Question

(a) If a complex absorbs light at \(610 \mathrm{~nm}\), what color would you expect the complex to be? (b) What is the energy in Joules of a photon with a wavelength of \(610 \mathrm{~nm}\) ? (c) What is the energy of this absorption in \(\mathrm{kJ} / \mathrm{mol}\) ?

Short answer

(a) The complex appears orange as it absorbs light at \(610 nm\). (b) The energy of a photon with a wavelength of \(610 nm\) is approximately \(3.26 \times 10^{-19} J\). (c) The energy of this absorption is approximately \(196.23 kJ/mol\).

Step-by-step solution

Determine the Color

Using the electromagnetic spectrum, we can determine the following visible light wavelength ranges and their corresponding colors: Violet: \(380 - 450 nm\) Blue: \(450 - 495 nm\) Green: \(495 - 570 nm\) Yellow: \(570 - 590 nm\) Orange: \(590 - 620 nm\) Red: \(620 - 750 nm\) Since the complex absorbs light at 610 nm, the wavelength falls in the orange color range. Therefore, we expect the complex to appear orange.

Calculate Photon Energy

We will use the Planck's equation to calculate the energy of a photon with a wavelength of 610 nm. The equation is given by: \(E = \dfrac{hc}{\lambda}\) Where \(E\) is the energy, \(h\) is Planck's constant (\(6.626 \times 10^{-34} Js\)), \(c\) is the speed of light (\(3.00 \times 10^8 ms^{-1}\)), and \(\lambda\) is the wavelength (in meters). First, convert the wavelength from nanometers to meters: \(610 nm = 610 \times 10^{-9} m\) Next, plug the values into the equation: \(E = \dfrac{(6.626 \times 10^{-34} Js) (3.00 \times 10^8 ms^{-1})}{610 \times 10^{-9} m}\) Solve for \(E\): \(E \approx 3.26 \times 10^{-19} J\) The energy of a photon with a wavelength of 610 nm is approximately \(3.26 \times 10^{-19} J\).

Calculate Energy per Mole

To convert the energy of a single photon to energy per mole, we will use Avogadro's number (\(6.022 \times 10^{23} mol^{-1}\)). The equation to find energy per mole is given by: \(E_{per mole} = E_{single} \times N_A\) Where \(E_{per mole}\) is the energy per mole, \(E_{single}\) is the energy of a single photon, and \(N_A\) is Avogadro's number. Plug the values into the equation: \(E_{per mole} = (3.26 \times 10^{-19} J) (6.022 \times 10^{23} mol^{-1})\) Solve for \(E_{per mole}\): \(E_{per mole} \approx 196230 J/mol\) To convert Joules to kilojoules, divide by 1000: \(E_{per mole} = \dfrac{196230 J/mol}{1000}\) \(E_{per mole} = 196.23 kJ/mol\) The energy of this absorption is approximately \(196.23 kJ/mol\).

Key concepts

Electromagnetic Spectrum

The electromagnetic spectrum is a way to classify different types of electromagnetic radiation by their wavelengths or frequencies. It spans from very short gamma rays to very long radio waves, with visible light—a narrow band of wavelengths visible to the human eye—sitting somewhere in the middle.

When delving into how substances absorb light, it's essential to look at this spectrum because a compound will absorb specific wavelengths. The wavelengths not absorbed are reflected or transmitted, and this is what gives the compound its color. For example, if a complex absorbs light at 610 nm, which falls within the orange range, we expect the complex to appear as the complementary color—or blue-green.

Planck's Equation

To understand photon energy in-depth, we turn to Planck's equation, which is fundamental in the realm of quantum mechanics. Max Planck proposed that energy is quantized and can be released or absorbed in discrete units called quanta.

Planck's equation, represented as
\[ E = \frac{hc}{\lambda} \]
reveals the relationship between a photon's energy (E), Planck's constant (h), the speed of light (c), and the wavelength (\(\lambda\)). Here, Planck's constant, \(6.626 \times 10^{-34} Js\), is a fundamental physical constant, and the speed of light, \(3.00 \times 10^8 ms^{-1}\), underscores the uniformity of light's speed in a vacuum.

Photon Energy Calculation

A crucial aspect of understanding light absorption is being able to calculate the energy of a photon. This process involves using the previously mentioned Planck's equation to determine the amount of energy carried by a single photon of a specific wavelength.

For a photon with a wavelength of 610 nm, the calculation follows a straightforward conversion from nanometers to meters and then the application of the Planck’s equation. This provides us with an energy approximation which, in our context, equals to about \(3.26 \times 10^{-19} J\) for each photon. Knowing the individual photon's energy is the key to further calculations, such as determining the energy absorbed or emitted when substances interact with light.

Avogadro's Number

Avogadro’s number, also known as Avogadro's constant, is the number of constituent particles (usually atoms or molecules) in one mole of a given substance. Its value is an approximate \(6.022 \times 10^{23} mol^{-1}\), and it's a cornerstone of chemistry when it comes to relating the macroscopic and atomic scale quantities.

By combining the energy of a single photon we've calculated with Avogadro's number, we can obtain the total energy per mole of photons, thus bridging the quantum level to a chemical scale practical for real-world applications. This method is used to compute the total energy necessary for chemical reactions and processes triggered or influenced by light absorption.