Given:
Wavelength of incident radiation (λ) = 300 nm
Kinetic energy of emitted electrons (KE) = 1.68 × 10^5 J/mol

To find:
The maximum wavelength that will cause a photoelectron to be emitted.

Explanation:

When electromagnetic radiation falls on the surface of a metal, it can transfer its energy to the electrons in the metal. If the energy of the incident radiation is sufficient to overcome the binding energy of the electrons, they are emitted from the metal surface. This phenomenon is known as the photoelectric effect.

The energy of a photon is given by the equation:

E = hc/λ

Where:
E is the energy of the photon
h is Planck's constant (6.626 × 10^-34 J·s)
c is the speed of light (3.0 × 10^8 m/s)
λ is the wavelength of the radiation

When the incident radiation has sufficient energy to overcome the binding energy, the excess energy is converted into the kinetic energy of the emitted electrons. The kinetic energy of the emitted electrons can be determined using the equation:

KE = E - Φ

Where:
KE is the kinetic energy of the emitted electrons
E is the energy of the incident photon
Φ is the work function (binding energy) of the metal

In this case, the kinetic energy of the emitted electrons is given as 1.68 × 10^5 J/mol.

Step 1: Calculate the energy of the incident photon using the given wavelength.

E = hc/λ
E = (6.626 × 10^-34 J·s × 3.0 × 10^8 m/s) / (300 × 10^-9 m)
E = 6.626 × 10^-19 J

Step 2: Calculate the work function of sodium.

Φ = E - KE
Φ = (6.626 × 10^-19 J) - (1.68 × 10^5 J/mol)
Φ ≈ 6.626 × 10^-19 J

Step 3: Calculate the maximum wavelength that will cause a photoelectron to be emitted.

E = hc/λ
λ = hc/E
λ = (6.626 × 10^-34 J·s × 3.0 × 10^8 m/s) / (6.626 × 10^-19 J)
λ = 3.0 × 10^-7 m

Therefore, the maximum wavelength that will cause a photoelectron to be emitted is approximately 300 nm.