Question

The ratio of the total energy E of the electron to its kinetic energy K in hydrogen atom is

• A. -1
• B. 1/2
• C. 2
• D. 1
• E. -1/2

Answer 1

The Correct Option is A

The total energy E of an electron in a hydrogen atom is the sum of its kinetic energy K and its potential energy U. According to the Bohr model of the hydrogen atom, the total energy of the electron is given by:

E = K + U

The kinetic energy K is equal in magnitude to the potential energy, but opposite in sign. Therefore, we have:

K = -U

The total energy of the electron is:

E = K + (-K) = -K

Thus, the ratio of the total energy E to the kinetic energy K is:

E / K = -K / K = -1 

Correct Answer:

Correct Answer: (A) -1

Answer 2

Step 1: Consider the total energy and kinetic energy of an electron in a hydrogen atom using Bohr's model. 

The electron experiences an electrostatic force due to the nucleus, which provides the centripetal force required for circular motion: \[ \frac{k e^2}{r^2} = \frac{mv^2}{r} \]

Step 2: Multiply both sides by \( r \): \[ \frac{k e^2}{r} = mv^2 \]

Kinetic energy of the electron is: \[ K = \frac{1}{2}mv^2 = \frac{1}{2} \cdot \frac{k e^2}{r} = \frac{k e^2}{2r} \]

Step 3: Potential energy of the electron in the electrostatic field is: \[ U = -\frac{k e^2}{r} \]

Total energy of the electron is: \[ E = K + U = \frac{k e^2}{2r} - \frac{k e^2}{r} = -\frac{k e^2}{2r} \]

Step 4: Now take the ratio of total energy to kinetic energy: \[ \frac{E}{K} = \frac{-\frac{k e^2}{2r}}{\frac{k e^2}{2r}} = -1 \]

Final Answer: -1