Analytical II Exam 2

Atom Bandwidth

0.001 nm

Advantages of Furnace over Flames

01. 1-2mL sample volume is the minimum required for a flame compared to 1uL for a furnace 02. Flames hold the sample in the optical path for 1s as it rises through the flame; however, a furnace confines the sample in the light path for several seconds 03. Multiple aliquots of sample can be injected onto the furnace platform and evaporated, pre-concentrating the sample in the furnace prior to analysis 04. Flames require nebulization which can dilute the sample, there is no nebulization step using a flame***

Photomultiplier Tube Process

01. A photon strikes the photosensitive cathode surface and electrons are ejected 02. The electrons are accelerated and strike the dynode with more than their original kinetic energy 03. Each of the 9 dynodes is 90 mV more positive than the previous to allow electrons to flow through 04. Each electron causes emission of several additional electrons 05. 10^6-10^7 electrons reach the anode for each incident photon ... results in a current gain

5 Parts of Inferometer

01. Collimated light source 02. Stationary mirror 03. Movable mirror 04. Beam splitter 05. Detector

Disadvantages of Furnace

01. Graphite furnaces have limited lifetimes 02. Memory effect-- interference from previous runs

How Temperature Affects Atomic Spectroscopy

01. It determines the degree to which a sample breaks down into atoms 02. It determines the extent to which an atom is found in the ground, excited, or ionized state Both of these affect the strength of the observed signal Temperature is not that important for absorbance because it measures from the ground state, but temperature is important for emission because uniform temperature is ideal because measuring from the excited state

Why is Plasma Used for Atomic Emission?

01. It is so hot that there is a more substantial population of atoms in the excited state 02. Each atoms emits many photons per second because atoms are rapidly promoted back to the excited state due to high temps As deltaE decreases ... more atoms in excited state As temp increases ... more atoms in excited state

Theory of Monochromators

01. Light enters the entrance slit in all directions, hits the first concave mirror, which collimates it (makes it parallel) 02. The mirror directs the light onto the grating, which reflects light as well as disperses light based on *diffraction* 03. The grating has equally spaced grooves, distance "d" apart, that each act as a source of diffraction (red light is diffracted at a greater angle than blue) 04. By turning the grating, we can choose the wavelength to make it through the exit slit 05. The light hits another mirror that focuses the light onto a focal point 06. Destructive of constructive interference can occur at the focal point 07. Light that enters the exit slit reaches the sample, then the detector

Inferometer Mechanism

01. Light from the source strikes the beam splitter at point "O" 02. 50% of light goes to stationary mirror (distance OS), 50% of light goes to movable mirror (distance OM) 03. Light reflects back to the beam splitter ... 50% of each is lost and 50% of each go to the detector where they combine 04. Paths OS and OM are not the same ... and each beam of light travels these twice 05. The difference between the paths = 2(OM-OS) = retardation of light (sigma) 06. Construction interference occurs if sigma is an integer (n) multiple of wavelength; Destructive interference occurs if sigma is a half-integer (n/2) multiple of wavelength 07. The production of an interference pattern occurs at the detector and an interferogram is graphed (*interferogram = light intensity vs. sigma*) 08. The computer applies a Fourier transform to get a spectrum (Intensity vs. wavenumber) from the interferogram

Process of Lasers

01. Pump energy through the side of the lasing medium ... molecules jump from Eo to E3 02. E3 to E2 is fast and nonradiative 03. E2 has a long lifetime, but E1 to Eo is fast and nonradiative 04. n2 > n1 (population of E2 > E1) = population inversion ... must have this because need more molecules releasing than absorbing protons) 05. E2 to E1 = LASER LIGHT ... molecules falling from E2 to E1 bounce in lasing cavity between mirrors 06. The left mirror of the laser is 0% T and the right mirror is 1% T (only 1% of the reflected light is the light useful light of the laser, the rest of the photons are used to stimulate others) 07. The photon can stimulate other molecules to emit photons = stimulated emission

Types of Interference with Atomic Spectroscopy

01. Spectral Interference 02. Chemical Interference 03. Ionization Interference *Standard addition compensates for many types of interference by adding known quantities of analyte to the unknown

Advantages of Plasma over Combustion Flame

01. Twice as hot ... more atoms in the excited state 02. High temp, stability, relatively inert Ar environment eliminate interferences encountered with the flame (does not generate additional molecules) 03. Simultaneous multi-elemental analysis ... emission only so there is no need for a lamp

Molecule Bandwidth

10-100 nm

Aerosol

A fine suspension of liquid (or solid) particles in a gas

Plasma

A gas hot enough to contain ions and free electrons

Atomic Absorption Spectroscopy

A liquid sample is aspirated through a plastic tube into the flame (or other atomizer) ... The liquid evaporates and the remaining solid is atomized in the flame

Pneumatic Nebulizers

A nebulizer dilutes the analyte*** A.) Concentric tube in which liquid sample is drawn through a capillary tube by high pressure gas flowing around the tip of the tube (aspiration) and high velocity gas breaks the liquid into droplets of various sizes that are then carried on to atomizer B.) Cross-Flow nebulizer in which the gas and sample meet at a right angle and do not mix C.) The sample is pumped onto a fritted disk through which carrier gas flows giving a much finer aerosol than the previous two ... not good for high salt concentrations D.) "Babington" nebulizer ... A hollow sphere, in which high pressure gas is pumped through a small orifice ... This expanding gas nebulizes the sample flowing in a thin film over the sphere's surface ... less clogging, so good with high salt samples

Detection Limits with Atomic Absorption

A radial slit is used for normal analysis and complex materials ... few matrix effects and interference An axial slit produces more intense signal and can detect lower concentrations (low detection limits) The detection limit for furnaces is typically two orders of magnitude lower than that observed with a flame because the sample is confined in a small volume of the furnace for a relatively long time Sensitivity of ICP is close to that of graphite furnace

Matrix Modifier

A substance added to the sample to reduce the loss of the analyte during the charring by making the matrix more volatile or the analyte less volatile (Allows for better separation)

Advantages/Disadvantages of Photomultiplier Tubes

Advantage = Great signal amplification Disadvantage = Scan through the spectrum one wavelength at a time ... Dispersive instruments have diffraction gratings, so the monochromator must be rotated to measure all wavelengths, which takes time

Advantages/Disadvantages of Photodiode Arrays

Advantage = Records all wavelengths of a spectrum at once; Allows faster spectral acquisition than dispersive instruments; Has almost no moving parts Disadvantage = Not good at amplification; There is higher resolution with a dispersive instrument; Stray light is less in a dispersive instrument

Advantages/Disadvantages of Charge Coupled Devices

Advantage = Records entire spectrum at once; Amplification occurs Disadvantage = Expensive

Most Common Flame

Air/Acetylene

1/f Noise; Pink Noise

Also called "drift" Greatest at zero frequency and decreases in proportion to 1/f ... Due to flickering of a flame or drifting of a light source

Line Noise

Also called "interference" or "whistle noise" At discrete frequencies such as 60 Hz ... there is a transmission-line frequency

Furnaces

An electrically heated graphite furnace is more sensitive than a flame and requires less sample 1-100 uL of sample is injected into the furnace through a hole at the center Light from a hollow cathode lamp travels through the window at each end of the graphite tube Requires more operator skill than flame to program: 01.) Dry sample at 125C for 20s to remove solvent 02.) 60s charring at 1400C to remove organic matter 03.) Atomization of sample at 2100C for 10 s ... Abs is measured 04.) Heat 2500C for 3s to clean *Do not need to nebulize*** The precision with manual injection is 5-10% at best ... for automatic injection it is 1%

Charge Coupled Device

An extremely sensitive detector that stores photo-generated charge in a two-dimensional array; Constructed of p-doped Silicon and an n-doped substrate ... Capped with an insulating layer of SiO2, on top of which is a pattern of conducting Si electrodes Light absorbed in the p-doped region introduces an electron into the conduction band and a hole is left in the valence band ... The electron is attached to and stored at the positive electrode ... The hole migrates to the n-doped substrate Electrodes store 10^5 electrons before they spill out into adjacent elements Electrons stored in each pixel of the top row are moved into the serial register at the top and the charge is read The next row moves up and is read until the entire array has been read Charge transfer is efficient (only lose 5 of every 10^6 electrons)

Atomic Fluorescent Spectroscopy

Atoms in the flame are irradiated by a laser to promote them to an excited state from which they can fluoresce to return to the ground state

Deuterium Lamp Background Correction

Broad emission from a D2 lamp is passed through the flame in alternation with that from the hollow cathode The D2 source can only produce signals from broadband absorption and scattering from the matrix ... The absorption from the atoms is so small compared to the absorption of the matrix, that it is insignificant, and does not show up on the signal Light from the HCL is absorbed by the analyte as well as absorbed and scattered by the background ... Light from the D2 lamp is only absorbed and scattered by the background HCL - D2 = analyte absorbance

Ionization Interference

Can be a problem in the analysis of alkali metals, because they have low ionization potentials Ionized atoms have different energy levels than neutral atoms, so the desired signal is decreased An *ionization suppressor* decreases the extent of ionization of the analyte

Chemical Interference

Caused by a component of the sample that decreases the extent of atomization of the analyte Use *releasing agent* to decrease the chemical interference

Pressure Broadening

Collisional broadening ... Collisions of the emitting or absorbing species with other atoms or ions in a hot environment ... Collisions between atoms shorten the lifetime of the excited state = more uncertainty ... The collision frequency is proportional to the pressure This yields linewidths of 10^-3 to 10^-2, so this limits us the most To combat this ... the pressure can be lowered using a vacuum

Photomultiplier Tube

Contains a photoemissive surface (photocathode) as well as several additional surfaces (dynodes) that emit a cascade of electrons when struck by an electron from a photosensitive area

Double Beam Spectrophotometer With Atomic Absorption

Disadvantage that the reference beam does not pass through the flame and therefore does not correct for loss of radiant power due to absorbance or scattering by the flame itself Monochromators are also present in atomic absorption instruments in order to select one line from the HCL and to reject as much emission from the flame or furnace as possible

Dispersion Equation

Dispersion (radians/distance) = (delta(phi)/ delta(wavelength)) = (n/(d X cos(phi))

Line Sources of Light

Emit a limited number of lines, or bands of radiation, each of which spans a limited range of wavelengths; Used in atomic absorption and atomic emission Can use molecules as light sources by having low pressures

Continuum Sources of Light

Emit radiation that changes in intensity slowly with wavelength; Used in molecular absorption and fluorescence spectroscopy Can use atoms as light sources by having high pressures so they become less ideal

Monochromator Diagram

Entrance slit -> concave mirror -> reflection grating -> concave mirror -> exit slit

Photodiode Array

Essential for rapid spectroscopy; A photodiode array spectrophotometer records the entire spectrum at once in a fraction of a second; White light is passed through a sample, then a *polychromator* disperses light into its component wavelengths and directs the light at a photodiode array ... Each diode receives a different wavelength 01. Rows of p-type silicon on a substrate of n-type silicon create a series of pn junction diodes 02. A *reverse bias* is applied to each diode, drawing electrons and holes away from the junction ... there is a *depletion region* at each junction, in which there are few electrons and holes 03. The junction acts as a capacitor with charge stores on either side of the depletion region 04. At the beginning of each measurement, the diode is fully charged 05. Photons absorbed in the semiconductor create mobile electron-hole pairs 06. The more radiation that strikes each diode, the less charge remains at the end of the measurement 07. The state of each diode is determined at the end of the cycle by measuring the current needed from the instrument to recharge the diode ... tells us how much light struck the diode -> T -> A -> C

Ferroelectric Material

Ex. DTGS (deuterated triglycine sulfate) In these materials, the dipole moments of molecules remain aligned in the absence of an external field; This gives the material a permanent electric polarization; When IR radiation is absorbed, the polarization changes and a voltage develops across the material; DTGS is a common detector in Fourier transfer spectrometers IR -> Change in polarization -> Change in Voltage

Rich Flame

Excess fluid; Increases sensitivity because excess carbon can reduce the metal oxides and hydroxides

Lean Flame

Excess oxidant; Produces a hotter flame

Disadvantage of Plasma Atomizer

Expensive to purchase and operate

Plasma Atomizer

Few molecules are able to withstand the high temperature of the plasma ... so no interference High-purity argon is fed into the plasma gas inlet at 14-18L per minute The top of the torch is a tube ... a water coiled induction coil powered by a radio frequency generator producing 2 kW of energy at around 27 MHz frequency A spark from a Tesla coil initiates ionization of At forming Ar+ and electrons Ions are agitated by the strongly fluctuating field and are forced to rapidly follow a circular path Friction heats the gas to the point where it forms a plasma Ar+ ions are stable enough that they will continue to absorb energy from the coil and stay hot enough to support the plasma indefinitely

Continuous Atomizers

Flames and Plasmas Samples are introduced in a steady manner *Require nebulizers*

Discrete Atomizers

Furnace (electrothermal atomizer) Samples are introduced in a discontinuous manner with a syringe or autosampler

What limits atomic linewidths?

Heisenberg Uncertainty Principle Doppler Broadening Pressure Broadening

UV-vis Line Light Sources

Hollow cathode lamps or Lasers

What do we use to generate line radiation sources?

Hollow-Cathode Lamps Monochromators generally cannot isolate lines narrower than 10^-3 to 10^-2

How does n contribute to diffraction?

If n = 0 ... no diffraction If n = -1 ... the diffracted angle is smaller If n = 1 ... the diffracted angle is larger As n increases, the diffracted angle increases, but the light is less intense

Zeeman Correction

Important for a graphite furnace correction When a sample is placed in a magnetic field and its absorption is observed in plane polarized light, the normal atomic signal (pi absorb light parallel to field, sigma absorb light perpendicular to field) ... molecules show no Zeeman splitting The strong magnetic field is pulsed on and off while light from the HCL is passed through a fixed polarizer When the magnetic field is off ... both analyte and background signals exist (no splitting) When magnetic field is on ... only background signal exists Magnetic field off - Magnetic field off = analyte signal

Doppler Broadening

In a collection of atoms in a hot environment, atomic motion occurs in every direction ... An atom moving toward the radiation source "sees" a higher frequency light than one moving away The Doppler effect occurs for atoms moving with the highest velocity toward or away from the source There is no shift for atoms moving perpendicular to the source and the remaining atoms show intermediate shifts ... This results in a bell curve of wavelengths The magnitude of uncertainty is 10^-3, so the Doppler effect limits us more than Heisenberg, but less than Pressure Broadening

Beam Chopping Background Correction

In atomic absorption, the electrical modulation of the hollow-cathode lamp can distinguish the signal of the flame from the atomic line at the same wavelength The signal reaching the detector while the beam is blocked by the chopper = signal from flame emission The signal reaching the detector while the beam is not blocked = signal from flame and the lamp Subtract 'blocked' from 'not blocked' = analytical signal *Does not account for scattering*

Thermocouplers

Junctions of specific allows that have a predictable and reproducible relationship between temperature and voltage; If a thermal couple is blackened to absorb radiation, its temperature and therefore voltage becomes sensitive to radiation; These convert a temperature gradient into electricity (current) IR -> Temp Change -> Current

If the exit slit width is decreased ...

Less wavelengths can pass through, which increases resolution, but less light goes through ... leads to less power reaching the detector and a large amount of stray light (noisy***)

LASER Acronym

Light Amplification by Stimulated Emission of Radiation

Disadvantage of Flame Atomizer

Many elements form oxides and hydroxides in the outer cone (complete combustion occurs here, where surrounding air is drawn into the flame) ... These molecules have different spectra than atoms

Resolution

Measure of the ability to separate two closely spaced peaks; The greater the resolution, the smaller the difference between the two wavelengths that can be distinguished from each other; The greater number of grooves (smaller the spacing), the higher the resolution*

Dispersion

Measure of the ability to separate wavelengths by change in wavelength through the difference angle (delta phi); Greater number of grooves (smaller the spacing), the greater the dispersion

Beam Chopper

Mirror that rotates (bc it is attached to a motor) to direct light to either the sample of the reference cuvette; If the light hits the transparent side, it reaches the detector via the sample If light hits the mirror side, it reaches the detector via the reference

Laser Light is ...

Monochromatic Extremely bright Collimated (parallel and non-diverging) Polarized (one direction) Coherent (unidirectional)

Hollow Cathode Lamp (HCL)

Most common source for atomic absorption; Consists of a tungsten anode and a cylindrical cathode (constructed of the metal whose spectrum is desired), sealed in a glass tube that is filled with neon or argon at a low pressure A potential difference of 500V is applied across the electrodes, which generates a current of 5-15 mA as the inert gas is ionized ... Gaseous cations can acquire enough kinetic energy to dislodge some of the metal atoms from the cathode surface producing an electron cloud, in a process called *sputtering* A portion of the sputtered metal atoms are in their excited state and emit characteristic radiation of one wavelength as they return back to their ground state The metal atoms eventually diffuse back to the cathode, or deposit onto the glass walls of the tube

Disadvantage of Monochromator

Must move the monochromator in order to get all wavelengths through the exit slit

Number of Illuminated Grooves Equation

N = 1/d

Background Correction in Atomic Spectroscopy

Necessary to distinguish the analyte signal from absorption, emission, and optical scattering of the sample matrix, the flame, plasma, or white-hot graphite furnace Background absorbance is usually around 0.3 Background correction is critical for graphite furnaces due to residual smoke from charring, which causes scattering 01. Beam chopping (flame flickering correction) 02. Deuterium Lamp (scattering) 03. Zeeman Correction

IR Continuum Light Sources

Nichrome wire or Globar (Silicon Carbide)

White Noise

Normal, random, Guassian noise arises from causes such as random motions of electrons within a circuit

Tubes of Plasma Torche

Outer tube (Coolant tube) = coolant gas that cools torch Middle tube (Plasma tube) = Ar is fed @ 14-18L per minute into plasma gas inlet Inner tube (Sample tube) = sample is aspirated into the plasma after being nebulized by an ultrasonic nebulizer

Spectral Interference

Overlap of the analyte signal with signals due to other elements or molecules in the sample or with signals due to the flame or furnace (subtracted out by D2 or Zeeman background correction) To deal with overlap between lines of different elements in the sample, choose another wavelength for analysis ... High-resolution spectrometers can resolve closely spaced lines Also, elements that form very stable diatomic oxides are not completely atomized ... Molecular spectra are much broader than atomic spectra

UV-vis Detectors

Photomultiplier tube, photodiode array, charged couple device Produces an electrical signal when it is struck by photons ... A detector response is a function of wavelength of incident light

Inferometer

Produces an interference pattern that contains all of the IR data; We use it to look at all wavelengths at one time ...

Transversely Heated Furnace

Provides a nearly constant temperature across the entire furnace

Atomic Emission Spectroscopy

Radiation from hot atoms whose electrons have been promoted to an excited state in the flame or plasma is emitted ... no lamp is neeeded*

Resolution Equation

Resolution = (average wavelength/change in wavelength) = (diffraction order X number of grooves)

Furnace Process

Sample is injected onto platform The furnace wall is heated ant this radiant energy heats the L'vov platform The sample is vaporized when wall reaches a constant temperature Everything in the sample besides the analyte is called the matrix-- Ideally the matrix decomposes and vaporizes during the charring step

Photoconductive Detectors

Semiconductors that have a small band gap that corresponds to IR energies; When they absorb a photon, an electron is excited into the conduction band and collected at an electrode; For IR ... Mercury/Cadmium Telluride (MCT) For NIR ... Lead Sulfide (PbS) and Indium Antimonide (InSb)

Signal vs. Noise Definition

Signal = the middle of the baseline noise to the middle of the noisy peak Noise = the top of one peak to the bottom of that peak

Dealing with Noise

Signal averaging can improve the quality of data More scans = better signal to noise ratio ... Want the experiment to be rapid so that we can increase the number of scans *Must define noise* Collecting "n" spectra increases the signal by "n" times ... Noise is random so it may be positive or negative at any point ... If "n" spectra are recorded, the noise increases in proportion to the square root of n

Lamp for IR

Silicon carbide glowbar: emits approximately the same spectrum as a blackbody at 1000K ... 4000-200 cm-1

Root-Mean-Square (rms) Noise

Standard deviation of the noise calculated by a computer The rms noise is 5X less than the peak-to-peak noise

Double-Beam Spectrophotometers

The beam is alternately sent through the reference and sample cells; It is chopped several times a second; When light passes through the sample, P is measured ... when light passes through the reference, Po is measured; P is compared to Po, T is obtained and converted to A 2 lamps** DB spectrophotometers avoid the issue of having to run a blank throughout the experiment, but you must have 2 matching cuvettes (expensive)

Longitudinally Heated Furnace

The center of the furnace is hotter than the ends ... Atoms from the central region condense at the ends, where they can vaporize during the next sample run Not ideal for furnace* Transversely heated furnaces reduce this *memory effect*, or the interference from previous runs

Nebulization

The formation of small droplets

Pyrolytic Graphite

The furnace is coated with this dense layer of graphite to seal the relatively porous graphite, so that it cannot absorb foreign atoms

Boltzmann Distribution

The relative populations of different states at thermal equilibrium At higher temps ... the more atoms in the excited state, which is better for emission Flame fluctuation does not influence absorption, but is not suitable for emission readings *Plasma temperature is more stable than the flame*

Disadvantages of Single-Beam Spectrophotometer

The sample and the reference must be placed alternately in the beam ... The reference (Po) should be measured frequently throughout the experiment because the ratio of P/Po is changing not due to the analyte For measurements at multiple wavelengths, the reference must be run at each wavelength Poor for measuring absorbance versus time for kinetics studies because both the source and the detector response slowly *drift*

Heisenberg Uncertainty Principle

The shorter the lifetime of the excited state, the more uncertain its energy ... Spectral lines have finite widths because the lifetimes of states are finite ... The excited state is short and the ground state is long As the lifetime of the excited state decreases, the uncertainty in the energy increases (as does the uncertainty of the wavelength Relative uncertainty in energy is the same as the relative uncertainty in wavelength The inherent linewidth of an atomic absorption of emission signal is 10^-4 nm because of the short lifetime of the excited state ... This is inherent uncertainty of the instrument and cannot be reduced Not the biggest issue because the linewidth must be 10^-3

Infrared Detectors

Thermocouplers, Ferroelectric material, Photoconductive detectors Infrared photons are not energetic enough to eject electrons from a photosensitive surface or promote electrons from the valence band of Si to the conduction band (unless if the band gap is smaller)

Lamps for UV-vis

Tungsten lamp: source of continuous visible and near-IR radiation, giving useful radiation in the range 320-2500 nm Deuterium arc lamp: an electric discharge (spark) causes D2 to dissociate and emit UV radiation from 110 to 400 nm A change is made between deuterium and tungsten lamps when passing through *360nm* so that the lamp of highest intensity is always used

Detector Types

UV-vis = photomultiplier tube, photodiode array, charged couple device IR = Thermocouplers, ferroelectric material, photoconductive detectors

How to get rid of unwanted diffraction orders?

Use a filter ... only allows certain bands of wavelengths to pass through

Ultrasonic Nebulizer

Used in plasma atomizers ... The sample is directed onto a piezoelectric crystal oscillating at 1 MHz A piezoelectric crystal is one whose dimensions changed in an applied electric field ... a sinusoidal voltage applied between two faces of the crystal causes it to oscillate The vibrating crystal makes a fine aerosol that is carried by a stream of argon into a heated tube where the solvent evaporates This stream then moves through a cool zone where the solvent condenses and is removed The analyte then reaches the plasma as an aerosol of dry, solid particles *Plasma energy is not needed to evaporate the solvent, so more energy is available for atomization* *Also, a larger fraction of the sample reaches the plasma than with conventional nebulizers* The concentration of analyte needed for adequate signal is reduced by an order of magnitude using the ultrasonic nebulizer

Flame

Uses a premix burner, in which fuel, oxidant, and sample are mixed prior to introduction into the flame ... This is pressure-driven (the oxidant forces the sample through the capillary and the oxidant and sample mix) Sample solution is drawn into the *pneumatic nebulizer* by rapid flow of oxidant (usually air) past the tip of the sample capillary Liquid breaks into a fine mist as it leaves the capillary The spray is directed against a glass bead, upon which the droplets break into smaller particles The nebulizer creates an aerosol from a liquid sample The mist, oxidant, and fuel flow past baffles that promote further mixing and block large droplets of liquid ... Excess liquid collects at the bottom of the spray chamber and flows out to a drain Aerosol reaching the flame contains only about 5% of the initial sample

Stray Light

Wavelengths outside the bandwidth expected from the monochromator; Comes from unwanted diffraction orders or outside the instrument when the sample compartment is not perfectly sealed; Error from stray light is most serious when sample absorbance is high***

UV-vis Continuum Light Sources

Xe lamp, H2 or D2 lamp, Tungsten lamp

Reflection Grating of Monochromator Symbols

a = greater distance traveled by beam 2 than beam 1 *before* it strikes the grating b = greater distance traveled by beam 1 than beam 2 *after* strikes the grating theta = angle of incident light phi = angle of diffracted light

Maximum Constructive Interference

n(lambda) = d(sin(theta) + sin(phi)) Also known as the grating equation

Signal to Noise Ratio

n/(n^1/2) = n^1/2 So, in order to increase the signal-to-noise ratio by a factor of 10 ... we must average 100 spectra

Beer's Law requires that the linewidth of the radiation source should be substaintially _______ than the linewidth of the absorbing sample

narrower; If this is not true, the measured absorbance will not be proportional to the sample concentration