 ## Quiz - quantum numbers

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1. If $$n=5$$, what are the possible values for $$\ell$$?

$$\ell$$ permitted values are $$0,1,2,3\ldots\left(n-1\right)$$. If $$n=5$$, $$\ell$$ can be $$0,1,2,3,4$$.
1. For a certain electron $$\ell=0$$. What is the shape of the subshell electron cloud of that electron?

The orbital quantum number $$\ell$$ indicates the shape of the subshell electron cloud
(orbital shape).
$$\ell$$ Orbital shape
$$0$$ $$s$$ orbital
$$1$$ $$p$$ orbital
$$2$$ $$d$$ orbital
$$3$$ $$f$$ orbital

$$\ell=0$$ indicates $$s$$ orbital, which has a spherical shape.

1. What are the $$n$$ and $$\ell$$ values for $$4d$$ electrons?

$$n=4$$ and $$\ell=2$$.

1. For an electron, principle quantum number, $$n$$ is $$3$$. What are the possible values for $$\ell$$ and $$m_{\ell}$$?

For $$\ell$$ permitted values are $$0,1,2,3\ldots\left(n-1\right)$$. If $$n=3$$, $$\ell$$ can be $$0,1,2$$.
For $$m_{\ell}$$ permitted values are $$0,\pm1,\pm2\ldots\pm\ell.$$ The following table shows the permitted values of $$m_{\ell}$$ for $$\ell=0,1,2$$
$$\ell$$ $$m_{\ell}$$
$$0$$ $$0$$
$$1$$ $$0,+1,-1$$
$$2$$ $$0,+1,-1,+2,-2$$

1. What are the names of the orbitals with the priciple quantum number $$n=3$$?

For $$\ell$$ permitted values are $$0,1,2,3\ldots\left(n-1\right)$$. If $$n=3$$, $$\ell$$ can be $$0,1,2$$.
For $$m_{\ell}$$ permitted values are $$0,\pm1,\pm2\ldots\pm\ell.$$
$$\ell$$ $$m_{\ell}$$ Name of the orbitals
$$0$$ $$0$$ $$3s$$
$$1$$ $$0,+1,-1$$ $$3p$$
$$2$$ $$0,+1,-1,+2,-2$$ $$3d$$
1. What are the possible values for $$\ell$$, $$m_{\ell}$$ and $$m_{s}$$ for an electron in $$n=1$$ state?

For $$\ell$$ permitted values are $$0,1,2,3\ldots\left(n-1\right)$$. If $$n=1$$, $$\ell$$ can be $$0$$.
For $$m_{\ell}$$ permitted values are $$0,\pm1,\pm2\ldots\pm\ell.$$ The following table shows the permitted values of $$m_{\ell}$$ for $$\ell=0$$
$$\ell$$ $$m_{\ell}$$
$$0$$ $$0$$ ($$1s$$ orbital)

$$m_{s}=+\frac{1}{2}$$, $$-\frac{1}{2}$$

1. How many valid combinations of quantum numbers exist for $$2p$$ electrons?

Principle quantum number $$\left(n\right)$$ and orbital quantum number $$\left(\ell\right)$$ are given in the question.
For $$2p$$ electrons: $$n=2$$ and $$\ell=1$$
What are the possible values for $$m_{\ell}$$ and $$m_{s}$$?
Quantum numbers $$n$$ $$\ell$$ $$m_{\ell}$$ $$m_{s}$$
Possible values $$2$$ $$1$$ $$0,+1,-1$$ $$-\frac{1}{2},+\frac{1}{2}$$
How many possible values $$1$$ $$1$$ $$3$$ $$2$$

Total valid combinations of quantum numbers $$=\text{3}\times2=6$$

$$n$$ $$\ell$$ $$m_{\ell}$$ $$m_{s}$$
$$2$$ $$1$$ $$0$$ $$-\frac{1}{2}$$
$$2$$ $$1$$ $$0$$ $$+\frac{1}{2}$$
$$2$$ $$1$$ $$+1$$ $$-\frac{1}{2}$$
$$2$$ $$1$$ $$+1$$ $$+\frac{1}{2}$$
$$2$$ $$1$$ $$-1$$ $$-\frac{1}{2}$$
$$2$$ $$1$$ $$-1$$ $$+\frac{1}{2}$$
1. Can an electron exist with $$n=4$$, $$\ell=2$$ and $$m_{\ell}=4$$ quantum numbers? Explain using possible quantum number combinations.

For $$n=4$$, $$\ell$$ can be $$0,1,2,3$$. Therefore, $$\ell=2$$ is valid.
For $$\ell=2$$, $$m_{\ell}$$ can be $$0,+1,-1,+2,-2$$. The valid combinations are given below.
$$n$$ $$\ell$$ $$m_{\ell}$$
$$4$$ $$2$$ $$0$$
$$4$$ $$2$$ $$+1$$
$$4$$ $$2$$ $$-1$$
$$4$$ $$2$$ $$+2$$
$$4$$ $$2$$ $$-2$$

Therefore, $$n=4$$, $$\ell=2$$ and $$m_{\ell}=4$$ is not a valid combination.

1. Can two electrons have the following combination of quantum numbers $$n=2$$, $$\ell=0$$ and $$m_{\ell}=0$$ ?

Yes. Because one electron can have the upward spin $$m_{s}$$ $$=$$ $$+\frac{1}{2}$$ and the other can have the downward spin $$m_{s}=-\frac{1}{2}$$.
1. Which type of orbital has the following combination of quantum numbers? $$n=4,\ell=2,m_{\ell}=1$$.
$$n=4$$ means fourth shell
$$\ell=2=d$$ orbital
For $$n=4$$ and $$\ell=2$$, $$m_{\ell}$$ can be $$0,+1,-1,+2,-2$$. Therefore, $$n=4,\ell=2,m_{\ell}=1$$ indicates one of the $$4d$$ orbitals.