Photoelectric Effect and Line Spectra (4E) - Chem & Physics

1. State the formula for photon energy emitted or absorbed for a hydrogen transition $n_i\to n_f$.

   $\Delta E=13.6\ \text{eV}\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$. Energy difference between levels determines photon energy in transitions, positive for emission when $n_i>n_f$.

2. Which quantity must exceed the work function $\phi$ for photoemission to occur?

   Photon energy $hf$ must be $\geq \phi$. Photoemission requires photon energy at least equal to work function to overcome electron binding energy.

3. Which hydrogen series corresponds to transitions ending at $n_f=1$?

   Lyman series. Lyman series involves transitions to ground state, producing UV lines.

4. Which hydrogen series corresponds to transitions ending at $n_f=3$?

   Paschen series. Paschen series involves transitions to n=3, producing infrared lines.

5. Which direction of transition produces emission: $n_i>n_f$ or $n_i<n_f$?

   Emission occurs for $n_i>n_f$. Electron dropping to lower energy level releases photon energy equal to level difference.

6. What is the Bohr energy formula for hydrogen energy level $n$?

   $E_n=-\frac{13.6\ \text{eV}}{n^2}$. Bohr model quantizes hydrogen electron energies, negative due to bound states relative to ionization.

7. What is the physical meaning of the stopping potential $V_s$ in a photoelectric experiment?

   Retarding voltage that reduces photocurrent to zero. Stopping potential opposes kinetic energy of fastest photoelectrons, halting them at collector.

8. What is the physical meaning of the work function $\phi$ in the photoelectric effect?

   Minimum energy required to eject an electron from a metal. Work function represents binding energy of least-bound electrons in metal surface.

9. What is the threshold frequency $f_0$ in terms of work function $\phi$?

   $f_0=\frac{\phi}{h}$. Threshold frequency is work function divided by Planck's constant, where photon energy just equals $\phi$.

10. What is the photoelectric equation relating $K_{\max}$, $hf$, and work function $\phi$?

    $K_{\max}=hf-\phi$. Maximum kinetic energy of photoelectrons equals photon energy minus work function, per Einstein's explanation.

11. What is the slope and $x$-intercept of a plot of stopping potential $V_s$ versus $f$?

    Slope $\frac{h}{e}$; $x$-intercept $f_0=\frac{\phi}{h}$. From $V_s=\frac{h}{e}f - \frac{\phi}{e}$, slope is $h/e$ and x-intercept is threshold frequency.

12. Which direction of transition produces absorption: $n_i>n_f$ or $n_i<n_f$?

    Absorption occurs for $n_i<n_f$. Electron jumping to higher energy level requires absorbing photon energy matching level difference.

13. What is the formula for photon energy in terms of frequency $f$?

    $E=hf$. Photon energy equals Planck's constant times frequency, linking wave and particle properties of light.

14. What is the stopping potential relation between $K_{\max}$ and $V_s$ for an electron?

    $K_{\max}=eV_s$. Maximum kinetic energy equals electron charge times stopping potential, converting KE to potential energy.

15. What is the threshold wavelength $\lambda_0$ in terms of work function $\phi$?

    $\lambda_0=\frac{hc}{\phi}$. Threshold wavelength is Planck's constant times speed of light divided by work function, longest $\lambda$ for emission.

16. What is the formula for $V_s$ in terms of $hf$ and $\phi$?

    $V_s=\frac{hf-\phi}{e}$. Stopping potential equals photon energy minus work function divided by electron charge, from $K_{\max}=eV_s$.

17. Identify the photon with higher energy: one with $\lambda=400\ \text{nm}$ or $\lambda=800\ \text{nm}$.

    $\lambda=400\ \text{nm}$ photon has higher energy. Photon energy inversely proportional to wavelength, so shorter $\lambda$ has higher energy.

18. Identify what changes when light intensity increases at fixed $f>f_0$ in the photoelectric effect.

    Photoelectron number (current) increases; $K_{\max}$ unchanged. Higher intensity provides more photons, ejecting more electrons and increasing current, but photon energy fixes $K_{\max}$.

19. Identify what changes when frequency $f$ increases at fixed intensity in the photoelectric effect.

    $K_{\max}$ and $V_s$ increase; emission requires $f\geq f_0$. Higher frequency increases photon energy, raising $K_{\max}$ and $V_s$ if above threshold for emission.

20. Which graph is linear for photoelectric data: $K_{\max}$ vs $f$ or $K_{\max}$ vs intensity?

    $K_{\max}$ vs $f$ is linear with slope $h$. $K_{\max}$ increases linearly with frequency per photoelectric equation, unlike independence from intensity.

21. What is the formula for the speed of light in terms of $\lambda$ and $f$?

    $c=\lambda f$. Speed of light equals wavelength times frequency for electromagnetic waves in vacuum.

22. Which hydrogen series corresponds to transitions ending at $n_f=2$ (visible region)?

    Balmer series. Balmer series involves transitions to n=2, yielding visible wavelengths.

23. What is the Rydberg formula for the wavelength of a hydrogen spectral line?

    $\frac{1}{\lambda}=R\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$. Rydberg formula derives from Bohr energy differences, with $R$ as constant for hydrogen spectral lines.

24. What is the slope and $y$-intercept of a plot of $K_{\max}$ versus $f$?

    Slope $h$; intercept $-\phi$. From $K_{\max}=hf-\phi$, slope is Planck's constant and y-intercept is negative work function.

25. What is the formula for photon energy in terms of wavelength $\lambda$?

    $E=\frac{hc}{\lambda}$. Photon energy is Planck's constant times speed of light divided by wavelength, derived from $E=hf$ and $c=\lambda f$.

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