The photo-electrons emitted from a surface of sodium metal are such that

Options

A. All are of the same frequency
B. Have the same kinetic energy
C. Have the same de Broglie wavelength
D. Have their speeds varying from zero to a certain maximum

Answer

To solve the question regarding the photoelectrons emitted from the surface of sodium metal, we will analyze the behavior of these photoelectrons based on the principles of photoelectric effect.

### Step-by-Step Solution:

1. **Understanding the Photoelectric Effect**:
- When light (photons) strikes a metal surface, it can cause the emission of electrons from that surface. The energy of the incident photons must be greater than the work function (Φ) of the metal for electrons to be emitted.

2. **Energy of Incident Photons**:
- The energy of a photon is given by the equation:
\[ E = h \cdot f \]
where \(E\) is the energy of the photon, \(h\) is Planck's constant, and \(f\) is the frequency of the incident light.

3. **Kinetic Energy of Emitted Electrons**:
- The kinetic energy (KE) of the emitted electrons can be expressed as:
\[ KE = E - \Phi \]
where \(E\) is the energy of the incident photon and \(\Phi\) is the work function of the metal.

4. **Range of Kinetic Energies**:
- Since the energy of the incident photons is fixed, the kinetic energy of the emitted electrons will vary depending on how much energy is used to overcome the work function. Thus, the kinetic energy of the emitted electrons will range from 0 (if all energy goes to work function) to a maximum value (if the entire energy above the work function is converted to kinetic energy).

5. **Speed of Emitted Electrons**:
- The relationship between kinetic energy and speed (v) is given by:
\[ KE = \frac{1}{2} mv^2 \]
where \(m\) is the mass of the electron. Rearranging gives:
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]
- Since KE varies from 0 to a maximum value, the speed of the emitted electrons will also vary from 0 to a maximum speed.

6. **Conclusion**:
- The correct statement regarding the photoelectrons emitted from sodium metal is that "they have their speeds varying from zero to a certain maximum."

### Final Answer:
- The correct option is: **have their speeds varying from zero to a certain maximum.**