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Chapter 27: Problem 48

The Bohr theory of the hydrogen atom ignores gravitational forces between the electron and the proton. Make a calculation to justify this omission. [Hint: Find the ratio of the gravitational and electrostatic forces acting on the electron due to the proton.]

Short Answer

Expert verified
Answer: The gravitational force between an electron and a proton is negligible compared to the electrostatic force. The ratio of gravitational force to electrostatic force is approximately \(3.61\times10^{-37}\), which is extremely small. This justifies the omission of gravitational forces in the Bohr theory of the hydrogen atom.

Step by step solution

01

Find the electrostatic force

The electrostatic force (also called the Coulomb force) between an electron and proton is given by Coulomb's Law: \(F_e =\dfrac{ke^2}{r^2}\) where \(F_e\) is the electrostatic force, \(k\) is the Coulomb constant \((8.9875\times10^9\, Nm^2C^{-2})\), \(e\) is the elementary charge \((1.602\times10^{-19}\, C)\), and \(r\) is the distance between the electron and the proton.
02

Find the gravitational force

The gravitational force between an electron and proton is given by Newton's Law of Universal Gravitation: \(F_g = \dfrac{Gm_em_p}{r^2}\) where \(F_g\) is the gravitational force, \(G\) is the gravitational constant \((6.674\times10^{-11}\, Nm^2kg^{-2})\), \(m_e\) is the mass of the electron \((9.11\times10^{-31}\, kg)\), and \(m_p\) is the mass of the proton \((1.673\times10^{-27}\, kg)\).
03

Calculate the ratio of the gravitational and electrostatic forces

Divide the gravitational force by the electrostatic force to find the ratio: \(R = \dfrac{F_g}{F_e} = \dfrac{\dfrac{Gm_em_p}{r^2}}{\dfrac{ke^2}{r^2}}\) Notice that \(r^2\) can be canceled from both the numerator and denominator: \(R = \dfrac{Gm_em_p}{ke^2}\) Now, plug in the values of all the constants and calculate the ratio: \(R = \dfrac{(6.674\times10^{-11})(9.11\times10^{-31})(1.673\times10^{-27})}{(8.9875\times10^9)(1.602\times10^{-19})^2}\) \(R \approx 3.61\times10^{-37}\)
04

Conclusion

The ratio of gravitational force to electrostatic force between the electron and proton is approximately \(3.61\times10^{-37}\). This number is extremely small, which means the gravitational force is negligible compared to the electrostatic force. Therefore, it's justified to omit the gravitational forces in the Bohr theory of the hydrogen atom.

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