Question

1. a police officer uses a radar gun to catch heisenberg speeding. the gun operates at a frequency of 22.235 x 10^9 hz. find the wavelength in nanometers of this radiation.
2. what are the possible values of n, l and m_l? what does each of these determine?
3. what is the order of the spectrum from highest to lowest energy? radio, microwaves, infrared, visible light, ultraviolet, x - rays, gamma rays

(a)
(b)

1. which has more energy, a or b? describe why. b has more energy because the waves are closer together, therefore they have more energy because they have a higher frequency.
2. the ____________ quantum number defines the shape of an orbital (hint: question 14).
3. the n = 1 shell contains ____________ p orbitals.
4. the lowest energy shell that contains f orbitals is the shell with n = ____________.
5. ____________ - orbitals are spherical.
6. each p - subshell can accommodate a maximum of ____________ electrons.
7. how many quantum numbers are necessary to designate a particular electron in an atom?

Explanation:

Step1: Recall the wave - speed formula

The speed of light $c = \lambda
u$, where $c = 3\times10^{8}\ m/s$ (speed of light in vacuum), $\lambda$ is the wavelength, and $
u$ is the frequency. We need to solve for $\lambda$, so $\lambda=\frac{c}{
u}$.

Step2: Convert the given frequency and calculate wavelength

The given frequency $
u = 22.235\times 10^{9}\ Hz$. Substitute $c = 3\times10^{8}\ m/s$ and $
u$ into the formula:
$\lambda=\frac{3\times 10^{8}\ m/s}{22.235\times 10^{9}\ Hz}\approx1.35\times 10^{-2}\ m$.
To convert to nanometers, use the conversion factor $1\ m = 10^{9}\ nm$. So $\lambda = 1.35\times 10^{-2}\times10^{9}\ nm = 13500\ nm$.

Step3: Quantum number values and their significance

• Principal quantum number ($n$): $n = 1,2,3,\cdots$. It determines the energy level or shell of the electron.
• Angular - momentum quantum number ($l$): $l=0,1,\cdots,n - 1$. It determines the shape of the orbital ($l = 0$ is $s$ - orbital, $l = 1$ is $p$ - orbital, $l = 2$ is $d$ - orbital, $l=3$ is $f$ - orbital).
• Magnetic quantum number ($m_l$): $m_l=-l,-l + 1,\cdots,0,\cdots,l - 1,l$. It determines the orientation of the orbital in space.

Step4: Order of spectrum by energy

The energy of electromagnetic radiation is given by $E = h
u$, where $h$ is Planck's constant. Since $
u=\frac{c}{\lambda}$, higher frequency (shorter wavelength) means higher energy. The order from highest to lowest energy is: gamma rays, x - rays, ultraviolet, visible light, infrared, microwaves, radio.

Step5: Quantum number for orbital shape

The angular - momentum quantum number ($l$) defines the shape of an orbital.

Step6: Number of p - orbitals in $n = 1$ shell

For $n = 1$, $l$ can only be $0$ (an $s$ - orbital). So the $n = 1$ shell contains $0$ p - orbitals.

Step7: Shell with f - orbitals

For $f$ - orbitals, $l = 3$. Since $l$ ranges from $0$ to $n - 1$, the lowest value of $n$ for which $l$ can be $3$ is $n = 4$.

Step8: Shape of s - orbitals

$s$ - orbitals are spherical.

Step9: Maximum electrons in p - subshell

Each orbital can hold a maximum of 2 electrons. A p - subshell has 3 orbitals ($m_l=- 1,0,1$ when $l = 1$). So each p - subshell can accommodate a maximum of $6$ electrons.

Step10: Number of quantum numbers for an electron

Four quantum numbers ($n$, $l$, $m_l$, $m_s$) are necessary to designate a particular electron in an atom. The spin quantum number ($m_s=\pm\frac{1}{2}$) determines the spin of the electron.

Answer:

1. $13500\ nm$
2. $n = 1,2,\cdots$ (energy level), $l = 0,1,\cdots,n - 1$ (orbital shape), $m_l=-l,\cdots,l$ (orbital orientation)
3. gamma rays, x - rays, ultraviolet, visible light, infrared, microwaves, radio
4. $l$
5. $0$
6. $4$
7. $s$
8. $6$
9. $4$