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In the Bohr model of the hydrogen atom, a single electron
Chapter 23, Problem 56P(choose chapter or problem)
In the Bohr model of the hydrogen atom, a single electron revolves around a single proton in a circle of radius r. Assume that the proton remains at rest. (a) By equating the electric force to the electron mass times its acceleration, derive an expression for the electron’s speed. (b) Obtain an expression for the electron’s kinetic energy, and show that its magnitude is just half that of the electric potential energy. (c) Obtain an expression for the total energy, and evaluate it using r = 5.29 × 10?11 m. Give your numerical result in joules and in electron volts.
Questions & Answers
QUESTION:
In the Bohr model of the hydrogen atom, a single electron revolves around a single proton in a circle of radius r. Assume that the proton remains at rest. (a) By equating the electric force to the electron mass times its acceleration, derive an expression for the electron’s speed. (b) Obtain an expression for the electron’s kinetic energy, and show that its magnitude is just half that of the electric potential energy. (c) Obtain an expression for the total energy, and evaluate it using r = 5.29 × 10?11 m. Give your numerical result in joules and in electron volts.
ANSWER:Introduction We have to derive the expression for the velocity of an electron orbiting around the proton with radius r. Then we have to derive the expression electrons kinetic energy and we have to show that the kinetic energy is half of the electric potential energy at that point. Finally we have to obtain the expression of total energy and then we have to calculate the total energy using r = 5.29 × 1011m. Step 1 Let us consider that the mass of the electron is m and charge of the electron is e, and the charge of the proton is + e. The electrostatic force will be The negative sign indicates that the force is attractive. And the centrifugal force is For stable orbit, the magnitude of this two force will be same. Hence we have