To determine the specific electron configuration from the given quantum numbers, we can analyze the provided values:
- \( n = 5 \): This is the principal quantum number, indicating that the electron is in the fifth energy level.
- \( l = 2 \): The azimuthal quantum number (\( l \)) of 2 corresponds to the d sublevel (where \( l = 0 \) is s, \( l = 1 \) is p, \( l = 2 \) is d, and \( l = 3 \) is f).
- \( m_l = -1 \): This is the magnetic quantum number, indicating a specific orbital within the d sublevel (there are five d orbitals, with \( m_l \) ranging from -2 to +2).
- \( m_s = -\frac{1}{2} \): This is the spin quantum number, indicating that the electron has a spin of -1/2.
Given this information, we can determine which element these quantum numbers represent. The d sublevel can hold up to 10 electrons, but for this electron configuration, we will place the electron in the 5d sublevel (since \( n=5 \) and \( l=2 \), corresponding to the d sublevel).
To find the element associated with these quantum numbers, we need to know the total number of electrons filling the 5th period, particularly focusing on the d block elements.
In the fifth period, the d block elements are part of the transition metals, specifically starting from the element Zirconium (Zr) with atomic number 40 and going to the element Cadmium (Cd) with atomic number 48. The last element of the d block is Mercury (Hg), which has atomic number 80.
However, since the electron configuration we're discussing is for the specific 5d sublevel, we can identify that the element must be part of the 5d transition metals:
- The 5d block starts with Hafnium (Hf) at atomic number 72. The filling of the 5d sublevel continues through Rutherforidium (Rf).
If we consider that we are filling the 5d sublevel and that the specified quantum numbers refer to the last electron:
- For Hafnium \( (Hf) \), the electron configuration is \( [Xe] 6s^2 5d^2 \).
- For Rutherforidium \( (Rf) \), the electron configuration is \( [Rn] 7s^2 5f^{14} 6d^2 \).
Given that we are targeting an electron with those specific quantum numbers, we conclude that the electron configuration highlights a possible placement in the 5d sublevel for the transition metals in the fifth period.
Thus, the answer is:
a. The specific electron configuration implied by the quantum numbers is in the 5d sublevel.
b. The likely element associated with these quantum numbers would be Rutherforidium (Rf). However, depending on the count of electrons up until we hit the specific m_l value, we may adjust the named element, but it generally fits within the group of 5d transition metals. If we focus solely on general occurrences, we could assume Hafnium (Hf), but the designation depends on where the program considers the electron decimal point placed. So, based on all information: element corresponds with general classification: Hafnium (Hf), Rutherforidium (Rf).
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