Which orbital is represented by the complete wave function, Psi_410? — Atomic Structure Chemistry Question

💡 Solution & Explanation

### Step 1 - Identifying Subscript Notation of Hydrogenic Wave Functions In quantum mechanics, the complete spatial wave function of a hydrogen-like atom in polar coordinates $(r, \theta, \phi)$ is uniquely specified by three quantum numbers, which are written as subscripts in a standard order: $$\Psi_{n, l, m_l}(r, \theta, \phi)$$ Where: * $n$ is the principal quantum number, representing the main energy level or shell. * $l$ is the azimuthal (or orbital angular momentum) quantum number, representing the subshell or orbital shape. * $m_l$ is the magnetic quantum number, representing the spatial orientation of the orbital in space. --- ### Step 2 - Extracting the Quantum Numbers from $\Psi_{410}$ By comparing the given wave function $\Psi_{410}$ with the standard subscript notation $\Psi_{n, l, m_l}$, we can identify the specific values of the three quantum numbers: * Principal quantum number: $$n = 4$$ * Azimuthal quantum number: $$l = 1$$ * Magnetic quantum number: $$m_l = 0$$ --- ### Step 3 - Decoding the Subshell Designation The principal quantum number $n = 4$ indicates that the electron resides in the fourth energy shell. The azimuthal quantum number $l$ determines the type of subshell according to the following standard spectroscopic notation: * If $l = 0$, the subshell is designated as $s$. * If $l = 1$, the subshell is designated as $p$. * If $l = 2$, the subshell is designated as $d$. * If $l = 3$, the subshell is designated as $f$. Since the azimuthal quantum number for our wave function is $l = 1$, the electron is in a $p$-subshell. Combining the shell and subshell, we find that the wave function represents a $4p$ orbital (specifically, the $4p_z$ orbital because $m_l = 0$). --- ### Step 4 - Analysis of the Options Let us analyze each of the given choices systematically: * **Option (A) $4s$:** For a $4s$ orbital, the quantum numbers are $n = 4$ and $l = 0$. The corresponding wave function would be represented as $\Psi_{400}$. Thus, this option is incorrect. * **Option (B) $3p$:** For a $3p$ orbital, the principal quantum number is $n = 3$ and the azimuthal quantum number is $l = 1$. The corresponding wave function would be of the type $\Psi_{31m_l}$ (such as $\Psi_{310}$). Thus, this option is incorrect. * **Option (C) $4p$:** For a $4p$ orbital, the quantum numbers are $n = 4$ and $l = 1$. This matches our wave function parameters exactly. Thus, this option is correct. * **Option (D) $4d$:** For a $4d$ orbital, the principal quantum number is $n = 4$ and the azimuthal quantum number is $l = 2$. The corresponding wave function would be of the type $\Psi_{42m_l}$. Thus, this option is incorrect. $$\text{Correct Option: } \boxed{\text{(C)}}$$