Question

1. how many sublevels are contained in each of these principal energy levels?

a. ( n = 1 )
b. ( n = 2 )
c. ( n = 3 )
d. ( n = 4 )

1. how many orbitals are in each of the following sublevels?

a. ( 6p ) sublevel
b. ( 7s ) sublevel
c. ( 5f ) sublevel
d. ( 4d ) sublevel
e. ( 1s ) sublevel
f. ( 2p ) sublevel

1. how do the bohr model and the quantum mechanical model differ in the way they describe the arrangement of electrons in atoms?

1. what is the formula to determine the number of orbitals in a principal energy level?

1. what is the maximum number of electrons in an orbital?
2. what is the formula to determine the number of electrons that can occupy a principal energy level? ( 2(n)^2 )

Explanation:

Step1: Determine the number of sublevels for each principal energy level

The number of sublevels in a principal energy level \(n\) is equal to \(n\).

• For \(n = 1\), the number of sublevels is \(1\) (only \(s\) sublevel).
• For \(n = 2\), the number of sublevels is \(2\) (\(s\) and \(p\) sublevels).
• For \(n = 3\), the number of sublevels is \(3\) (\(s\), \(p\), and \(d\) sublevels).
• For \(n = 4\), the number of sublevels is \(4\) (\(s\), \(p\), \(d\), and \(f\) sublevels).

Step2: Determine the number of orbitals for each sublevel

The number of orbitals in a sublevel is given by \(2l + 1\), where \(l\) is the azimuthal quantum number (\(l=0\) for \(s\), \(l = 1\) for \(p\), \(l=2\) for \(d\), \(l = 3\) for \(f\)).

• For \(6p\) sublevel (\(l = 1\)), the number of orbitals is \(2(1)+1=3\).
• For \(7s\) sublevel (\(l = 0\)), the number of orbitals is \(2(0)+1 = 1\).
• For \(5f\) sublevel (\(l=3\)), the number of orbitals is \(2(3)+1=7\).
• For \(4d\) sublevel (\(l = 2\)), the number of orbitals is \(2(2)+1=5\).
• For \(1s\) sublevel (\(l = 0\)), the number of orbitals is \(2(0)+1=1\).
• For \(2p\) sublevel (\(l = 1\)), the number of orbitals is \(2(1)+1=3\).

Step3: Compare Bohr model and quantum mechanical model

• Bohr model: Describes electrons as moving in fixed circular orbits around the nucleus. The energy of electrons is quantized, and electrons can jump between orbits by absorbing or emitting specific amounts of energy.
• Quantum mechanical model: Describes electrons in terms of probability distributions (orbitals) around the nucleus. It uses quantum numbers (\(n\), \(l\), \(m_l\), \(m_s\)) to describe the state of an electron. Orbitals have different shapes (\(s\)-spherical, \(p\)-dumbbell, \(d\)-clover - leaf, \(f\)-complex) and orientations.

Step4: Formula for number of orbitals in a principal energy level

The formula to determine the number of orbitals in a principal energy level \(n\) is \(n^{2}\).

Step5: Maximum number of electrons in an orbital

The maximum number of electrons in an orbital is \(2\) (due to the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of four quantum numbers. Electrons in the same orbital have opposite spins (\(m_s=+\frac{1}{2}\) and \(m_s =-\frac{1}{2}\))).

Answer:

10.
a. \(1\)
b. \(2\)
c. \(3\)
d. \(4\)
11.
a. \(3\)
b. \(1\)
c. \(7\)
d. \(5\)
e. \(1\)
f. \(3\)

1. Bohr model: electrons in fixed circular orbits. Quantum mechanical model: electrons in probability - distribution orbitals with different shapes and quantum - number descriptions.
2. \(n^{2}\)
3. \(2\)
4. \(2n^{2}\)