The energy levels of a hydrogen atom are given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
For \( n = 4 \):
\[ E_4 = -\frac{13.6 \, \text{eV}}{4^2} = -\frac{13.6 \, \text{eV}}{16} = -0.85 \, \text{eV} \]
For \( n = 3 \):
\[ E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6 \, \text{eV}}{9} = -1.5111 \, \text{eV} \]
The energy of the photon emitted during the transition from \( n = 4 \) to \( n = 3 \) is the difference in energy levels:
\[ \Delta E = E_3 - E_4 = -1.5111 \, \text{eV} - (-0.85 \, \text{eV}) = -1.5111 \, \text{eV} + 0.85 \, \text{eV} = -0.6611 \, \text{eV} \]
Convert the energy from electron volts to joules:
\[ 1 \, \text{eV} = 1.6022 \times 10^{-19} \, \text{J} \]
\[ \Delta E = -0.6611 \, \text{eV} \times 1.6022 \times 10^{-19} \, \text{J/eV} = -1.0588 \times 10^{-19} \, \text{J} \]
Since energy is emitted, we take the absolute value:
\[ \Delta E = 1.0588 \times 10^{-19} \, \text{J} \]
\(\boxed{\Delta E = 1.0588 \times 10^{-19} \, \text{J}}\)
The frequency \( \nu \) of the photon can be found using the relation:
\[ E = h \nu \]
where \( h \) is Planck's constant (\( h = 6.6261 \times 10^{-34} \, \text{J} \cdot \text{s} \)).
\[ \nu = \frac{E}{h} = \frac{1.0588 \times 10^{-19} \, \text{J}}{6.6261 \times 10^{-34} \, \text{J} \cdot \text{s}} = 1.597 \times 10^{14} \, \text{Hz} \]
\(\boxed{\nu = 1.597 \times 10^{14} \, \text{Hz}}\)
The wavelength \( \lambda \) of the photon can be found using the speed of light \( c \) and the frequency \( \nu \):
\[ \lambda = \frac{c}{\nu} \]
where \( c = 3.00 \times 10^8 \, \text{m/s} \).
\[ \lambda = \frac{3.00 \times 10^8 \, \text{m/s}}{1.597 \times 10^{14} \, \text{Hz}} = 1.878 \times 10^{-6} \, \text{m} \]
Convert the wavelength from meters to nanometers:
\[ 1 \, \text{m} = 10^9 \, \text{nm} \]
\[ \lambda = 1.878 \times 10^{-6} \, \text{m} \times 10^9 \, \text{nm/m} = 1878 \, \text{nm} \]
\(\boxed{\lambda = 1878 \, \text{nm}}\)
a. The energy of the photon emitted in Joules (J) is \(\boxed{1.0588 \times 10^{-19} \, \text{J}}\).
b. Its frequency in Hz is \(\boxed{1.597 \times 10^{14} \, \text{Hz}}\).
c. Its wavelength in nanometers (nm) is \(\boxed{1878 \, \text{nm}}\).
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