Problem 46 - Chemistry The Central Science

Question

(a) A complex absorbs light in the range of \(200-300 \mathrm{nm} . \mathrm{Do}\) you expect it to have visible color? (b) A solution of a compound appears green. Does this observation necessarily mean that all colors of visible light other than green are absorbed by the solution? Explain. (c) What information is usually presented in a visible absorption spectrum of a compound? (d) What energy is associated with an absorption at \(440 \mathrm{nm}\) in \(\mathrm{kJ} / \mathrm{mol} ?\)

Step-by-Step Solution

Answer

(a) We cannot determine its visible color since the complex absorbs light in the 200-300 nm range, which is outside the range of visible light (380-750 nm). (b) A green-colored solution does not necessarily mean that all colors other than green are absorbed by the solution; other colors might still be partially transmitted or reflected but to a much lesser extent. (c) A visible absorption spectrum usually presents the absorption bands, peak wavelengths, and molar absorptivity of a compound. (d) The energy associated with an absorption at 440 nm is approximately 272.7 kJ/mol.

Step 1: Visible Light Range

Visible light has a wavelength range of approximately 380-750 nm. Any color we can see falls within this range.

Step 2: Determining the Visible Color

The compound absorbs light in the range of 200-300 nm, which is outside the range of visible light (380-750 nm). Therefore, we cannot determine its visible color based on this information. #Part (b)#

Step 3: Color Perception

A compound appears green if it reflects or transmits green light and absorbs most of the other colors in the visible spectrum.

Step 4: Conclusion

The observation of a green-colored solution does not necessarily mean that all colors other than green are absorbed by the solution. Some of the other colors might still be partially transmitted or reflected, but to a much lesser extent than the green light. #Part (c)#

Step 5: Visible Absorption Spectrum

In a visible absorption spectrum, the following information is usually presented: 1. The absorption bands of the compound, 2. The peak wavelengths where maximum absorption occurs, 3. The molar absorptivity (a measure of the compound's ability to absorb light) associated with each peak. #Part (d)#

Step 6: Formula for Energy

To calculate the energy associated with light absorption at 440 nm, we use the formula: \[E = \frac{hc}{\lambda}\] Where E is energy, h is Planck's constant (6.63 x 10^{-34} Js), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength of the light (in meters).

Step 7: Convert Wavelength

Convert the given wavelength (440 nm) to meters: \[\lambda = 440 \,\text{nm} \cdot \frac{1 \,\text{m}}{10^9\, \text{nm}} = 4.40 \times 10^{-7}\, \text{m}\]

Step 8: Calculate Energy per Photon

Use the formula to calculate the energy per photon: \[E = \frac{(6.63 \times 10^{-34}\, \text{Js})(3.00 \times 10^8\, \text{m/s})}{4.40 \times 10^{-7}\, \text{m}} = 4.53 \times 10^{-19}\,\text{J}\]

Step 9: Calculate Energy per Mole of Photons

Multiply the energy per photon by Avogadro's number (6.022 x 10^23 mol^{-1}) to find the energy per mole of photons: \[E_\text{mol} = (4.53 \times 10^{-19}\,\text{J}) \cdot (6.022 \times 10^{23}\, \text{mol}^{-1}) = 272.7\, \text{kJ mol}^{-1}\] The energy associated with an absorption at 440 nm is approximately 272.7 kJ/mol.

Key Concepts

Wavelength and Color

When it comes to light, "wavelength" refers to the distance between consecutive peaks of a light wave. Visible light ranges between approximately 380 to 750 nanometers (nm). Each color we see is associated with a specific portion of this range.
For example:

• Violet has shorter wavelengths, around 380-450 nm
• Blue is in the range of 450-495 nm
• Green occurs between 495-570 nm
• Yellow ranges from 570-590 nm
• Orange is within 590-620 nm
• Red spans from 620-750 nm

Understanding this spectrum helps us predict which colors might be visible. A compound absorbing light at wavelengths outside this visible range will not exhibit a color. For example, the absorption between 200-300 nm won't be visible to the naked eye, as these wavelengths correspond to the ultraviolet range.

Absorption Spectrum

An absorption spectrum showcases how a compound absorbs light across different wavelengths. It provides insights into the interaction between the compound and light.
Typically, it shows:

• Absorption bands – areas on the spectrum where the compound absorbs light strongly
• Peak wavelengths – specific points where light absorption is maximal

Through this spectrum, we discern which wavelengths are absorbed and which might be reflected or transmitted, helping to deduce the compound's perceived color. For instance, if a solution appears green, its absorption spectrum likely shows minimal absorption between 495-570 nm, while other wavelengths are more absorbed. This reflection or transmission of green light makes the solution appear green to us.

Energy Calculations

Calculating the energy associated with light absorption is crucial for understanding how light interacts with matter. For absorption at a specific wavelength like 440 nm, we have to calculate the energy involved in this process.
The formula used is:\[ E = \frac{hc}{\lambda} \]

• \(E\) is the energy in joules (J)
• \(h\) is Planck's constant \(6.63 \times 10^{-34} \text{Js}\)
• \(c\) is the speed of light \(3.00 \times 10^8 \text{m/s}\)
• \(\lambda\) is the wavelength in meters

For a wavelength of 440 nm, convert it to meters (\(440 \text{nm} = 4.40 \times 10^{-7} \text{m}\)). Plugging this into the formula gives an energy per photon of \(4.53 \times 10^{-19} \text{J}\). To obtain the energy per mole of photons, multiply by Avogadro’s number \((6.022 \times 10^{23} \text{mol}^{-1}\)), resulting in \(272.7 \text{kJ/mol}\). This calculation is key in fields like photochemistry.

Molar Absorptivity

Molar absorptivity, often denoted as \(\varepsilon\), is a measure of a compound's ability to absorb light at a particular wavelength. It finds a place on the absorption spectrum alongside peak wavelengths.
This value helps determine the concentration of a solution based on its absorbance using Beer's Law: \[ A = \varepsilon c l \]

• \(A\) is the absorbance
• \(\varepsilon\) is the molar absorptivity (in \( \text{L mol}^{-1} \text{cm}^{-1}\))
• \(c\) is the concentration of the solution in molarity (M)
• \(l\) is the path length through which the light passes (in cm)

In practical applications, knowing \(\varepsilon\) allows scientists to determine solution concentrations from measured absorbances, making it a pivotal factor in quantitative chemical analysis. Importantly, the larger the molar absorptivity, the more strongly the substance absorbs light at that wavelength.