Sodium and Copper have work functions 2.3 eV and 4.5 eV respectively. Then the ratio of their threshold wavelengths is nearly:
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A higher work function means electrons are tightly bound to the metal surface, requiring higher-energy photons (which correspond to shorter threshold wavelengths) to kick them out via the photoelectric effect.
Step 1: Link work function and threshold wavelength. The threshold wavelength is the longest wavelength that can just free an electron. From Einstein's photoelectric equation: \[ \phi = \frac{hc}{\lambda_0} \] So a larger work function means a smaller threshold wavelength. Step 2: Note the inverse relation. Because $\lambda_0 \propto \dfrac{1}{\phi}$, the ratio of wavelengths is the inverse of the ratio of work functions: \[ \frac{\lambda_{01}}{\lambda_{02}} = \frac{\phi_2}{\phi_1} \] Step 3: List the work functions. Sodium has $\phi_1 = 2.3$ eV and copper has $\phi_2 = 4.5$ eV. Step 4: Write the wavelength ratio. \[ \frac{\lambda_{Na}}{\lambda_{Cu}} = \frac{4.5}{2.3} \] Step 5: Work out the number. \[ \frac{4.5}{2.3} \approx 1.96 \] Step 6: Round to a simple ratio. This is very close to $2$, so: \[ \boxed{\lambda_{Na} : \lambda_{Cu} \approx 2 : 1} \]