Final answer:
The work function of the piece of metal is 1.77 eV.
Step-by-step explanation:
The work function, denoted by φ, represents the minimum energy required to free an electron from the surface of a material. The energy of a photon is given by the equation E = hc/λ, where E is the energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. When the wavelength changes from 445 nm to 225 nm, the maximum energy doubles. Using this information, we can calculate the work function in eV.
Let's assume the work function is W eV. When light of wavelength 445 nm is used, the energy of the photons is E = hc/λ = (6.63x10^-34 J s)(3x10^8 m/s)/(445x10^-9 m). This energy is equal to W + K, where K is the maximum kinetic energy of the electron. Similarly, when light of wavelength 225 nm is used, the energy of the photons is E' = hc/λ' = (6.63x10^-34 J s)(3x10^8 m/s)/(225x10^-9 m). This energy is equal to W + 2K, since the maximum energy doubles. Considering these equations, we can set up two equations and solve for the work function.
W + K = (6.63x10^-34 J s)(3x10^8 m/s)/(445x10^-9 m)
W + 2K = (6.63x10^-34 J s)(3x10^8 m/s)/(225x10^-9 m)
Subtracting the first equation from the second equation, we get K = (6.63x10^-34 J s)(3x10^8 m/s)((1/225)-(1/445)).
Solving for K, we find K = 3.28x10^-19 J. Converting this to eV, we can use the conversion factor 1 eV = 1.6x10^-19 J. Therefore, the work function is W = 2K = 2(3.28x10^-19 J)(1.6x10^-19 J/eV) = 1.77 eV.
Therefore, the work function of the piece of metal is 1.77 eV.