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To find the wavelength of a photon, we use the relationship between its frequency and the speed of light. The speed of light is a constant, denoted by \( c \), which is approximately \( 3.00 \times 10^8 \text{ m/s} \). The equation we use is \( c = \lambda u \), where \( \lambda \) is the wavelength and \( u \) is the frequency. By rearranging the equation to solve for the wavelength, we get \( \lambda = \frac{c}{u} \). We substitute the given frequency of \( 6.5 \times 10^9 \text{ Hz} \), and find the wavelength, \( \lambda = 4.615 \times 10^{-2} \text{ meters} \). To convert this to nanometers (nm), since 1 meter is \( 10^9 \text{ nm} \), we multiply by \( 10^9 \) yielding \( \lambda = 4.615 \times 10^7 \text{ nm} \). This shows us the vital step of unit conversion, which ensures our calculations match scientific measurements and standards.

The energy of a photon can be calculated using the famous equation \( E = hu \), where \( E \) is the energy, \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \), and \( u \) is the frequency. This equation highlights that the energy of a photon is directly proportional to its frequency.Using the given frequency, we substitute to find \( E = 6.626 \times 10^{-34} \times 6.5 \times 10^9 = 4.307 \times 10^{-24} \text{ J} \). This result reflects the basic principle that even very small particles like photons have a measurable energy dependent on their frequency. Understanding photon energy is essential in the study of light behavior and quantum mechanics.

Avogadro's number is a fundamental constant used in chemistry and physics. It's approximately \( 6.022 \times 10^{23} \text{ photons/mol} \) and represents the number of particles, such as atoms or molecules, in one mole of substance. In the context of photons, Avogadro's number tells us how many individual photons are in a mole.To calculate the energy of one mole of photons, multiply the energy of a single photon by Avogadro's number. The formula used is \( E_{\text{mole}} = E \times \text{Avogadro's number} \). Thus, for our photon energy of \( 4.307 \times 10^{-24} \text{ J} \), the energy of one mole is \( 25.93 \text{ J/mol} \). This demonstrates how small energy units add up to significant quantities when considered in bulk chemical reactions or light interactions.

The visible light spectrum refers to the range of electromagnetic wavelengths that are detectible by the human eye, generally spanning from about 400 nm to 750 nm. This forms the colorful rainbow that we perceive in everyday life.In the original exercise, the calculated wavelength for the photon was \( 4.615 \times 10^7 \text{ nm} \). This wavelength is far outside the visible spectrum as it is much larger than what our eyes can perceive. Understanding the visible light spectrum is crucial for applications in optics, photography, and color theory. It also explains why some electromagnetic waves, despite their presence, remain unnoticed by human senses.