The work function of a substance is 4.0 eV. Then longest wavelength of light that can cause photoelectron emission from this substance approximately (A) 540 nm (B) 400 nm (C) 310 nm (D) 220 nm

To determine the longest wavelength of light that can cause photoelectron emission from a substance with a work function of 4.0 eV, we can use the relationship between energy and wavelength derived from the photoelectric effect. The key concept here is that the energy of a photon must be equal to or greater than the work function in order to eject an electron from the material.

Understanding Key Concepts

The work function (φ) is the minimum energy needed to remove an electron from the surface of a substance. In this case, it is given as 4.0 eV. The energy of a photon can be calculated using the formula:

E = hc/λ

• E = energy of the photon (in eV)
• h = Planck's constant (4.135667696 × 10^-15 eV·s)
• c = speed of light (approximately 3.00 × 10⁸ m/s)
• λ = wavelength of light (in meters)

Calculating the Wavelength

To find the longest wavelength (λ) that can still cause photoelectron emission, we need to set the energy (E) equal to the work function (φ). Therefore, we set E to 4.0 eV:

4.0 eV = (hc)/λ

Rearranging this equation to find λ gives us:

λ = hc/E

Substituting Values

Now we will substitute the values for h and c, and the energy we need:

λ = (4.135667696 × 10^-15 eV·s) × (3.00 × 10⁸ m/s) / 4.0 eV

Calculating this yields:

λ = (1.245 × 10^-6 m·eV) / 4.0 eV

λ = 3.1125 × 10^-7 m

To convert this wavelength from meters to nanometers (1 nm = 1 × 10^-9 m), we multiply by 1 × 10⁹:

λ ≈ 311.25 nm

Comparing to Given Options

Now, if we compare this calculated wavelength (approximately 311 nm) to the options provided:

• (A) 540 nm
• (B) 400 nm
• (C) 310 nm
• (D) 220 nm

The closest match is option (C) 310 nm, which is just about equal to our calculated value. Thus, the answer is:

The longest wavelength of light that can cause photoelectron emission from the substance is approximately 310 nm (option C).