Use the Balmer equation to calculate the wavelength in nano-meters of the spectral line for hydrogen when n = 6 and m = 2. What is the energy in kilojoules per mole of the radiation corresponding to this line?

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First, use the Balmer equation to calculate the wavelength of the spectral line. The Balmer equation is given by \( \frac{1}{\lambda} = R \left( \frac{1}{m^2} - \frac{1}{n^2} \right) \), where \( R \) is the Rydberg constant (approximately 1.097 x 10^7 m^-1), \( m \) is the lower energy level (2 for Balmer series), and \( n \) is the higher energy level (6 in this case).

Substitute the values of \( R \), \( m \), and \( n \) into the Balmer equation to find \( \frac{1}{\lambda} \).

Calculate the wavelength \( \lambda \) by taking the reciprocal of the value obtained in the previous step. Convert this wavelength from meters to nanometers by multiplying by 10^9.

Next, calculate the energy of the photon emitted using the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (6.626 x 10^-34 J s), \( c \) is the speed of light (3.00 x 10^8 m/s), and \( \lambda \) is the wavelength you calculated.

Convert the energy from joules to kilojoules by dividing by 1000, and then use Avogadro's number (6.022 x 10^23 mol^-1) to convert the energy from per photon to per mole.

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Balmer Equation

The Balmer equation describes the wavelengths of the spectral lines of hydrogen. It is given by the formula λ = b(n² / (n² - m²)), where λ is the wavelength, n is the principal quantum number of the higher energy level, m is the lower energy level, and b is a constant (approximately 364.50682 nm). This equation is essential for calculating the wavelengths of light emitted when electrons transition between energy levels in hydrogen.

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01:00

Balmer Series Example

Energy of Photons

The energy of a photon can be calculated using the equation E = hc/λ, where E is the energy, h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength. This relationship is crucial for determining the energy associated with the spectral line calculated from the Balmer equation, allowing us to convert wavelength into energy.

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01:40

Photon Energy Formulas

Energy Conversion to Kilojoules per Mole

To express the energy of a photon in kilojoules per mole, we use the conversion factor that 1 mole of photons contains Avogadro's number (6.022 x 10^23) of photons. Thus, the energy in joules can be converted to kilojoules per mole by dividing the energy of a single photon by 1000 and then multiplying by Avogadro's number. This step is necessary for reporting energy in a standard unit used in chemistry.

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