Chemical Measurements and Quantitative Analysis

Biochemical Measurements with a Nanoelectrode

• Chemical and physical measurements span a vast range of sizes, from atoms to whole galaxies, often described by powers of 10.

• Nanoelectrodes are analytical tools with tips smaller than a single cell, used to measure neurotransmitter molecules released by single nerve cells in response to chemical stimuli.

• Relevant orders of magnitude for nanoelectrode measurements:

  • Dimension of active region: nanometers ($10^{-9}\,\text{m}$).

  • Electrical current: picoamperes ($10^{-12}\,\text{A}$).

  • Time duration: milliseconds ($10^{-3}\,\text{s}$).

SI Units (Système International d’Unités)

• SI units are the fundamental base units from which all other units are derived.

• The fundamental quantities and their units are:

  • Length: meter ($\text{m}$)

  • Mass: kilogram ($\text{kg}$)

  • Time: second ($s$)

  • Temperature: kelvin ($\text{K}$)

  • Amount of substance: mole ($\text{mol}$)

  • Electric current: ampere ($\text{A}$)

  • Luminous intensity: candela ($\text{cd}$)

• Sub-units or other quantities are defined in terms of fundamental quantities, such as force ($\text{Newtons, N}$), pressure ($\text{Pascals, Pa}$), and energy ($\text{Joules, J}$).

Redefinition of SI Units (2019)

As of 2019, SI units are based on six physical constants assigned exact values based on 2017 measurements. These values are fixed by convention:

• Elementary charge ($e$): $1.602\,176\,634 \times 10^{-19}\,\text{C}$

• Speed of light in vacuum ($c$): $2.997\,924\,58 \times 10^8\,\text{m/s}$

• Planck’s constant ($h$): $6.626\,070\,15 \times 10^{-34}\,\text{J s}$

• Avogadro’s number ($N_A$): $6.022\,140\,76 \times 10^{23}\,\text{mol}^{-1}$

• Boltzmann’s constant ($k$): $1.380\,649 \times 10^{-23}\,\text{J/K}$

• Unperturbed ground-state hyperfine transition frequency of $^{133}\text{Cs}$ ($\Delta\nu_{Cs}$): $9.192\,631\,770 \times 10^9\,s^{-1}$

Definitions of Fundamental SI Units

• Time (second, s): Defined as the duration of $9\,192\,631\,770$ periods of the radiation of the unperturbed ground-state hyperfine atomic transition of $^{133}\text{Cs}$.

• Length (meter, m): The distance light travels in a vacuum during $\frac{1}{299\,792\,458}$ of a second. It can be measured by counting wavelengths of red light from a stabilized helium-neon laser.

• Mass (kilogram, kg): Previously defined by a platinum-iridium cylinder (1889–2019), but mass diverged by $\sim 50\,\mu\text{g}$ due to atmospheric reactions or wear. Now defined by setting Planck’s constant exactly to $6.626\,070\,15 \times 10^{-34}\,\text{kg m}^2/s$. Measurements are performed using a Kibble balance, which balances electromagnetic force against the weight of a test mass in Earth's gravity.

• Electric Current (ampere, A): A current of $1\,\text{coulomb per second}$ based on the fixed elementary charge ($1.602\,176\,634 \times 10^{-19}\,\text{C}$).

• Temperature (kelvin, K): Defined by the fixed numerical value of Boltzmann’s constant ($1.380\,649 \times 10^{-23}\,\text{kg m}^2 s^{-2} K^{-1}$).

• Amount of Substance (mole, mol): Exactly Avogadro’s number ($6.022\,140\,76 \times 10^{23}$) of particles. Historically measured using silicon spheres enriched to $99.998\,\text{atom}\%\,^{28}\text{Si}$ using X-ray crystallography and laser interferometry.

• Luminous Intensity (candela, cd): Defined by the luminous efficacy of monochromatic radiation of frequency $540 \times 10^{12}\,\text{Hz}$ to be $683\,\text{lumens/watt}$.

• Angles: There are $2\pi$ radians in a circle and $4\pi$ steradians ($\text{sr}$) in a sphere.

SI-Derived Units with Special Names

Prefixes as Multipliers

Prefixes are used for every third power of ten to manage scale:

• Large Scales:

  • Yotta (Y): $10^{24}$

  • Zetta (Z): $10^{21}$

  • Exa (E): $10^{18}$

  • Peta (P): $10^{15}$

  • Tera (T): $10^{12}$

  • Giga (G): $10^{9}$

  • Mega (M): $10^{6}$

  • Kilo (k): $10^{3}$

• Small Scales:

  • Milli (m): $10^{-3}$

  • Micro ($\mu$): $10^{-6}$

  • Nano (n): $10^{-9}$

  • Pico (p): $10^{-12}$

  • Femto (f): $10^{-15}$

  • Atto (a): $10^{-18}$

  • Zepto (z): $10^{-21}$

  • Yocto (y): $10^{-24}$

Conversion Factors and Errors

• Atmospheric Example: Upper atmospheric ozone ($O_3$) absorbs UV radiation. Altitude is roughly $1.7 \times 10^4\,\text{m}$ and peak pressure is $0.019\,Pa$.

• Importance of Units: The $125 \text{ million}$ Mars Climate Orbiter was lost in 1999 because engineers used British pounds of force while JPL used newtons, causing the craft to enter the atmosphere $100\,\text{km}$ lower than intended.

• Key Conversions:

  • $1\,\text{calorie (cal)} = 4.184\,\text{J}$ (exact)

  • $1\,\text{mile} \approx 1.609\,\text{km}$

  • $1\,\text{pound (lb)} \approx 0.4536\,\text{kg}$

  • $1\,\text{bar} = 10^5\,\text{Pa}$ (exact)

  • $1\,\text{atm} = 101\,325\,\text{Pa}$ (exact) = $760\,\text{torr}$

  • $1\,eV = 1.602\,176\,634 \times 10^{-19}\,\text{J}$

Chemical Concentrations and Definitions

• Solution: A homogeneous mixture of two or more substances.

• Solute: Minor species in solution.

• Solvent: Major species in solution (water in aqueous solutions).

• Molarity (M): Moles of substance per liter of solution ($\text{mol/L}$).

• Molality (m): Moles of substance per kilogram of solvent ($\text{mol/kg}$).

• Formal Concentration (F): The molarity of a substance if it were not converted to other species. For example, for a strong electrolyte like $MgCl_2$, if $[MgCl_2] = 0.054\,M$, this is the formal concentration. In solution, $70\%$ may exist as free $Mg^{2+}$ and $30\%$ as $MgCl^+$.

• Atomic Mass: Grams of an element containing Avogadro’s number of atoms.

• Molecular Mass: Sum of atomic masses in a molecule.

• Formula Mass (FM): Molecular mass of a strong electrolyte.

Composition and Trace Analysis

• Weight Percent (wt%): $\frac{\text{mass of solute}}{\text{mass of total solution}} \times 100$.

• Parts per Million (ppm): $\frac{\text{mass of solute}}{\text{mass of solution}} \times 10^6$. In dilute aqueous solutions ($\text{density} \approx 1.00\,g/mL$), $1\,ppm = 1\,\mu\text{g/mL} = 1\,mg/L$.

• Parts per Billion (ppb): $\frac{\text{mass of solute}}{\text{mass of solution}} \times 10^9$. For dilute aqueous solutions, $1\,ppb = 1\,ng/mL = 1\,\mu\text{g/L}$.

• Gas Concentration: For gases, ppm usually refers to volume. For example, $39\,ppb\,O_3$ means $39\,nL$ of $O_3$ per liter of air.

Preparing Solutions

• From Solids: Weigh the reagent, dissolve in a volumetric flask with distilled or deionized water, and dilute to the mark.

• Distillation: Boils water to remove non-volatile impurities.

• Deionization: Passes water through a column to remove ions; nonionic impurities remain.

• Dilution: To prepare a dilute solution from concentrations, use Equation 1-5:     $M_1 V_1 = M_2 V_2$     The moles of solute from the concentrated solution must equal the moles in the final dilute solution.

Gravimetric Analysis: Analytical Procedures

Gravimetric analysis measures the mass of a substance (such as a precipitate) to complete the analysis.

Case Study: Iron in Dietary Supplements

1. Dissolution: Tablets are mixed with $0.100\,M\,HCl$ to dissolve $Fe^{2+}$. Insoluble binder is filtered out.

2. Oxidation: $Fe^{2+}$ is oxidized to $Fe^{3+}$ using hydrogen peroxide ($H_2O_2$).

3. Precipitation: Ammonium hydroxide precipitates hydrous iron(III) oxide. This gel is filtered and heated to form pure solid $Fe_2O_3$.

Stoichiometry Example (Iron Analysis): If 12 tablets yield $0.277\,g$ of $Fe_2O_3$ ($FM = 159.69\,g/mol$):

• Moles of $Fe_2O_3 = \frac{0.277\,g}{159.69\,g/mol} = 1.73 \times 10^{-3}\,\text{mol}$

• Moles of $Fe = 2 \times 1.73 \times 10^{-3}\,\text{mol} = 3.47 \times 10^{-3}\,\text{mol}$

• Mass of $Fe = 3.47 \times 10^{-3}\,\text{mol} \times 55.845\,g/mol = 0.194\,g$

• Average mass per tablet $= \frac{0.194\,g}{12} = 16.1\,mg$

Limiting Reagent Calculations

• The limiting reagent is the species consumed first in a reaction, causing the reaction to cease.

• To determine the limiting reagent:

  • Find the number of moles for each reagent.

  • Compare the moles present to the stoichiometric requirements.

• Example Calculation: Mixing $1.00\,g\,CaCl_2$ ($FM = 110.98$) and $1.15\,g\,Na_2C_2O_4$ ($FM = 134.00$).

  • Moles of $Ca^{2+} = \frac{1.00\,g}{110.98\,g/mol} = 9.01\,mmol$

  • Moles of $C_2O_4^{2-} = \frac{1.15\,g}{134.00\,g/mol} = 8.58\,mmol$

  • Since the reaction is $1:1$, oxalate is the limiting reagent. Residual $Ca^{2+} = 9.01 - 8.58 = 0.43\,mmol$, which represents $4.8\%\,unreacted$.