Pyrometry: A Practical Treatise on the Measurement of High Temperatures

by Charles R. Darling

Wien’s Law.—When the temperature of a substance increases, the enhanced brightness which results is shared by all parts of its spectrum; and if the substance were viewed through a glass prism, it would be noticed that every portion was brighter than before. Taking a ray of wave-length λ, the relation between its intensity and the temperature of the (black-body) source is given by Wien’s formula:—

J = c₁λ^-5 × e^-(c₂/λT)  (1)

where J = energy corresponding to wave-length λ; e = the base of the natural system of logarithms; T = absolute (thermodynamic) temperature of the black-body source, and c₁ and c₂ are constants, the values of which may be found by measuring J at two known temperatures for light of a known wave-length. Experiment has shown that this formula is correct for wave-lengths which lie in the visible spectrum, but does not hold for longer waves; and modifications of Wien’s equation have been given by Planck and others which are of more extended application. For the purposes of optical pyrometry, however, using red light of wave-length about 65 millionths of a centimetre, Wien’s law may be applied with great accuracy; and a calibration based upon this law agrees closely with the values obtained by other pyrometric methods.

Wien’s formula may be written in the form—

log₁₀ J = K₁ + K₂ (¹⁄_T)  (2)

where K₁ = (log c₁) - (5 log λ) and K₂ = c₂(^{log e}⁄_λ). This simplified expression shows a linear relation between log J and ¹⁄_T; and hence if the temperatures corresponding to two intensities be known, the results may be plotted on squared paper in the form of a straight line connecting T and J, from which line intermediate or extraneous readings of temperatures may be obtained for any given intensity. Another useful form of Wien’s equation, referring to the ratio of two intensities J₁ and J₂, is as under:—

			c2 log e
log	( J1 ⁄ J2 )	=	───	( 1 ⁄ T2 - 1 ⁄ T1 )  (3)
			λ

where T₂ and T₁ are the absolute temperatures corresponding to J₂ and J₁ The value of c₂ is 1450000, when λ is expressed in millionths of a centimetre. Evidently, if the ratio J₁ / J₂ and the value of c₂, λ, and T₂ be known, T₁ may be calculated. When λ is not known, as in the case of a piece of red glass for which its value has not been determined, two readings at known temperatures will establish the value of (c₂ log e) / λ, all the other results may then be calculated. Examples illustrating the application of the formula will now be given.