the possible values of \(m\) for \(\ell = 4\) are \(-4, -3, -2, -1, 0, 1, 2, 3, 4\). As \(m=-5\) is not within this range, this option is also invalid. c) For option (c), we have \(\ell = 3\) and \(m = -2\). The \(\ell\) value is within the range for \(n = 5\). The possible values of \(m\) for \(\ell = 3\) are \(-3, -2, -1, 0, 1, 2, 3\). As \(m=-2\) is within this range, this option is valid. d) For option (d), we have \(\ell = 4\) and \(m = -6\). Here, \(\ell = 4\) is within the range for \(n = 5\). However, the possible values of \(m\) for \(\ell = 4\) are \(-4, -3, -2, -1, 0, 1, 2, 3, 4\). As \(m=-6\) is not within this range, this option is invalid.

Conclusion

Based on the quantum number rules, only option (c) with \(\ell = 3\) and \(m = -2\) is valid for the electron in the hydrogen atom when it is in the \(n = 5\) state.

Key concepts

hydrogen atom

The hydrogen atom is the simplest of all atoms, consisting of just one proton and one electron. This simplicity makes it a perfect model for quantum mechanics, allowing us to explore fundamental principles like quantization of energy levels. In a hydrogen atom, the single electron orbits the nucleus in specific allowed paths, known as orbitals. These orbitals are defined by quantum numbers, each providing crucial information about the electron's state.

• The energy levels in a hydrogen atom are solely determined by the principal quantum number, denoted by the symbol \( n \). These levels get higher as \( n \) increases, leading to more complex possible arrangements for the electron.
• Despite its simplicity, the hydrogen atom exhibits rich and complex behavior due to quantum mechanical principles, which are essential for understanding reactions in chemistry and the characteristics of materials in physics.

The exploration of quantum numbers in hydrogen atoms offers insights into how electrons are arranged and how they interact within atoms.

angular momentum quantum number

The angular momentum quantum number, typically symbolized by \( \ell \), is crucial in determining the shape of an electron's orbital within an atom. It essentially describes the orbital's angular shape and is closely related to the electron's angular momentum.

• The angular momentum quantum number can take on integer values from 0 up to \( n-1 \), where \( n \) is the principal quantum number. This range indicates the variety of shapes an orbital can take depending on its energy level.
• Each value of \( \ell \) corresponds to a specific type of orbital: \( \ell = 0 \) denotes an s orbital, \( \ell = 1 \) denotes a p orbital, \( \ell = 2 \) denotes a d orbital, and so forth. Each of these orbitals has distinct spatial distributions and energies.

Understanding \( \ell \) helps us predict the orbital arrangement and chemical properties of an atom. It also influences the angular nodes present, which are specific regions where the probability of finding an electron is zero.

magnetic quantum number

The magnetic quantum number, symbolized as \( m \), provides information regarding the orientation of an electron's orbital in space. This is important for understanding how atoms interact in magnetic fields and with other atoms.

• The possible values of \( m \) depend on the angular momentum quantum number \( \ell \) and range from \( -\ell \) to \( \ell \), including zero. This means if \( \ell \) is 2, then \( m \) can be -2, -1, 0, 1, or 2.
• Each value of the magnetic quantum number corresponds to a specific orbital orientation in three-dimensional space, providing insights into how orbitals overlap and bond in molecules.

The magnetic quantum number's role is crucial when atoms are placed in external magnetic fields, influencing their energy levels and electron orientation. This plays a prominent role in spectroscopy and magnetic resonance imaging (MRI) technologies.

principal quantum number

The principal quantum number, represented by \( n \), is fundamental in defining an electron's energy level and the overall size of the orbital it occupies.

• Generally, \( n \) takes on integral values such as 1, 2, 3, and so forth. Each integer corresponds to a more energetic and larger-sized orbital.
• Higher values of \( n \) indicate that the electron is further from the nucleus, leading to increased energy states and more extensive electron cloud sizes.

Understanding \( n \) helps to explain why different atoms exhibit varying levels of reactivity and stability. For example, as \( n \) increases, electrons are less tightly bound to the nucleus, affecting ionization energy and chemical interactions.