Chapter 11, Problem 11.4TC

Interpretation Introduction

(a)

Interpretation:

The statement “There is one $s$ orbital when $n=1$, two $s$ orbitals when $n=2$, three $s$ orbitals when $n=3$, and so on” is to be identified as true or false and if false, true statement for that is to be stated.

Concept introduction:

According to the quantum mechanical model of an atom, there are principal energy levels in an atom and each principal energy level is represented by a principal quantum number ($n$). In each principal energy level, there are one or more sublevels present.

Answer to Problem 11.4TC

The statement “There is one $s$ orbital when $n=1$, two $s$ orbitals when $n=2$, three $s$ orbitals when $n=3$, and so on” is false. “There is only one s orbital for each value of n” is a true statement.

Explanation of Solution

The statement “There is one $s$ orbital when $n=1$, two $s$ orbitals when $n=2$, three $s$ orbitals when $n=3$, and so on” is a false statement. There is only one $s$ orbital for each value of $n$. It is only the number of sublevels in any principal energy level are equal to its principal quantum number.. That is for $n=3$, there are three sublevels in the principal energy level. Therefore, the correct statement is “There is only one $s$ orbital for each value of $n$.”

Conclusion

The statement “There is one $s$ orbital when $n=1$, two $s$ orbitals when $n=2$, three $s$ orbitals when $n=3$, and so on” is false. The correct or true statement is “There is only one $s$ orbital for each value of $n$”.

Interpretation Introduction

(b)

Interpretation:

The statement “All $n=3$ orbitals are at lower energy than all $n=4$ orbitals” is to be identified as true or false and if false, true statement for that is to be stated.

Concept introduction:

Aufbau gave the principle of the energy of empty orbitals present in an atom and also the trends they follow while filling the electrons. Under his statement, lower energy levels should be filled first than the higher energy levels. The energy of level is classified on the basis of $n+l$ value.

Answer to Problem 11.4TC

The statement “All $n=3$ orbitals are at lower energy than all $n=4$ orbitals” is false. “Not all $n=3$ orbitals are at lower energy than all $n=4$ orbitals” is a true statement.

Explanation of Solution

The statement “All $n=3$ orbitals are at lower energy than all $n=4$ orbitals” is a false statement. The energy of $3d$ orbitals is more than the $4s$ orbital on the basis of $n+l$ value as stated by Aufbau. Therefore, the correct statement is “Not all $n=3$ orbitals are at lower energy than all $n=4$ orbitals”.

Conclusion

The statement “All $n=3$ orbitals are at lower energy than all $n=4$ orbitals” is false. The correct or true statement is “Not all $n=3$ orbitals are at lower energy than all $n=4$ orbitals”.

Interpretation Introduction

(c)

Interpretation:

The statement “There is no d sublevel when $n=2$” to be identified as true or false and if false, true statement for that is to be stated.

Concept introduction:

According to the quantum mechanical model of an atom, there are principal energy levels in an atom and each principal energy level is represented by a principal quantum number ($n$). In each principal energy level, there are one or more sublevels present.

Answer to Problem 11.4TC

The statement “There is no d sublevel when $n=2$” is true.

Explanation of Solution

The statement “There is no d sublevel when $n=2$” is a true statement. The number of sublevels in any principal level always equals to the principal quantum number of that level. For example, the number of sublevels in the principal energy level having quantum number $\left(n\right)$ equals to $3$ have three sublevels in it and sublevels are $3s,3p$ and $3d$. Therefore, there is no d orbital in $n=2$ level. It has only s and p sublevels only.

Conclusion

The statement “There is no d sublevel when $n=2$” is correct.

Interpretation Introduction

(d)

Interpretation:

The statement “There are five d orbitals at both the fourth and sixth principal energy levels” to be identified as true or false and if false, a true statement for that is to be stated.

Concept introduction:

According to the quantum mechanical model of an atom, there are principal energy levels in an atom and each principal energy level is represented by a principal quantum number ($n$). In each principal energy level, there are one or more sublevels present. Each sublevel is also associated with some orbitals.

Answer to Problem 11.4TC

The statement “There are five d orbitals at both the fourth and sixth principal energy levels” is true.

Explanation of Solution

The statement “There are five d orbitals at both the fourth and sixth principal energy levels” is a true statement. The value of azimuthal quantum number $\left(l\right)$ for the sublevels $s,p,d$ and $f$ is $0,1,2$ and $3$ respectively. The orbitals are the region of space around an atom where the probability of finding the electrons is maximum. There are $2l+1$ orbitals in a particular sublevel. That is the number of orbitals in a sublevel is fixed and depends only on its azimuthal quantum number $\left(l\right)$. Therefore, there are only five orbitals in the d sublevel of any principal energy level.

Conclusion

The statement “There are five d orbitals at both the fourth and sixth principal energy levels” is correct.