Chapter 36, Problem 34P

(a)

To determine

The possible values of $n$ and $m_{\text{l}}$ for the given orbital number.

Answer to Problem 34P

The possible values of $n$ is $n\ge 4$ and $m_{\text{l}}$ is $-3,-2,-1,0,+1,+2,+3$ .

Explanation of Solution

Given:

The orbital number is $l=3$ .

Formula used:

The expression to calculate the possible values of principal quantum number is given by,

  $n=l+x,\left(x\in \text{N}\right)$

Here $x$ belongs to Natural number.

The expression to calculate the possible values of magnetic orbital quantum number is given by,

  $m=x,\left(-l_{\text{n}}\le x\le l_{\text{n}}\right)$

Here $x$ is the integer.

Calculation:

The possible values of the principal quantum number is calculated as,

  $n=3+x,\left(x\in \text{N}\right)$

So, the possible value of the orbital numbers are $3,4,5,\mathrm{6...}$ .

The possible values of $n$ is $n\ge 4$ .

The possible values of the magnetic orbital quantum number for $l=0$ is calculated as,

  $m=x,\left(0\le x\le 0\right)$

So, the possible value of the magnetic orbital numbers is $0$ .

The possible values of the magnetic orbital quantum number for $l=1$ is calculated as,

  $m_{\text{l}}=x,\left(-1\le x\le 1\right)$

So, the possible value of the magnetic orbital numbers is $-1,0,1$ .

The possible values of the magnetic orbital quantum number for $l=2$ is calculated as,

  $m_{\text{l}}=x,\left(-2\le x\le 2\right)$

So, the possible value of the magnetic orbital numbers is $-2,-1,0,1,2$ .

The possible values of the magnetic orbital quantum number for $l=3$ is calculated as,

  $m_{\text{l}}=x,\left(-3\le x\le 3\right)$

So, the possible value of the magnetic orbital numbers is $-3,-2,-1,0,1,2,3$ .

Conclusion:

Therefore, the possible values of $n$ is $n\ge 4$ and $m_{\text{l}}$ is $-3,-2,-1,0,+1,+2,+3$ .

(b)

To determine

The possible values of $n$ and $m_{\text{l}}$ for the given orbital number.

Answer to Problem 34P

The possible values of $n$ is $n\ge 5$ and $m_{\text{l}}$ is $-4,-3,-2,-1,0,+1,+2,+3,+4$ .

Explanation of Solution

Given:

The orbital number is $l=4$ .

Formula used:

The expression to calculate the possible values of principal quantum number is given by,

  $n=l+x,\left(x\in \text{N}\right)$

Here $x$ belongs to Natural number.

The expression to calculate the possible values of magnetic orbital quantum number is given by,

  $m=x,\left(-l_{\text{n}}\le x\le l_{\text{n}}\right)$

Here $x$ is the integer.

Calculation:

The possible values of the principal quantum number is calculated as,

  $n=4+x,\left(x\in \text{N}\right)$

So, the possible value of the orbital numbers are $5,6,7,\mathrm{8...}$ .

The possible values of $n$ is $n\ge 5$ .

The possible values of the magnetic orbital quantum number for $l=0$ is calculated as,

  $m=x,\left(0\le x\le 0\right)$

So, the possible value of the magnetic orbital numbers is $0$ .

The possible values of the magnetic orbital quantum number for $l=1$ is calculated as,

  $m_{\text{l}}=x,\left(-1\le x\le 1\right)$

So, the possible value of the magnetic orbital numbers is $-1,0,1$ .

The possible values of the magnetic orbital quantum number for $l=2$ is calculated as,

  $m_{\text{l}}=x,\left(-2\le x\le 2\right)$

So, the possible value of the magnetic orbital numbers is $-2,-1,0,1,2$ .

The possible values of the magnetic orbital quantum number for $l=3$ is calculated as,

  $m_{\text{l}}=x,\left(-3\le x\le 3\right)$

So, the possible value of the magnetic orbital numbers is $-3,-2,-1,0,1,2,3$ .

The possible values of the magnetic orbital quantum number for $l=4$ is calculated as,

  $m_{\text{l}}=x,\left(-4\le x\le 4\right)$

So, the possible value of the magnetic orbital numbers is $-4,-3,-2,-1,0,1,2,3,4$ .

Conclusion:

Therefore, the possible values of $n$ is $n\ge 5$ and $m_{\text{l}}$ is $-4,-3,-2,-1,0,+1,+2,+3,+4$ .

(c)

To determine

The possible values of $n$ and $m_{\text{l}}$ for the given orbital number.

Answer to Problem 34P

The possible values of $n$ is $n\ge 1$ and $m_{\text{l}}$ is $0$ .

Explanation of Solution

Given:

The orbital number is $l=3$ .

Formula used:

The expression to calculate the possible values of principal quantum number is given by,

  $n=l+x,\left(x\in \text{N}\right)$

Here $x$ belongs to Natural number.

The expression to calculate the possible values of magnetic orbital quantum number is given by,

  $m=x,\left(-l_{\text{n}}\le x\le l_{\text{n}}\right)$

Here $x$ is the integer.

Calculation:

The possible values of the principal quantum number is calculated as,

  $n=0+x,\left(x\in \text{N}\right)$

So, the possible value of the orbital numbers are $1,2,\mathrm{3...}$ .

The possible values of $n$ is $n\ge 1$ .

The possible values of the magnetic orbital quantum number for $l=0$ is calculated as,

  $m=x,\left(0\le x\le 0\right)$

So, the possible value of the magnetic orbital numbers is $0$ .

Conclusion:

Therefore, the possible values of $n$ is $n\ge 1$ and $m_{\text{l}}$ is $0$ .