Suppose you take a trip to a distant universe and find that the periodic table there is derived from an arrangement of quantum numbers different from the one on Earth. The rules in that universe are:

1. principal quantum number n = 1, 2, . . . (as on Earth);
2. angular momentum quantum number ℓ = 0, 1, 2,. . . , n – 1 (as on Earth);
3. magnetic quantum number mℓ = 0, 1, 2, . . . , ℓ (only positive integers up to and including ℓ are allowed);
4. spin quantum number ms = –1, 0, 1 (that is, three allowed values of spin).

(a) Assuming that the Pauli exclusion principle remains valid in the alternate universe, what is the maximum number of electrons that can populate a given orbital there?

(b) Write the electronic configuration of the element with atomic number 8 in the periodic table.Formatting: superscript numbers where appropriate but omit parentheses.

(c) What is the atomic number of the second noble gas?

a) 3

b) 1s^3 2s^3 2p^2
c) 12

(a) In the given universe, the maximum number of electrons that can populate a given orbital is determined by the number of unique combinations of quantum numbers allowed by the rules.

Using the given rules, the maximum number of electrons in an orbital is determined by the product of the allowed values for the quantum numbers: nℓmℓms.

For principal quantum number n = 1, there is only one possible value of angular momentum quantum number, ℓ = 0. The only possible value for the magnetic quantum number mℓ is also 0. And finally, for spin quantum number ms, there are three allowed values.

So, for one orbital in the alternate universe, the maximum number of electrons is 1 × 1 × 1 × 3 = 3.

(b) The electronic configuration of the element with atomic number 8 (oxygen) in the periodic table of the alternate universe would be:

1s^2 2s^2 2p^4

(c) The second noble gas in the periodic table would have a full electron configuration up to its outermost energy level. In the alternate universe, based on the given rules, the electron configuration would be:

1s^2 2s^2 2p^6

The atomic number of this second noble gas would be 2 + 2 + 6 = 10.

(a) In the alternate universe, the maximum number of electrons that can populate a given orbital can be calculated by using the formula 2(2ℓ + 1).

For example, if we consider the s orbital (where ℓ = 0), the maximum number of electrons in that orbital would be 2(2 * 0 + 1) = 2.

For the p orbital (where ℓ = 1), the maximum number of electrons in that orbital would be 2(2 * 1 + 1) = 6.

Similarly, for the d orbital (where ℓ = 2), the maximum number of electrons in that orbital would be 2(2 * 2 + 1) = 10.

Therefore, the maximum number of electrons that can populate a given orbital in the alternate universe depends on the value of the angular momentum quantum number (ℓ) and is given by 2(2ℓ + 1).

(b) The electronic configuration of the element with atomic number 8 in the periodic table can be determined by filling the orbitals in increasing order of energy. We start with the lowest energy orbital and continue to fill orbitals until we reach the atomic number.

In the alternate universe, the periodic table is arranged based on the quantum numbers mentioned above. Therefore, we follow the rules:

- The principal quantum number (n) is the shell or energy level of the atom.
- The angular momentum quantum number (ℓ) determines the shape of the orbital.
- The magnetic quantum number (mℓ) determines the orientation of the orbital.
- The spin quantum number (ms) represents the spin of an electron.

For the element with atomic number 8, we need to distribute the electrons across the available orbitals.

Using the given rules:
- The principal quantum number (n) can be 1 or 2.
- The angular momentum quantum number (ℓ) can be 0 (s orbital) or 1 (p orbital).
- The magnetic quantum number (mℓ) can be 0 (for s orbital) or -1, 0, 1 (for p orbital).
- The spin quantum number (ms) can be -1, 0, or 1.

Following the Pauli exclusion principle, each orbital can hold a maximum of 2 electrons with opposite spins.

Therefore, the electronic configuration of the element with atomic number 8 in the periodic table in the alternate universe is: 1s² 2s² 2p⁴.

(c) The second noble gas in the alternate universe can be determined by finding the noble gas with the highest atomic number that is smaller than the atomic number of the element in question.

In the periodic table, the noble gases are the elements in Group 18 (Group 8A), and they have completely filled electron shells.

Using the electronic configuration of the element with atomic number 8 (1s² 2s² 2p⁴), we can see that it has filled the first two energy levels (1s and 2s), and the p orbital of the second energy level (2p).

Therefore, the second noble gas in the alternate universe would be the element with atomic number 10, which is neon (Ne).

(a) To find the maximum number of electrons that can populate a given orbital in the alternate universe, we need to consider the quantum numbers and the Pauli exclusion principle.

In the given rules, the principal quantum number (n) represents the energy level of the orbital. The angular momentum quantum number (ℓ) determines the shape of the orbital, and the magnetic quantum number (mℓ) gives the orientation of the orbital. The spin quantum number (ms) indicates the direction of spin of an electron.

The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers.

In the alternate universe, the maximum number of electrons that can populate a given orbital is determined by the maximum values of ℓ and mℓ.

Since the maximum value of ℓ is n - 1, and the maximum value of mℓ is ℓ, we can calculate the maximum number of electrons using the formula:

maximum electrons = (2ℓ + 1) * 2

Therefore, in the alternate universe, the maximum number of electrons that can populate a given orbital is calculated by substituting the maximum values of ℓ and mℓ into the formula.

(b) To find the electronic configuration of an element in the periodic table in the alternate universe, we need to follow a specific order of filling the orbitals based on the quantum numbers.

The order of filling orbitals in the alternate universe can be determined by following the Aufbau principle, which states that electrons fill the lowest energy orbitals first.

Based on the rules provided, we start by filling orbitals with the lowest principal quantum number (n) and then proceed to higher values.

For the given element with atomic number 8, we can determine its electronic configuration by filling the orbitals using the quantum numbers in the alternate universe.

(c) To find the atomic number of the second noble gas in the alternate universe, we need to identify the second noble gas element and determine its atomic number.

In the periodic table, noble gases are found in Group 18, and they have full outer electron shells. Therefore, the second noble gas would be found at the end of Group 18.

We can determine the atomic number of the second noble gas by referring to the periodic table in the alternate universe and counting the elements in Group 18 until we reach the second noble gas.

2222