Video Transcript
The quantum numbers for the valence electron in an atom of lithium are 𝑛 equals two, 𝑙 equals zero, 𝑚 sub 𝑙 equals zero, and 𝑚 sub 𝑠 equals positive one-half. What are the quantum numbers for the second valence electron in an atom of beryllium? (A) 𝑛 equals three, 𝑙 equals zero, 𝑚 sub 𝑙 equals zero, 𝑚 sub 𝑠 equals negative one-half. (B) 𝑛 equals two, 𝑙 equals one, 𝑚 sub 𝑙 equals zero, 𝑚 sub 𝑠 equals negative one-half. (C) 𝑛 equals three, 𝑙 equals zero, 𝑚 sub 𝑙 equals zero, 𝑚 sub 𝑠 equals positive one-half. (D) 𝑛 equals two, 𝑙 equals zero, 𝑚 sub 𝑙 equals zero, 𝑚 sub 𝑠 equals negative one-half. (E) 𝑛 equals two, 𝑙 equals zero, 𝑚 sub 𝑙 equals one, 𝑚 sub 𝑠 equals positive one-half.
Quantum numbers are a set of four numbers that help us describe electrons in an atom. The principal quantum number, given the letter 𝑛, tells us the energy level the electron is in, as well as the size of an atomic orbital. The next quantum number is the subsidiary quantum number, which is given the letter 𝑙. This tells us the type of orbital the electron occupies. The next quantum number is the magnetic quantum number, given the letter 𝑚 sub 𝑙. This quantum number specifies the orientation of the atomic orbital. The final quantum number is the spin quantum number. This determines the spin state of the electron, that is, whether the electron is spin up or spin down.
Each atomic orbital holds one spin-up electron and one spin-down electron. So each orbital holds two electrons total. Electrons fill orbitals in order of increasing energy, which is summarized by this diagram. We can also map how electrons fill orbitals on the periodic table. The periodic table is divided into blocks corresponding to the type of orbital that’s filled there. We can go across periods and down groups to determine which orbitals electrons are occupying.
As an example, let’s zoom in on the portion of the periodic table where lithium is located. Lithium is located in the s block of the periodic table. So electrons fill s orbitals here. The energy level of the orbital matches the period number. Lithium has three electrons. The first two will occupy a 1s atomic orbital. Lithium’s third electron, which is its valence electron, will occupy the 2s atomic orbital. The number here tells us the energy level. So the electron is in the second energy level. The principal quantum number 𝑛 equals two, which matches what the question told us. The letter is for the orbital type. Each type of orbital has a corresponding value of 𝑙. 𝑙 equals zero for s-type orbitals, which again matches the information in the question.
The next quantum number 𝑚 sub 𝑙 can have values from negative 𝑙 to positive 𝑙. So if 𝑙 equals zero, like it does for the valence electron of lithium, 𝑚 sub 𝑙 can only equal zero. But if 𝑙 were to equal one, 𝑚 sub 𝑙 could take on a range of values from negative one to positive one. The final quantum number 𝑚 sub 𝑠 is equal to positive one-half here. Recall that the valence electron in lithium is the first electron in the 2s orbital. By convention, the first electron in the orbital is spin up and the second electron is spin down.
Now that we understand the quantum numbers for lithium, let’s figure out the quantum numbers for the second valence electron in an atom of beryllium. Beryllium is right next to lithium on the periodic table. Atoms of beryllium have four electrons. Just like in lithium, the first two electrons will fill the 1s atomic orbital. And beryllium’s two valence electrons will fill the 2s orbital. We know that this orbital is in the second energy level, so 𝑛 equals two. This means we can rule out answer choices (A) and (C). Since the valence electrons are in an s-type orbital, we know 𝑙 equals zero. So now we can rule out answer choice (B).
Since 𝑙 equals zero, 𝑚 sub 𝑙 only has one value, zero. So we’ve ruled out answer choice (E) now, which leaves us with one answer choice remaining. This answer choice says that 𝑚 sub 𝑠 equals negative one-half, which matches that this is the second electron in the orbital. So the quantum numbers for the second valence electron in an atom of beryllium are 𝑛 equals two, 𝑙 equals zero, 𝑚 sub 𝑙 equals zero, and 𝑚 sub 𝑠 equals negative one-half.
