According to the equation for the Balmer line spectrum of hydrogen, a value of n = 3 gives a red spectral line at 656.3 nm, a value of n = 4 gives a green line at 486.1 nm, and a value of n = 5 gives a blue line at 434.0 nm. Calculate the energy in kilojoules per mole of the radiation corresponding to each of these spectral lines.
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Identify the formula to calculate the energy of a photon: \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J s} \), \( c \) is the speed of light \( 3.00 \times 10^8 \text{ m/s} \), and \( \lambda \) is the wavelength in meters.
Convert the given wavelengths from nanometers to meters by using the conversion factor \( 1 \text{ nm} = 1 \times 10^{-9} \text{ m} \).
Substitute the values of \( h \), \( c \), and the converted \( \lambda \) into the energy formula to calculate the energy of a single photon for each wavelength.
Convert the energy from joules per photon to kilojoules per mole by using Avogadro's number \( 6.022 \times 10^{23} \text{ mol}^{-1} \) and the conversion factor \( 1 \text{ J} = 0.001 \text{ kJ} \).
Repeat the calculation for each wavelength (656.3 nm, 486.1 nm, and 434.0 nm) to find the energy in kilojoules per mole for each spectral line.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Balmer Series
The Balmer series describes the wavelengths of light emitted by hydrogen when an electron transitions from a higher energy level (n ≥ 3) to the second energy level (n = 2). Each transition corresponds to a specific wavelength, producing visible spectral lines. The series is crucial for understanding the emission spectrum of hydrogen and is foundational in quantum mechanics.
The energy of a photon can be calculated using the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This relationship shows that energy is inversely proportional to wavelength; shorter wavelengths correspond to higher energy photons. This concept is essential for calculating the energy associated with the spectral lines in the Balmer series.
To find the energy per mole of photons, the energy calculated for a single photon must be multiplied by Avogadro's number (approximately 6.022 x 10²³ mol⁻¹). This conversion is necessary to express energy in kilojoules per mole, which is a standard unit in chemistry for quantifying energy changes in reactions and processes involving large numbers of particles.
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