To solve the problem regarding the velocity of the electron in the first orbit of a hydrogen atom, we need to start by understanding the relationship between kinetic energy (KE) and potential energy (PE) in the context of the Bohr model of the atom. According to the Bohr model, the electron orbits the nucleus in discrete energy levels, and we can derive the necessary equations from the information provided.

Understanding Energy Relationships

In the Bohr model, the kinetic energy of an electron in orbit is given by:

• Kinetic Energy (KE) = (1/2)mv²

Where m is the mass of the electron and v is its velocity. The potential energy (PE) of the electron due to the electrostatic attraction to the nucleus is given by:

• Potential Energy (PE) = - (k * e²) / r

Here, k is Coulomb's constant, e is the charge of the electron, and r is the radius of the orbit.

Relationship Between KE and PE

According to the problem, we are told that the kinetic energy is half of the potential energy:

• KE = (1/2) PE

Substituting the expressions for KE and PE, we have:

• (1/2)mv² = (1/2) * (- (k * e²) / r)

This simplifies to:

• mv² = - (k * e²) / r

Finding the Velocity

From the Bohr model, we also know that the radius of the first orbit (n=1) for hydrogen is given by:

• r = (ε₀h²) / (πk * e²)

Where ε₀ is the permittivity of free space and h is Planck's constant. For hydrogen, we can use the known values to find the velocity of the electron in the first orbit.

Using the Bohr Model Equation

From the Bohr model, we also know that the velocity of the electron in the first orbit can be expressed as:

• v = (Z * e²) / (2 * ε₀ * h)

For hydrogen, where Z = 1, this simplifies to:

• v = (e²) / (2 * ε₀ * h)

Now, we can relate this to the speed of light c. The fine structure constant α is defined as:

• α = e² / (4πε₀hc)

Thus, we can express the velocity in terms of c:

• v = (αc) / 2

Final Calculation

Given that the fine structure constant α is approximately 1/137, we can substitute this value into our equation:

• v = (1/137 * c) / 2 = (c / 2) * (1/137)

This leads us to the conclusion that the velocity of the electron in the first orbit of the hydrogen atom is:

• v = (c / 2) * (1/137)

Choosing the Correct Answer

From the options provided, the correct answer is:

• (c/2) * 137

Thus, the answer is option (c) (c/2) * 137. This reflects the relationship between the electron's velocity and the speed of light in the context of the Bohr model.