A neutral lithium atom has three electrons. As you will discover in the textbook

Question

A neutral lithium atom has three electrons. As you will discover in the textbook, two of these electrons form an “inner core,” but the third—the valence electron—orbits at much larger radius. From the valence electron’s perspective, it is orbiting a spherical ball of charge having net charge +1e (i.e., the three protons in the nucleus and the two inner-core electrons). The energy required to ionize a lithium atom is 5.14 eV.

Part A

According to Rutherford’s nuclear model of the atom, what is the orbital radius of the valence electron?

Part B

According to Rutherford's nuclear model of the atom, what is the speed of the valence electron?

Explanation / Answer

Part B: To solve this question, you need to use Fnet=ma.
0= (centripetal force) + (force due to electron repulsion).
0= (mv^2/r) - (kqQ/r^2) where qQ = e^2 (charge of the electron)
solving for v= ((ke^2)/(mr))^1/2

where E=5.11 , eV = 8.18x10^-19 J
U=-qV=-ke^2/r

=-16.34x10^-19 J [k=1/(4??o)]

=9x10^9 Nm^2/C^2
then
v=[2(8.18+16.34)x10^-19/9.1x10^-31]^1/2
v=1.34*10^6m/s

Part A : Due to shielding, the total energy of the orbiting valence electron is …
… E = ½ m v ² - ( k e ² / r ) … the force in circular motion is given by …
… F = m v ² / r … while the electric interaction due to shielding gives a
force with magnitude … F = k e ² / r ² … so that equality of the two
forces gives … m v ² / r = k e ² / r ² … which reduces to …
m v ² = k e ² / r … the total energy of the orbiting valence electron is
therefore … E = ½ m v ² - ( k e ² / r ) = ½ ( k e ² / r ) – ( k e ² / r )
……………………………………….... = - ( k e ² / 2 r ) … solving for
the orbital radius r, we get … r = - k e ² / [ 2 E ] … where …
… E = - 5.25 eV = - ( 5.25 eV ) (1.6 × 10 ? ¹? J / eV ) = - ( 5.25 e ) J …
negative energy, as in the total energy considered earlier, since the
valence electron is a bound state … it follows that the orbital radius is …

r = - k e ² / [ 2 E ] = - k e ² / [ 2 ( -5.25 e ) J ] = k e / [ 2 (5.25 ) J ]
= ( 8.99 × 10 ? N • m ² ) ( 1.6 × 10 ? ¹? ) / [ 2 (5.25 ) N • m ]

= 1.41 × 10 ? ¹? m

= 1.41 nm

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