Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Chapter 9, Problem 9.32E

Calculate the power of light in the wavelength range λ = 350 nm 351 nm (that is, let d λ be Δ λ = 1 nm in the Planck’s law, and let λ be 350.5 nm ) at temperatures of 1000 K , 3000 K and 10 , 000 K .

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Interpretation Introduction

Interpretation:

The power of light in the wavelength range λ=350351nm at temperatures 1000K, 3000K and 10,000K is to be calculated.

Concept introduction:

Planck’s equation can also be represented in the form of power flux or infinitesimal power per unit area for a black body radiation at a given temperature and wavelengths as shown below.

dξ=2πhc2λ5(1ehc/λkT1)dλ

Answer to Problem 9.32E

The power of light in the wavelength range λ=350351nm at the temperatures 1000K, 3000K and 10,000K is 1.01×1010W/m2, 80.21W/m2 and 1.18×106W/m2 respectively.

Explanation of Solution

It is given that wavelength range is 350nm351nm, the value of dλ is 1nm and λ is 350.5nm.

The formula to calculate the power of light is,

dξ=2πhc2λ5(1ehc/λkT1)dλ

Where,

ξ is the power flux.

λ is the wavelength.

h is the Planck’s constant.

k is the Boltzmann constant.

c is the speed of light.

Substitute the values of constants and temperature 1000K to calculate the power of light in the given formula.

dξ=(2(3.14)(6.626×1034Js)(3×108m s1)2(350.5×109 m)5(1e6.626×1034Js×3×108ms1350.5×109m×1.38×1023JK-1×1000K1)×(1×109m))=3.745×10165.289×1033×(1e1.9878×10254.8369×10271)×1×109 Js1m2=3.745×10165.289×1033×17.00×10171×1×109 Js1m2=7.08×1016×1.428×1018×1×109 Js1m2

On simplifying the above equation,

1 W=1 Js1

dξ=1.01×1010W/m2

Thus, the value of power flux is 1.01×1010W/m2.

Substitute the values of constants and temperature 3000K to calculate the power of light in the given formula.

dξ=(2(3.14)(6.626×1034Js)(3×108ms1)2(350.5×109)5(1e6.626×1034Js×3×108ms1350.5×109m×1.38×1023JK-1×3000K1)(1×109m))=3.745×10165.289×1033×(1e1.9878×10251.451×10261)×1×109 Js1m2=3.745×10165.289×1033×18.8×1051×1×109 Js1m2=7.08×1016×1.133×106×1×109 Js1m2

On simplifying the above equation,

1 W=1 Js1

dξ=1.01×1010W/m2

Thus, the value of power flux is 80.21W/m2.

Substitute the values of constants and temperature 10,000K to calculate the power of light in the given formula.

dξ=(2(3.14)(6.626×1034Js)(3×108ms1)2(350.5×109)5(1e6.626×1034Js×3×108ms1350.5×109m×1.38×1023JK-1×10000K1)(1×109m))=3.745×10165.289×1033×(1e1.9878×10254.8369×10261)×1×109 Js1m2=3.745×10165.289×1033×160.881×1×109 Js1m2=7.08×1016×0.0167×1×109 Js1m2

On simplifying the above equation,

1 W=1 Js1

dξ=1.01×1010W/m2

Thus, the value of power flux is 1.18×106W/m2.

Conclusion

The power of light at the temperatures 1000K, 3000K and 10,000K is 1.01×1010W/m2, 80.21W/m2 and 1.18×106W/m2 respectively.

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