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QuestionA photon with wavelength 1.57 nm1.57 \mathrm{~nm} emits an electron with 61.5eV61.5 \mathrm{eV}. Find the binding energy in J.

Studdy Solution

STEP 1

Assumptions1. The wavelength of the photon is 1.57 nm1.57 \, nm . The kinetic energy of the emitted electron is 61.5 eV61.5 \, eV
3. The binding energy of the electron is what we need to find4. The conversion rate between electron volts and Joules is 1 eV=1.602×10−19 J1 \, eV =1.602 \times10^{-19} \, J
5. We use the photoelectric effect formula photon=binding+kinetic_{photon} =_{binding} +_{kinetic}

STEP 2

First, we need to convert the wavelength of the photon to energy. We can do this using the formula for the energy of a photon, which is photon=hcλ_{photon} = \frac{hc}{\lambda}, where hh is Planck's constant (6.626×10−34 J⋅s6.626 \times10^{-34} \, J \cdot s), cc is the speed of light (.00×108 m/s.00 \times10^{8} \, m/s), and λ\lambda is the wavelength.
photon=hcλ_{photon} = \frac{hc}{\lambda}

STEP 3

Now, plug in the given values for Planck's constant, the speed of light, and the wavelength to calculate the energy of the photon. Note that we need to convert the wavelength from nanometers to meters.
photon=(6.626×10−34 J⋅s)(3.00×108 m/s)1.57×10−9 m_{photon} = \frac{(6.626 \times10^{-34} \, J \cdot s)(3.00 \times10^{8} \, m/s)}{1.57 \times10^{-9} \, m}

STEP 4

Calculate the energy of the photon.
photon=(6.626×10−34 J⋅s)(3.00×108 m/s)1.57×10−9 m=1.266×10−18 J_{photon} = \frac{(6.626 \times10^{-34} \, J \cdot s)(3.00 \times10^{8} \, m/s)}{1.57 \times10^{-9} \, m} =1.266 \times10^{-18} \, J

STEP 5

Next, we need to convert the kinetic energy of the electron from electron volts to Joules. We can do this using the conversion rate.
kinetic=61.5 eV×1.602×10−19 J/eV_{kinetic} =61.5 \, eV \times1.602 \times10^{-19} \, J/eV

STEP 6

Calculate the kinetic energy of the electron in Joules.
kinetic=61.5 eV×1.602×10−19 J/eV=9.854×10−18 J_{kinetic} =61.5 \, eV \times1.602 \times10^{-19} \, J/eV =9.854 \times10^{-18} \, J

STEP 7

Now that we have the energy of the photon and the kinetic energy of the electron, we can find the binding energy of the electron. We can do this using the photoelectric effect formula photon=binding+kinetic_{photon} =_{binding} +_{kinetic}.
binding=photon−kinetic_{binding} =_{photon} -_{kinetic}

STEP 8

Plug in the values for the energy of the photon and the kinetic energy of the electron to calculate the binding energy.
binding=1.266×10−18 J−.854×10−18 J_{binding} =1.266 \times10^{-18} \, J -.854 \times10^{-18} \, J

STEP 9

Calculate the binding energy of the electron.
binding=.266×−18 J−9.854×−18 J=3.12×−19 J_{binding} =.266 \times^{-18} \, J -9.854 \times^{-18} \, J =3.12 \times^{-19} \, JThe binding energy of the electron is 3.12×−19 J3.12 \times^{-19} \, J.

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