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29 January, 13:50

Compare the wavelengths of an electron (mass = 9.11 * 10-31 kg) and a proton (mass = 1.67 * 10-27 kg), each having (a) a speed of 8.83 * 106 m/s and (b) a kinetic energy of 7.81 * 10-15 J. Enter your answers in scientific notation.

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  1. 29 January, 15:22
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    a) Wavelength of electron = 8.21 * 10⁻¹¹ m = 821 pm

    Wavelength of proton = 4.50 * 10⁻¹³ m = 0.45 pm

    The electron has a higher wavelength than the electron.

    b) Wavelength of electron = 4.61 * 10⁻¹⁰ m = 461 pm

    Wavelength of proton = 1.30 * 10⁻¹³ m = 0.130 pm

    The electron once again has a higher wavelength.

    Note 1 pm = 10⁻¹² m

    Explanation:

    a) The relationship between wavelength, mass of a particle and its speed is given in the De Broglie's equation

    λ = h/mv

    For the two particles

    h = Planck's constant = 6.63 * 10⁻³⁴ J. s

    v = 8.83 * 10⁶ m/s

    For an electron

    m = 9.11 * 10⁻³¹ kg

    Wavelength = (6.63 * 10⁻³⁴) / (9.11 * 10⁻³¹ * 8.83 * 10⁶) = 8.21 * 10⁻¹¹ m = 821 pm

    For a proton

    m = 1.67 * 10⁻²⁷ kg

    Wavelength = (6.63 * 10⁻³⁴) / (1.67 * 10⁻²⁷ * 8.83 * 10⁶) = 4.50 * 10⁻¹³ m = 0.45 pm

    b) The velocity needs to be first obtained from the kinetic energy relation. Before using the De Broglie's equation.

    K. E = mv²/2

    v = √ (2K. E/m)

    For the electron,

    m = 9.11 * 10⁻³¹ kg

    v = √[ (2 * 7.81 * 10⁻¹⁵) / (9.11 * 10⁻³¹) ]

    v = 1.31 * 10⁸ m/s

    Wavelength = h/mv

    Wavelength = (6.63 * 10⁻³⁴) / (9.11 * 10⁻³¹ * 1.31 * 10⁸) = 4.61 * 10⁻¹⁰ m = 461 pm

    For the proton,

    m = 1.67 * 10⁻²⁷ kg

    v = √[ (2 * 7.81 * 10⁻¹⁵) / (1.67 * 10⁻²⁷) ]

    v = 3.06 * 10⁶ m/s

    Wavelength = h/mv

    Wavelength = (6.63 * 10⁻³⁴) / (1.67 * 10⁻²⁷ * 3.06 * 10⁶) = 1.30 * 10⁻¹³ m = 0.130 pm
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