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?

Verified step by step guidance

1. 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).
2. Substitute the values of \( R \), \( m \), and \( n \) into the Balmer equation to find \( \frac{1}{\lambda} \).
3. 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.
4. 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.
5. 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.

Key Concepts

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.

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.

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.