Photoelectric Effect and Line Spectra (4E) - MCAT Chemical and Physical Foundations of Biological Systems

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

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

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

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

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.

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

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

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

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

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

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

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

$K_{\max}$ 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.

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

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

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

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

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

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

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

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

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

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

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