Step 1: Apply the Photoelectric Equation
The photoelectric equation is defined as: \[ E_{photon} = W + K.E. \] Where: \( E_{photon} \) represents the incoming photon's energy ( \( E_{photon} = h \cdot f \) ). \( W \) denotes the material's work function. \( K.E. \) is the maximum kinetic energy of the emitted photoelectron. At the threshold frequency, \( K.E. = 0 \), meaning the photon energy equals the work function: \( E_{photon} = W \).
Step 2: Compute the Longest Wavelength
The photon energy can also be expressed as \( E_{photon} = h \cdot c / \lambda \), where \( \lambda \) is the light's wavelength, \( h \) is Planck's constant, and \( c \) is the speed of light.
Rearranging for \( \lambda \): \[ \lambda = \frac{h \cdot c}{W} \]
Substitute the given constants: \(h = 6.626 \times 10^{34} \, \text{J.s}\), \( c = 3.0 \times 10^8 \, \text{m/s} \), and \( W = 4.0 \, \text{eV} = 4.0 \times 1.602 \times 10^{19} \, \text{J} \).
Calculation: \[ \lambda = \frac{6.626 \times 10^{34} \times 3.0 \times 10^8}{4.0 \times 1.602 \times 10^{19}} = 310 \, \text{nm} \]
Final Answer: The maximum wavelength capable of causing photoelectron emission is approximately 310 nm.