Physical Chemistry 2 Quiz: Dimensional Analysis In Spectroscopy

In photoelectron spectroscopy, the kinetic energy of ejected electrons is given by $KE = h\nu - BE$, where $BE$ is the binding energy. If X-rays with energy $1486.6 \text{ eV}$ are used and an electron is detected with kinetic energy $1200.3 \text{ eV}$, what is the binding energy in both eV and wavenumbers?

A researcher measures the rotational constant $B_e$ of a diatomic molecule as $1.93 \text{ cm}^{-1}$. Using the relationship $B_e = \frac{h}{8\pi^2 I c}$, what is the moment of inertia in both $\text{kg}\cdot\text{m}^2$ and $\text{amu}\cdot\text{Å}^2$?

A researcher calculates the oscillator strength $f$ using the formula $f = \frac{4.32 \times 10^{-9}}{n} \int \varepsilon(\tilde{\nu}) \, d\tilde{\nu}$, where $n$ is the refractive index, $\varepsilon$ is the molar absorptivity, and $\tilde{\nu}$ is the wavenumber. If $\varepsilon$ has units $\text{M}^{-1}\text{cm}^{-1}$ and the integral evaluates to $2.5 \times 10^6 \text{ M}^{-1}\text{cm}^{-2}$, what are the units of the numerical constant $4.32 \times 10^{-9}$?

In surface-enhanced Raman spectroscopy (SERS), the enhancement factor $G$ is defined as $G = \frac{I_{\text{SERS}}/N_{\text{SERS}}}{I_{\text{normal}}/N_{\text{normal}}}$ where $I$ represents intensity and $N$ represents the number of molecules. If both intensities have units $\text{counts}\text{s}^{-1}$ and the molecular surface density is $\rho_s = 10^{12} \text{ molecules}\text{cm}^{-2}$, what must be the units of the effective area $A_{\text{eff}}$ used in calculating $N_{\text{SERS}} = \rho_s \cdot A_{\text{eff}}$?

In Z-scan measurements, the nonlinear absorption coefficient $\beta_{\text{eff}}$ is determined from $T = \frac{1}{1 + \beta_{\text{eff}} I_0 L_{\text{eff}}}$ where $T$ is transmittance, $I_0$ is peak intensity, and $L_{\text{eff}} = \frac{1-e^{-\alpha_0 L}}{\alpha_0}$ is the effective length with $\alpha_0$ being the linear absorption coefficient. If $\alpha_0 = 0.1 \text{ cm}^{-1}$, $L = 2 \text{ mm}$, and $I_0 = 10^8 \text{ W}\text{cm}^{-2}$, what are the units of $\beta_{\text{eff}}$?

In terahertz time-domain spectroscopy, the complex refractive index $\tilde{n}(\omega) = n(\omega) + i\kappa(\omega)$ is extracted from the transmission $\tilde{T}(\omega) = \frac{4\tilde{n}}{(\tilde{n}+1)^2} e^{i\omega \tilde{n} d/c}$ where $d$ is sample thickness and $c$ is the speed of light. If the extinction coefficient $\kappa$ is related to the absorption coefficient by $\alpha = \frac{4\pi\kappa\nu}{c}$, what are the units of the expression $\frac{\alpha \cdot c \cdot d}{\nu \cdot \kappa}$?

In electroabsorption spectroscopy, the change in absorption coefficient under an applied electric field $F$ is given by $\Delta \alpha = \alpha_0 + \alpha_1 F + \alpha_2 F^2$ where $\alpha_0$, $\alpha_1$, and $\alpha_2$ are field-independent coefficients. If $\Delta \alpha$ and the baseline absorption $\alpha_0$ both have units $\text{cm}^{-1}$ and the electric field strength $F = 10^5 \text{ V}\text{cm}^{-1}$, what are the units of the quadratic electroabsorption coefficient $\alpha_2$?

In single-molecule fluorescence correlation spectroscopy, the autocorrelation function $G(\tau) = 1 + \frac{1}{N} \left(1 + \frac{\tau}{\tau_D}\right)^{-1} \left(1 + \frac{\tau}{\omega^2 \tau_D}\right)^{-1/2}$ describes molecular diffusion. If the diffusion coefficient $D$ is related to the diffusion time by $\tau_D = \frac{w_0^2}{4D}$ where $w_0$ is the beam waist radius, and $D = 2.3 \times 10^{-10} \text{ m}^2\text{s}^{-1}$, what are the units of the ratio $\frac{\tau_D \cdot D}{w_0^2}$?

In NMR spectroscopy, the chemical shift in ppm is related to the absolute frequency difference by $\delta = \frac{\nu - \nu_{\text{ref}}}{\nu_0} \times 10^6$. If a proton signal appears at $\delta = 7.25 \text{ ppm}$ in a $400 \text{ MHz}$ spectrometer, what is the absolute frequency difference from TMS in both Hz and rad/s?

A researcher measures the fluorescence lifetime of a compound and reports the value as $2.5 \times 10^{-8}$ seconds. When calculating the radiative rate constant using $k_r = \phi_f / \tau$, where $\phi_f = 0.85$ is the quantum yield, what are the correct units for $k_r$ and its numerical value?

In infrared spectroscopy, the wavenumber of a vibrational transition is given by $\tilde{\nu} = \frac{1}{2\pi c}\sqrt{\frac{k}{\mu}}$, where $k$ is the force constant and $\mu$ is the reduced mass. For a C-H stretch with $\tilde{\nu} = 2900 \text{ cm}^{-1}$, what is the force constant in both N/m and mdyn/Å?

A mass spectrometer operates with an electric field of $2.5 \times 10^4 \text{ V/m}$ and accelerates ions through a potential difference of $3000 \text{ V}$. An ion with $m/z = 120$ reaches the detector. What is the final velocity of this ion in m/s, and what assumption about initial conditions is required for dimensional consistency?

In electron spin resonance (ESR), the resonance condition is $h\nu = g\mu_B B$. If the g-factor is measured as $2.003$ and the magnetic field is $0.335 \text{ T}$, what microwave frequency should be used? Express the answer in both GHz and wavenumbers.

A researcher uses time-resolved fluorescence to measure excited-state dynamics. The fluorescence intensity follows $I(t) = I_0 e^{-t/\tau}$ with $\tau = 4.2 \text{ ns}$. If the detection system has a time resolution of $100 \text{ ps}$, what fraction of the total fluorescence signal is collected within the first $500 \text{ ps}$, and what are the appropriate units for the rate constant $k = 1/\tau$?

A UV-Vis absorption spectrum shows a peak with molar absorptivity $\varepsilon = 2.4 \times 10^4 \text{ L mol}^{-1} \text{cm}^{-1}$ at $\lambda = 280 \text{ nm}$. Using the relationship $f = 4.32 \times 10^{-9} \int \varepsilon(\tilde{\nu}) d\tilde{\nu}$, what are the units of the oscillator strength $f$ and the integral?