7.38			SECTION 7
TABLE 7.19 Sensitive Lines of the Elements (Continued)
Wavelength, nm	Element		Sensitivity	Wavelength, nm	Element		Sensitivity
196.03	Se		U2	183.00	I		U2
194.23	Hg	II	V1	182.59	B	II	V2
193.76	As		U1	180.73	S		U1
193.09	C		U1	178.38	I		U1
190.86	Tl	II	V1	178.28	P		U1
189.99	Sn	II	V1	154.07	Br	II	V4
189.04	As		U2	134.72	Cl	II	V1

are steady-state techniques, are compared with the electrothermal or furnace technique which uses the entire sample and detects an absolute amount of the analyte element. To compare the several methods on the basis of concentration, the furnace detection limits assume a 20- L sample.

Data for the several flame methods assume an acetylene–nitrous oxide flame residing on a 5- or 10-cm slot burner. The sample is nebulized into a spray chamber placed immediately ahead of the burner. Detection limits are quite dependent on instrument and operating variables, particularly the detector, the fuel and oxidant gases, the slit width, and the method used for background correction and data smoothing.

7.4.1Some Common Spectroscopic Relationships

7.4.1.1 Electromagnetic Radiation. Electromagnetic radiation travels in straight lines in a uniform medium, has a velocity of 299 792 500 m · s 1 in a vacuum, and possesses properties of both a wave motion and a particle (photon). Wavelength is the distance from crest to crest; frequency v is the number of waves passing a fixed point in a unit length of time. Wavelength and frequency are related by the relation

c v

where c is the velocity of light (in a vacuum). In any material medium the speed of propagation is smaller than this and is given by the product nc, where n is the refractive index of the medium.

Radiation is absorbed or emitted only in discrete packets called photons and quanta:

E hv

where E is the energy of the quantum and h is Planck’s constant.

The relation between energy and mass is given by the Einstein equation:

E mc2

where E is the energy release and m is the loss of mass. Strictly, the mass of a particle depends on its velocity, but here the masses are equated to their rest masses (at zero velocity).

The Wien displacement law states that the wavelength of maximum emission, m, of a blackbody varies inversely with absolute temperature; the product mT remains constant. When m is expressed in micrometers, the law becomes

mT 2898

SPECTROSCOPY

7.39

In terms of m, the wavenumber of maximum emission:

m 3.48T

Another useful version is hvm 5kT, where k is the Boltzmann constant.

Stefan’s law states that the total energy J radiated by a blackbody per unit time and area (power per unit area) varies as the fourth power of the absolute temperature:

J aT 4

where a is a constant whose value is 5.67 10 8 W · m 2 · K 4.

The relationship between the voltage of an X-ray tube (or other energy source), in volts, and the wavelength is given by the Duane-Hunt equation:

hc 12 398 eV V

where the wavelength is expressed in angstrom units.

7.4.1.2 Laws of Photometry. The time rate at which energy is transported in a beam of radiant energy is denoted by the symbol P0 for the incident beam, and by P for the quantity remaining unabsorbed after passage through a sample or container. The ratio of radiant power transmitted by the sample to the radiant power incident on the sample is the transmittance T:

T P

P0

The logarithm (base 10) of the reciprocal of the transmittance is the absorbance A: A log T log T1

When a beam of monochromatic light, previously rendered plane parallel, enters an absorbing medium at right angles to the plane-parallel surfaces of the medium, the rate of decrease in radiant power with the length of light path (cuvette interior) b, or with the concentration of absorbing material C (in grams per liter) will follow the exponential progression, often referred to as Beer’s law:

T 10 abC or A abC

where a is the absorptivity of the component of interest in the solution. When C is expressed in moles per liter,

T 10 bC or A bC

where is the molar absorptivity.

The total fluorescence (or phosphorescence) intensity is proportional to the quanta of light absorbed, P0 P, and to the efficiency , which is the ratio of quanta absorbed to quanta emitted:

F (P0 P) P0 (1 e bC)

When the terms bC is not greater than 0.05 (or 0.01 in phosphorescence),

F k P0 bC

7.40

SECTION 7

where the term k has been introduced to handle instrumental artifacts and the geometry factor because fluorescence (and phosphorescence) is emitted in all directions but is viewed only through a limited aperture.

The thickness of a transparent film or the path length of infrared absorption cells b, in centimeters, is given by

	b	1		n
		2nD	v¯1 v¯2

where n is the number of fringes (peaks or troughs) between two wavenumbers v¯1 and v¯2, and nD is the refractive index of the sample material (unity for the air path of an empty cuvette). If measurements are made in wavelength, as micrometers, the expression is

	b	1		n 1 2
		2nD	2 1

7.4.1.3 Grating Equation. The light incident on each groove is diffracted or spread out over a range of angles, and in certain directions reinforcement or constructive interference occurs, as stated in the grating formula:

m b(sin i sin r)

where b is the distance between adjacent grooves, i is the angle of incidence, r is the angle of reflection (both angles relative to the grating normal), and m is the order number. A positive sign applies where incoming and emergent beams are on the same side of the grating normal.

The blaze wavelength is that wavelength for which the angle of reflectance from the groove face and the angle of reflection (usually the angle of incidence) from the grating are identical.

The Bragg equation

m 2d sin

states the condition for reinforcement of reflection from a crystal lattice, where d is the distance between each set of atomic planes and is the angle of reflection.

7.4.1.4 Ionization of Metals in a Plasma. A loss in spectrochemical sensitivity results when a free metal atom is split into a positive ion and an electron:

M M e

The degree of ionization, i, is defined as

[M ]

i [M ] [M]

At equilibrium, when the ionization and recombination rates are balanced, the ionization constant Ki (in atm) is given by

Ki	[M ][e ]			i2		p M
			1 i2
	[M]

where p M (in atm) is the total atom concentration of metal in all forms in the plasma.