Question

Before the development of quantum theory, Ernest Rutherford's experiments with gold atoms led him to propose the so-called Rutherford Model of atomic structure. The basic idea is that the nucleus of the atom is a very dense concentration of positive charge, and that negatively charged electrons orbit the nucleus in much the same manner as planets orbit a star. His experiments appeared to show that the average radius of an electron orbit around the gold nucleus must be about 10−1010−10 m. Stable gold has 79 protons and 118 neutrons in its nucleus.

What is the strength of the nucleus' electric field at the orbital radius of the electrons?
What is the kinetic energy of an electron in a circular orbit around the gold nucleus?

Answer 1

1.
`E = 1.14 \cdot 10^(13) N/C`

2.
`E_(k) = 9.1 \cdot 10^(-17) J`

Step-by-step explanation:

1. The strength of the nucleus' electric field (E):

`E = (kq)/(r^(2))`

Where:

k: is the Coulomb constant = 9x10⁹ Nm²/C²

q: is the proton charge = 1.6x10⁻¹⁹ C

r: is the radius = 10⁻¹⁰ m

`E = (kq)/(r^(2)) = (9\cdot 10^(9) Nm^(2)/C^(2)*79*1.6 \cdot 10^(-19) C)/((10^(-10) m)^(2)) = 1.14 \cdot 10^(13) N/C`

2. The kinetic energy (Ek) of an electron is the following:

`E_(k) = (1)/(2)mv^(2)`

Where:

m is the electron's mass = 9.1x10⁻³¹ kg

v: is the speed of the electron

We can find the speed of the electron by equaling the centripetal force (Fc) and the electrostatic force (Fe):

`F_(c) = F_(e)`

`(mv^(2))/(r) = (kq^(2))/(r^(2)) = qE`

`v^(2) = (qEr)/(m) = (1.6 \cdot 10^(-19) C*1.14 \cdot 10^(13) N/C*10^(-10) m)/(9.1 \cdot 10^(-31) kg) = 2.00 \cdot 10^(14) m^(2)/s^(2)`

Now, we can find the kinetic energy:

`E_(k) = (1)/(2)mv^(2) = (1)/(2)9.1 \cdot 10^(-31) kg*2.00 \cdot 10^(14) m^(2)/s^(2) = 9.1 \cdot 10^(-17) J`

I hope it helps you!