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Wavelength is an essential concept in understanding the behavior of light. When calcium chloride is introduced into a nonluminous flame, the wavelength of the emitted orange light can be measured. In this exercise, the wavelength is given as \(422.7\, \mathrm{nm}\). To use this in calculations, it's essential to convert it from nanometers to meters because the standard unit in physics for wavelength is meters. To convert nanometers to meters, use the conversion factor:

• 1 nanometer = \(1 \times 10^{-9}\) meters.

This means:\[ \lambda = 422.7 \, \mathrm{nm} \times \left(1 \times 10^{-9} \, \mathrm{m/nm}\right) = 422.7 \times 10^{-9} \, \mathrm{m} \]Converting to meters makes it straightforward to plug into other formulas, like calculating frequency.

Frequency is a measure of how many wavelengths pass a given point per unit of time. It is denoted by the Greek letter nu (\(u\)). To find the frequency of light emitted from the excited calcium, we use the speed of light equation: \[ c = \lambda \times u \]Where:

• \( c \) is the speed of light \(\left(3 \times 10^8 \, \mathrm{m/s} \right)\).
• \( \lambda \) is the wavelength in meters \(\left(422.7 \times 10^{-9} \, \mathrm{m}\right)\).

Rearrange the equation to solve for frequency:\[ u = \frac{c}{\lambda} \]Substitute in the values:\[ u = \frac{3 \times 10^8 \, \mathrm{m/s}}{422.7 \times 10^{-9} \, \mathrm{m}} \approx 7.10 \times 10^{14} \, \mathrm{Hz} \]This means that the frequency of the emitted light is \(7.10 \times 10^{14}\) Hertz, indicating how rapidly the light waves are oscillating.

The energy of a photon relates directly to its frequency. We use Planck's equation to find the energy associated with one photon of calcium's light emission:\[ E = h \times u \] Where:

• \( E \) is the energy of the photon.
• \( h \) is Planck's constant \(\left(6.626 \times 10^{-34} \, \mathrm{Js}\right)\).
• \( u \) is the frequency \(\left(7.10 \times 10^{14} \, \mathrm{Hz}\right)\).

Substitute in these values:\[ E = 6.626 \times 10^{-34} \, \mathrm{Js} \times 7.10 \times 10^{14} \, \mathrm{Hz} = 4.71 \times 10^{-19} \, \mathrm{J} \]This result gives the energy for an individual photon. To determine the energy for a mole of these photons, multiply by Avogadro's number (\(6.022 \times 10^{23} /\mathrm{mol}\)):\[ E_1 = 6.022 \times 10^{23} \times 4.71 \times 10^{-19} \, \mathrm{J} = 2.84 \times 10^5 \, \mathrm{J/mol} \]This value signifies the collective energy carried by one mole of these light particles, termed an Einstein.

Chemical excitation occurs when electrons in atoms absorb energy and move to a higher energy level. This process is temporary, and when these electrons transition back to their original states, they release energy in the form of light. For calcium in a flame test, the atoms become excited by absorbing thermal energy. When returning to the ground state, the orange light is emitted. This emission corresponds to the energy gap between the excited and ground states. In practical terms, for the calcium atom:

• The energy difference (or gap) given by the photon's energy in this process is \(4.71 \times 10^{-19} \, \mathrm{J}\).

The distinct color emitted helps us identify specific elements, showcasing the significance in fields such as analytical chemistry and astrophysics.