Question

Use De Broglie's equation (see below) to determine the velocity of a proton if it has a wavelength of 2.13E-15 meters. Planck's constant = 6.626E-34 J*s Speed of light = 3.00E8 m/s 1 cal = 4.184 J 1 g = 6.022E23 amu Mass of electron: 5.4858E-4 amu Mass of neutron: 1.00866 amu Mass of proton: 1.00728 amu λ = h mv Where λ has units of meters m is mass in kg v is velocity in m/s

Answer 1

Given that;

wavelength of 2.13E-15 meters.

Planck's constant = 6.626E-34 J*s

Speed of light = 3.00E8 m/s 1 cal = 4.184 J

1 g = 6.022E23 amu

Mass of electron: 5.4858E-4 amu

Mass of neutron: 1.00866 amu

Mass of proton: 1.00728 amu OR 1.672627644e-27 kg

λ = h mv

Where λ has units of meters m is mass in kg v is velocity in m/s

Step I:

Use the de Broglie equation to determine the energy (not momentum) of the atom [note the appearence of the mass (in kg) of a He atom]:

λ = h/p

λ = h/√(2Em)

2.13 x 10¯¹⁵ m = 6.626 x 10¯³⁴ J s / √[(2) (x) (1.67 x 10¯²⁷ kg)]

Or without unit;

2.13 x 10¯¹⁵ times √[(2) (x) (1.67 x 10¯²⁷) = 6.626 x 10¯³⁴]

√[(2) (x) (1.67x 10¯²⁷)] = 6.626 x 10¯³⁴ / 2.13 x 10¯¹⁵

√[(2) (x) (1.67 x 10¯²⁷)] = 3.11 x 10¯¹⁹

(2) (x) (1.67 x 10¯²⁷) = 9.67 x 10¯³⁸

x = 2.895 x 10¯¹² J

2) Use the kinetic energy equation to get the velocity:

KE = (1/2)mv²

2.895 x 10¯¹² = (1/2) (1.67 x 10¯²⁷) v²

v² = 8.67 x 10¹⁴

v = 2.94 x 10⁷ m/s