Compare the electric force holding the electron in orbit (r=0.53×10−10m) around the proton nucleus of the hydrogen atom, with the gravitational force between the same electron and proton. What is the ratio of these two forces?
Electrical force:
Coulomb's law: F = kqQ / r^2
q = Q = 1.6*10^-19 Coulombs
r = 5.9*10^-11 meters
k = 8.99 x 10^9 N m^2 / C^2
Gravitational force: F = GmM /r^2
G = 6.67*10^-11
m = 9.1*10^-31 kg
M = 1.67*10^-27 kg
same r so it cancels when you do ratio
Electricity wins :)
To compare the electric force and gravitational force between the electron and proton in a hydrogen atom, we can use the following formulas:
Electric force (F_e) = k * (q1 * q2) / r^2
Gravitational force (F_g) = G * (m1 * m2) / r^2
Where:
- k is the electrostatic constant (9 * 10^9 Nm^2/C^2)
- q1 and q2 are the charges of the electron and proton respectively (e.g. 1.6 * 10^-19 C)
- r is the distance between the electron and proton (0.53 * 10^-10 m)
- G is the gravitational constant (6.67 * 10^-11 Nm^2/kg^2)
- m1 and m2 are the masses of the electron and proton respectively (e.g. 9.11 * 10^-31 kg)
Now we can calculate the ratio of the electric force to the gravitational force:
Ratio = F_e / F_g
First, let's calculate the electric force:
F_e = (9 * 10^9 Nm^2/C^2) * (1.6 * 10^-19 C)^2 / (0.53 * 10^-10 m)^2
F_e = (9 * 10^9 Nm^2/C^2) * (2.56 * 10^-38 C^2) / (0.28 * 10^-20 m^2)
F_e = (9 * 2.56) / (0.28) * (10^9 Nm^2/C^2) / (10^20 m^2)
F_e = 8.64 * 10^-12 N
Next, let's calculate the gravitational force:
F_g = (6.67 * 10^-11 Nm^2/kg^2) * (9.11 * 10^-31 kg) * (1.67 * 10^-27 kg) / (0.53 * 10^-10 m)^2
F_g = (6.67 * 9.11 * 1.67) / (0.53)^2 * (10^-11 * 10^-31 * 10^-27 Nm^2/kg^2) / (10^-20 m^2)
F_g = 8.91 * 10^-8 N
Now, let's calculate the ratio:
Ratio = F_e / F_g
Ratio = (8.64 * 10^-12 N) / (8.91 * 10^-8 N)
Ratio = 0.097
Therefore, the ratio of the electric force to the gravitational force between the electron and proton in a hydrogen atom is approximately 0.097.
To compare the electric force and the gravitational force between the electron and proton in a hydrogen atom, we need to calculate each force separately and then find their ratio.
First, let's calculate the electric force (Fe) holding the electron in orbit around the proton:
The electric force between two charged particles is given by Coulomb's law:
Fe = (k * |q1 * q2|) / r^2
Where:
- Fe is the electric force
- k is the electrostatic constant, with a value of 9 * 10^9 Nm^2/C^2
- q1 and q2 are the charges of the particles, in this case, the charge of an electron (e) is -1.6 * 10^-19 C, and the charge of a proton is +1.6 * 10^-19 C
- r is the distance between the two particles, which is given as 0.53 * 10^-10 m
Plugging in the values:
Fe = (9 * 10^9 * |-1.6 * 10^-19 * 1.6 * 10^-19|) / (0.53 * 10^-10)^2
Next, let's calculate the gravitational force (Fg) between the electron and proton:
The gravitational force between two masses is given by Newton's law of universal gravitation:
Fg = (G * |m1 * m2|) / r^2
Where:
- Fg is the gravitational force
- G is the gravitational constant, with a value of 6.67 * 10^-11 Nm^2/kg^2
- m1 and m2 are the masses of the particles, in this case, the mass of an electron (me) is 9.11 * 10^-31 kg, and the mass of a proton (mp) is 1.67 * 10^-27 kg
- r is the distance between the two particles, which is given as 0.53 * 10^-10 m
Plugging in the values:
Fg = (6.67 * 10^-11 * |9.11 * 10^-31 * 1.67 * 10^-27|) / (0.53 * 10^-10)^2
Now we can find the ratio of the electric force to the gravitational force:
Ratio = Fe / Fg
Plug in the calculated values of Fe and Fg and calculate the ratio.
By following these steps, you can compare the electric force holding the electron in orbit with the gravitational force between the electron and proton in the hydrogen atom.