To determine secondary information, use constant values and calculate the energy in 1 mole of the given photons and conversion factors to predict the colour of light emitted.

“Heated lithium atoms emit photons of light with an energy of 2.961 × 10⁻¹⁹ J. Calculate the frequency and wavelength of one of these photons. What is the total energy in 1 mole of these photons? What is the colour of the emitted light?”

We can use this information to solve our question.

Frequency:

Using Planck’s equation, we can rearrange and solve for the wave’s frequency (v). We are given the energy, and we know Planck’s constant (h = 6.626 × 10^-34 Js).

[latex]\begin{aligned} \mathrm{E}&=\mathrm{h}v\\ 2.691 \times 10^{-19}\mathrm{~J}&=\left(6.626\times10^{-34}\mathrm{Js}\right)v\\ v&=\frac{2.691\times 10^{-19}\mathrm{~J}}{6.626\times10^{-34} \mathrm{Js}}\\ v& =4.469\times 10^{14} \mathrm{~s}^{-1} \end{aligned}[/latex]

Answer: 4.469 × 1014 s⁻¹

We can now rearrange and solve for our wavelength (λ) using the frequency we just found and the speed of light (c = 3.00 × 10⁸ m/s) in the equation c = λv.

[latex]\begin{aligned} \mathrm{c}&=\lambda v\\ 3.00\times 10^8\mathrm{~m}/\mathrm{s}&=\left(4.469\times 10^{14}\mathrm{~s}^{-1}\right)\\ \lambda&=\frac{3.00\times 10^8\mathrm{~m}/\mathrm{s}}{4.469\times10^{14}\mathrm{~s}^{-1}}\\ \lambda&=6.713\times10^{-7}\mathrm{~m} \end{aligned}[/latex]

Answer: 6.713 × 10⁻⁷ m

To find the energy in one mole of these photons, we need to use Avogadro’s Number as a conversion factor.

There are 2.066 × 10²³ photons in 1 mole. To set up a conversion properly, be sure that the numerator has the unit you want and the denominator has the unit that you must cancel out!

[latex]\dfrac{2.691\times 10^{-19}\mathrm{~J}}{1\text{ photon }}\times\dfrac{6.02\mathrm{P}\times 10^{23}\text{ photons }}{1\mathrm{~mole}}=1.783\times 10^5\mathrm{~J}/\mathrm{mol}[/latex]

Answer: 1.783 × 10⁵ J

To find the colour of light emitted, first convert from metres to nanometres (this makes it easier to compare the calculated value to known colour ranges in the visible light spectrum).

Recall that each colour on the light spectrum has a different wavelength, typically described in nanometres. To find the colour of light, convert your wavelength into nanometres.

There are 10⁹ nanometres in 1 metre; alternately 1 nanometre is 10^-9 metres. Either conversion factor will work; just be sure that the units cancel out appropriately.

• Violet — 400–420 nm
• Indigo — 420–440 nm
• Blue — 440–490 nm
• Green — 490–570 nm
• Yellow — 570–585 nm
• Orange — 585–620 nm
• Red — 620–780 nm

[latex]\begin{aligned} &\quad6.713\times10^{-7}\mathrm{~m}\times\frac{1\times10^9\mathrm{~nm}}{1\mathrm{n}}=671\mathrm{~nm}=\text{Red Light} \end{aligned}[/latex]

The calculated value falls in the range of red light, which is between 620 and 780 nm.

Answer: Red