The dye acriflavine when dissolved in water has its maximum light absorption at \(4530 \AA\) and has maximum florescence emission at \(5080 \AA\). The number of fluorescence quanta is about \(53 \%\) of the number of quanta absorbed. What percentage of absorbed light energy is emitted as fluorescence? (a) \(41 \%\) (b) \(47 \%\) (c) \(74 \%\) (d) \(63 \%\)

Planck's equation is a fundamental principle in quantum mechanics, linking the energy of a photon to the frequency of its electromagnetic wave. Mathematically, it is expressed as \( E = hu \), where \( E \) is the photon energy, \( h \) is Planck's constant (\(6.62607015 \times 10^{-34} \text{m}^2\text{kg/s}\)), and \( u \) is the frequency of the light. An alternative form of this equation uses wavelength \( \lambda \) instead of frequency, where \( u = \frac{c}{\lambda} \) and \( c \) is the speed of light in a vacuum (\(3 \times 10^8 \text{m/s}\)). Thus, the equation can be rewritten as \( E = \frac{hc}{\lambda} \). This equation is central to understanding the quantized nature of light absorption and emission in substances like dyes. Planck's equation helps us calculate the precise amount of energy associated with photons absorbed or emitted by a material, which is particularly useful in exercises where we need to understand the energy changes during processes involving light, such as fluorescence in dyes.

The wavelength-energy relationship is a direct consequence of Planck’s equation and reveals how the energy of a photon is inversely proportional to its wavelength. This means that as the wavelength of light increases, the energy of each photon decreases, and vice versa. This relationship can be seen in the modified Planck's equation \( E = \frac{hc}{\lambda} \). When addressing problems in fluorescence where different wavelengths are involved, it’s critical to recognize that the energy content of the light emitted as fluorescence will be different from the energy of the light absorbed if their wavelengths are different. For instance, in the given exercise, acriflavine absorbs light at \(4530 \AA\) and emits at \(5080 \AA\). Applying the wavelength-energy relationship shows us that the emitted light has less energy per photon due to its longer wavelength.

Quantum efficiency is a measure of how effectively a substance converts absorbed photons into emitted photons. Specifically, it is the ratio of the number of photons emitted to the number of photons absorbed. In the context of fluorescence, this efficiency is reflected in the percentage of absorbed light that is re-emitted as fluorescent light. A 100% quantum efficiency means every absorbed photon results in an emitted photon, but in real-world substances, some energy is always lost to non-radiative processes like heat. Thus, quantum efficiency is always less than 100%. The given exercise states that there is a \(53\%\) quantum efficiency in the dye acriflavine, indicating that for every 100 photons absorbed, approximately 53 photons are emitted as fluorescence. This is a critical factor when calculating the energy output as fluorescence compared to the energy input from light absorption.

Absorption and emission of light are two fundamental processes that occur when light interacts with matter. Absorption is when a material takes in photons, which often results in an increase in the material's internal energy. Emission, on the other hand, is when a material releases energy in the form of light. In a fluorescent substance like acriflavine, absorption occurs when it takes in light at its resonance frequency (or maximum absorption wavelength), and then, after a brief delay, it releases part of this energy as light at a different wavelength during emission. This delay is usually because the absorbed energy transitions the molecule to an excited state, and when it returns to a lower energy state, light is emitted. The difference in energy (and consequently wavelength) between the absorbed and emitted light is known as the Stokes shift. Understanding these processes is key when calculating the efficiency of energy transfer from absorbed to emitted light in fluorescent materials.