Physical Chemistry and Instrumental Analysis Mock Exam Notes

Electrochemistry: Calculation of Equilibrium Cell Voltage

• Cell Description and Symbolism

  • The system contains an electrochemical cell at a temperature of $T = 25\,^{\circ}\text{C}$ (which equals $298.15\,K$).
  • The cell notation is provided as: $Ag \mid Ag^{+} (aq, c = 10^{-3}\,mol/l) \parallel KCl (aq, c = 10^{-2}\,mol/l), AgCl (s) \mid Ag$.
  • The left side of the notation represents the anode (oxidation), and the right side represents the cathode (reduction).

• Relevant Physicochemical Constants

  • Gas Constant ($R$): $8.314\,VAs/(K\,mol)$.
  • Faraday Constant ($F$): $96485.309\,As/mol$.
  • Solubility Product ($K_L$) of $AgCl$: $2 \times 10^{-10}\,mol^2/l^2$.

• Calculation of the Equilibrium Cell Voltage ($U_{eq}$)

  • Step 1: Identifying the Anode Potential
  • At the anode, silver is in contact with silver ions: $Ag \rightarrow Ag^{+} + e^{-}$.
  • The concentration of silver ions at the anode ($c_{Anode}$) is given as $10^{-3}\,mol/l$.
  • The potential is determined by the Nernst equation: $E_{Anode} = E^0(Ag/Ag^{+}) + \frac{RT}{F} \ln(c_{Anode})$.
  • Step 2: Identifying the Cathode Potential
  • The cathode is a Silver/Silver Chloride electrode (an electrode of the second kind).
  • The reaction is: $AgCl(s) + e^{-} \rightarrow Ag(s) + Cl^{-}(aq)$.
  • The silver ion concentration at the cathode ($c_{Cathode}$) is governed by the solubility product of $AgCl$ and the concentration of chloride ions provided by the $KCl$.
  • The chloride concentration $c(Cl^{-})$ is $10^{-2}\,mol/l$.
  • Using the solubility product $K_L = c(Ag^{+}) \times c(Cl^{-})$, the silver ion concentration is calculated:       $c_{Cathode}(Ag^{+}) = \frac{K_L}{c(Cl^{-})} = \frac{2 \times 10^{-10}\,mol^2/l^2}{10^{-2}\,mol/l} = 2 \times 10^{-8}\,mol/l$.
  • The potential is: $E_{Cathode} = E^0(Ag/Ag^{+}) + \frac{RT}{F} \ln(c_{Cathode})$.
  • Step 3: Calculating Cell Voltage
  • $U_{eq} = E_{Cathode} - E_{Anode}$.
  • The standard potential $E^0(Ag/Ag^{+})$ cancels out: $U_{eq} = \frac{RT}{F} \ln\left(\frac{c_{Cathode}}{c_{Anode}}\right)$.
  • Substituting the values: $U_{eq} = \frac{8.314 \times 298.15}{96485.309} \ln\left(\frac{2 \times 10^{-8}}{10^{-3}}\right)$.

Reaction Kinetics: Alkaline Hydrolysis of Esters

• The Chemical Reaction

  • The alkaline hydrolysis of an ester ($RCOOR'$) in aqueous solution at $25\,^{\circ}\text{C}$ follows the equation:     $RCOOR' + OH^{-} \rightarrow RCOO^{-} + R'OH$

• Differentiation and Hypothesis of Reaction Order

  • The differential rate law (differentielle Zeitgesetz) for this bimolecular reaction is hypothesized to be of the second order ($n = 2$) overall (first order with respect to each reactant).
  • The rate law is expressed as: $v = -\frac{d[RCOOR']}{dt} = k \times [RCOOR'] \times [OH^{-}]$.
  • If the concentrations are stoichiometric or if one is considered in the context of specific experimental conditions, it may simplify to: $v = k \times [RCOOR']^2$ or similar.

• Analysis of Experimental Data and Identification of Reaction Order

  • Three linear regression diagrams were provided to determine the reaction order by plotting concentration-time data in different forms:
  1. 0th Order Plot: $[RCOOR']$ vs. $t$ in min.
     • Equation: $y = -0.0003x + 0.0178$
     • Coefficient of Determination ($R^2$): $0.7126$
  2. 1st Order Plot: $\ln[RCOOR']$ vs. $t$ in min.
     • Equation: $y = -0.0332x - 4.0086$
     • Coefficient of Determination ($R^2$): $0.9256$
  3. 2nd Order Plot: $1/[RCOOR']$ vs. $t$ in min.
     • Equation: $y = 4.9075x + 39.818$
     • Coefficient of Determination ($R^2$): $0.9999$
  • Conclusion: Since the $R^2$ value is highest (closest to 1) for the plot of $1/[RCOOR']$ versus time, the reaction is confirmed to be second order ($n = 2$).
  • Rate Constant ($k$): For a second-order reaction, the slope of the line in the $1/c$ vs. $t$ plot is equal to the rate constant $k$. Therefore, $k = 4.9075\,l\,mol^{-1}\,min^{-1}$.

Enzyme Kinetics: Michaelis-Menten and Lineweaver-Burk

• The Catalyst

  • Hydrolases are enzymes used to mediate hydrolysis reactions.

• Analysis via Lineweaver-Burk Plot

  • The Lineweaver-Burk equation is the linear reciprocal form of the Michaelis-Menten equation:     $\frac{1}{v_0} = \frac{K_m}{V_{max}} \times \frac{1}{[S_0]} + \frac{1}{V_{max}}$
  • The provided linear regression from the data is: $y = 12884x + 156592$.
  • Here, $y = 1/v_0$ and $x = 1/[S_0]$.

• Parameter Determination

  • Calculating Maximum Velocity ($V_{max}$):
  • The y-intercept is $\frac{1}{V_{max}} = 156592\,ls/mol$.
  • $V_{max} = \frac{1}{156592} \approx 6.386 \times 10^{-6}\,mol\,l^{-1}\,s^{-1}$.
  • Calculating Michaelis Constant ($K_m$):
  • The slope is $\frac{K_m}{V_{max}} = 12884\,s$.
  • $K_m = \text{slope} \times V_{max} = 12884 \times \frac{1}{156592} \approx 0.08227\,mol/l$.

Temperature Dependence of Reaction Rates: Arrhenius Equation

• Initial Conditions ($T_1$)

  • Temperature ($T_1$): $290\,K$.
  • Rate Constant ($k_1$): $5\,mol/(l\,min)$.

• Target Conditions ($T_2$)

  • Temperature ($T_2$): $300\,K$.
  • Task: Calculate the value of the rate constant $k_2$.

• Physical Constants and Energy Parameters

  • Activation Energy ($E_a$): $54\,kJ/mol = 54000\,J/mol$.
  • Gas Constant ($R$): $8.314\,J/(mol\,K)$.

• The Arrhenius Equation (Two-Point Form)

  • $\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \times \left(\frac{1}{T_1} - \frac{1}{T_2}\right)$.
  • Substituting values:     $\ln\left(\frac{k_2}{5}\right) = \frac{54000}{8.314} \times \left(\frac{1}{290} - \frac{1}{300}\right)$.
  • $\ln\left(\frac{k_2}{5}\right) = 6495.068 \times (0.003448 - 0.003333)$.
  • $\ln\left(\frac{k_2}{5}\right) = 6495.068 \times 0.0001149 \approx 0.7465$.
  • $\frac{k_2}{5} = e^{0.7465} \approx 2.1097$.
  • $k_2 = 5 \times 2.1097 = 10.5485\,mol/(l\,min)$.

Spectroscopy: Lambert-Beer Law

• Definition and Formula

  • The Lambert-Beer Law describes the attenuation of light as it passes through a substance.
  • $A = \epsilon \times c \times d$

• Explanation of Symbols

  • $A$: Absorbance (dimensionless units, also known as Extinction).
  • $\epsilon$: Molar Decadic Extinction Coefficient ($l\,mol^{-1}\,cm^{-1}$). It is a substance-specific constant that depends on the wavelength.
  • $c$: Concentration of the absorbing substance in the solution ($mol/l$).
  • $d$: Path length (thickness) of the cuvette or sample through which light passes ($cm$).

• Conditions for Validity

  • The law is valid under the following conditions:
  • Monochromatic light is used (only one specific wavelength).
  • The solution is dilute (usually $c < 0.01\,mol/l$) to avoid interactions between particles.
  • The sample is homogeneous and non-scattering.
  • No chemical changes occur in the sample due to light irradiation (photochemical stability).
  • The temperature remains constant, as refractive index and volume are temperature-dependent.

Mass Spectrometry and NMR

• Ionization Techniques in Mass Spectrometry

  • Based on the context of providing a typical spectrum analysis, participants must distinguish between:
  • (A) Hard Ionization: (e.g., Electron Ionization/EI) often leads to significant fragmentation.
  • (B) Ion Trap (Iontrap): This is a type of mass analyzer, not an ionization method.
  • (C) Soft Ionization: (e.g., ESI or MALDI) preserves the molecular ion with minimal fragmentation.
  • (D) Synchrotron: An electromagnetic radiation source often used as a light source for various spectroscopic methods.

• Nuclear Magnetic Resonance (NMR) Detectable Nuclei

  • To be detectable via NMR, an atomic nucleus must possess a non-zero magnetic moment, which requires a non-zero nuclear spin ($I \neq 0$).
  • Three common examples of such nuclei are:
  1. $^{1}H$ (Proton): The most common nucleus in NMR spectroscopy.
  2. $^{13}C$ (Carbon-13): Used extensively in organic chemistry to determine carbon skeletons.
  3. $^{15}N$ (Nitrogen-15): Common in biomolecular studies.
  • (Additional common examples include $^{19}F$ and $^{31}P$).