Determine the wavelength of an electron that has a kinetic energy of 1.15 × 10^{-19} J. Report your answer in units of nm to the correct num

Question

Determine the wavelength of an electron that has a kinetic energy of 1.15 × 10^{-19} J. Report your answer in units of nm to the correct number of significant figures.

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Hải Đăng 4 years 2021-08-29T00:07:55+00:00 1 Answers 114 views 0

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  1. Giakhanh
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    2021-08-29T00:08:59+00:00
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    Answer:

    The electron has a wavelength of 1448 nanometers.

    Explanation:

    The wavelength of the electron can be determined by means of the de Broglie wavelength.

    \lambda = \frac{h}{p}  (1)

    Where h is the Planck’s constant and p is the momentum

    \lambda = \frac{h}{mv} (2)

    The velocity can be found by means of the kinetic energy equation.

    KE = \frac{1}{2}mv^{2}

    2KE = mv^{2}

    v^{2} = \frac{2KE}{m}

    v = \sqrt{\frac{2KE}{m}} (3)

    Therefore, equation 3 can be replaced in equation 2

    \lambda = \frac{h}{\sqrt{2mKE}} (4)

    Where m is the mass of the electron and KE its kinetic energy.

    \lambda = \frac{6.624x10^{-34} J.s}{\sqrt{2(9.1x10^{-31}Kg)(1.15x10^{-19} J)}}

    But 1J = Kg.m^{2}/s^{2}

    \lambda = \frac{6.624x10^{-34} Kg.m^{2}/s^{2}.s}{\sqrt{2.093x10^{-49}Kg^{2}.m^{2}/s^{2}}}

    \lambda = \frac{6.624x10^{-34} Kg.m^{2}/s^{2}.s}{4.574x10^{-25}Kg.m/s}

    \lambda = 1.448x10^{-9}m

    Finally,  the wavelength can be expressed in terms of nanometers:

    \lambda = 1.448x10^{-9}m .\frac{1x10^{9}nm}{1m} ⇒ 1448nm

    Hence, the electron has a wavelength of 1448 nanometers.

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