Comprehensive Notes on Amount of Substance, Atomic Structure, and Time of Flight Mass Spectrometry

Quantitative Chemistry, Stoichiometry, and Solutions

• Empirical Formula:

  • Defined as the simplest whole number ratio of atoms of each element present in a compound.

• Percentage Atom Economy:

  • A measure of the proportion of reactant materials that end up as useful products in a chemical process.
  • Calculated using the relative formula masses ($M_r$) of reactants and products:     $\% \text{ Atom Economy} = \frac{\text{Sum of } M_r \text{ of Useful Products}}{\text{Sum of } M_r \text{ of all Reactants (or Products)}} \times 100$

• Percentage Yield:

  • Compares the actual experimental mass of a product obtained to the maximum possible theoretical mass calculated from stoichiometry:     $\% \text{ Yield} = \frac{\text{Actual Yield}}{\text{Theoretical Yield}} \times 100$

• Definitions in Solution Chemistry:

  • Solute: A substance that dissolves in a solvent (for example, salt).
  • Solvent: The liquid medium into which a solute dissolves (for example, water).
  • Solution: A homogeneous mixture formed when a solute dissolves into a solvent (for example, salty water).

Fundamental Atomic Structure and Subatomic Particles

• Subatomic Particle Properties:
  • Atoms consist of three fundamental subatomic particles: protons, neutrons, and electrons.
  • Proton:
  • Symbol / sign: positive ($+$)
  • Relative mass: $1$
  • Relative charge: $+1$
  • Actual mass: $1.673 \times 10^{-27}\,kg$
  • Neutron:
  • Symbol / sign: neutral
  • Relative mass: $1$
  • Relative charge: $0$
  • Actual mass: $1.675 \times 10^{-27}\,kg$
  • Electron:
  • Symbol / sign: negative ($\ominus$)
  • Relative mass: $\frac{1}{1840}$
  • Relative charge: $-1$
  • Actual mass: $0.911 \times 10^{-30}\,kg$

Isotopes and Their Physical and Chemical Behavior

• Definition of Isotopes:

  • Atoms that possess the exact same atomic number (number of protons) but have different mass numbers (due to a differing number of neutrons).

• Chemical Properties:

  • Isotopes of the same element exhibit virtually identical chemical behavior.
  • Chemical properties are defined entirely by electronic configuration, specifically the arrangement and number of outer shell electrons.

• Physical Properties:

  • Isotopes exhibit slight differences in their physical properties due to variations in atomic mass and atomic size.
  • Physical properties that vary between isotopes include:
  • Melting points
  • Boiling points
  • Densities
  • Rates of diffusion

Relative Atomic Mass

• Formal Definition of Relative Atomic Mass ($A_r$):

  • The average mass of an atom of an element divided by $1/12\text{th}$ the mass of a single atom of Carbon-12 ($^{12}\text{C}$):     $\text{Relative Atomic Mass } (A_r) = \frac{\text{Average mass of } 1 \text{ atom of an element}}{\frac{1}{12}\text{th mass of } 1 \text{ atom of Carbon-12}}$

• Calculation from Isotopic Data:

  • Relative atomic mass is computed as the weighted average of the masses of all stable isotopes taking into account their natural percentage abundances:     $A_r = \frac{(m_1 \times \%_1) + (m_2 \times \%_2) + \dots}{100}$
  • In this equation, $m_1, m_2, \dots$ denote the isotopic masses, and ${}\%_1, \%_2, \dots$ denote their corresponding percentage abundances.

Overview and Principles of Mass Spectrometry

• Primary Functions:

  • Used to separate and determine the individual abundances and masses of each isotope in an element, enabling the precise determination of relative atomic mass ($A_r$).
  • Used to determine the relative molecular mass ($M_r$) of substances that exist as molecules.

• Real-World Applications:

  • Utilized in space travel and extraterrestrial exploration due to the instrumentation being very lightweight.
  • Used in archaeological and geochronological carbon dating.

• Core Mechanism of Time of Flight (TOF) Mass Spectrometry:

  • Particles in an analyte sample are ionized to form positive ions.
  • All positive ions are accelerated such that they all acquire an identical kinetic energy.
  • The ions travel a defined distance along a flight tube; the time taken to traverse this distance is directly related to and used to determine the mass of each ion in the sample.

• The Four Sequential Stages of TOF Mass Spectrometry:

  1. Ionization (achieved via one of two distinct methods)
  2. Acceleration
  3. Separation (Time of Flight)
  4. Detection

Stage 1 of TOF Mass Spectrometry: Ionization Techniques

• Method A: Electrospray Ionization:

  • Characteristics and Suitability:
  • Acts as a soft, non-destructive ionization technique.
  • Used primarily for large, fragile molecules (e.g., proteins or polymers) to prevent them from fragmenting.
  • Process Steps:
  1. The sample is dissolved in a polar, volatile solvent (such as water or ethanol).
  2. The dissolved sample is injected into the mass spectrometer through a hypodermic needle connected to a high-voltage power supply, producing a fine mist/spray of charged droplets.
  3. As the droplets exit the needle, particles gain a proton ($H^+$) from the solvent:        $X + H^+ \rightarrow XH^+$        (where $X$ represents the large intact molecule).
  4. The volatile solvent rapidly evaporates, leaving isolated gaseous $XH^+$ ions.
  5. The resulting positive ions are attracted toward and accelerated by a negatively charged electric plate.

• Method B: Electron Impact Ionization (Electron Gun):

  • Characteristics and Suitability:
  • A more energetic, hard ionization method.
  • Frequently causes fragmentation by breaking down large molecules into smaller fragment ions, which can provide useful structural information.
  • Standard technique applied to elements and low formula mass substances.
  • Process Steps:
  1. The sample must be present in the gas phase:        $X(g)$
  2. An electron gun fires a stream of high-energy electrons directly at the vaporized substance.
  3. The high-energy electrons knock out an electron from the outer shell of the sample particles, generating positive ions with a single positive charge:        $X(g) \rightarrow X^+(g) + e^-$

Stage 2 of TOF Mass Spectrometry: Acceleration

• Electric Field Acceleration:

  • Two charged plates set up an electric field across which the positive ions are accelerated.
  • The electric field provides the same amount of work on all ions possessing the same charge, ensuring that all particles gain identical kinetic energy ($KE$).

• Mathematical Derivations for Ion Velocity:

  • Kinetic energy is expressed by the standard formula:     $KE = \frac{1}{2}mv^2$
  • Rearranging to express the relationship for velocity ($v$):     $2KE = mv^2$$v^2 = \frac{2KE}{m}$$v = \sqrt{\frac{2KE}{m}}$
  • Because all ions carry identical kinetic energy, the velocity ($v$) of an ion depends strictly on its mass ($m$):
  • Ions with smaller mass travel at higher velocities.
  • Ions with larger mass travel at lower velocities.

Stage 3 of TOF Mass Spectrometry: Separation

• Flight Tube Drift Region:

  • The positive ions exit the electric field and pass into a field-free drift region (flight tube).
  • Ions travel through this flight tube with the specific kinetic energy and velocity attained during acceleration.
  • Because heavier particles have less velocity and lighter particles have greater velocity, the ions separate along the tube according to their mass-to-charge ratio ($m/z$).

• Derivation of Time of Flight ($t$):

  • Velocity is defined as the fixed length of the flight tube ($d$) divided by the time taken to travel ($t$):     $v = \frac{d}{t}$$t = \frac{d}{v}$
  • Substituting the expression for velocity ($v = \sqrt{\frac{2KE}{m}}$) into the time equation yields:     $t = \frac{d}{\sqrt{\frac{2KE}{m}}}$$t = d\sqrt{\frac{m}{2KE}}$
  • Under standard operating conditions, the charge is always single ($z = 1$), meaning:     $\frac{m}{z} = \frac{m}{1} = m$
  • Since distance ($d$) and kinetic energy ($KE$) are held constant across all ions, the flight time is directly proportional to the square root of the ion mass:     $t \propto \sqrt{m}$
  • Lighter ions arrive at the end of the tube in a shorter time, whereas heavier ions take longer.

Stage 4 of TOF Mass Spectrometry: Detection

• Discharge Mechanism at the Detector Plate:

  • The separated positive ions reach the end of the flight tube and strike a negatively charged electric detector plate.
  • Upon colliding with the plate, each positive ion accepts/gains an electron from the plate, neutralizing the ion back into an uncharged atom or molecule.

• Signal Generation and Abundance Measurement:

  • The neutralization process causes a movement of electrons across the detector circuit to replenish the plate, generating a measurable electric current.
  • The specific arrival time of the current pulse identifies the mass-to-charge ratio ($m/z$) of the arriving ion or molecule.
  • The magnitude (size) of the generated electric current is directly proportional to the abundance of that specific ion in the analyzed sample (a higher number of ions striking the plate produces a correspondingly larger electric current).