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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition · 356 pages

ISBN: 9780201380279

Chapter 7: Q 7.37 (page 293)

Prove that the peak of the Planck spectrum is at x = 2.82.

Short Answer

Expert verified

Hence proved that plank's spectrum is at x = 2.82.

Step by step solution

01

Given information

The peak of the Planck spectrum is at e = 2.82.

02

Explanation

Set the derivative of $x^{3} / (e^{x} - 1)$ with respect to x equals to zero to get the maximum, then solve for:

$\frac{\partial}{\partial x} (\frac{x^{3}}{e^{x} - 1}) = 0 \frac{3 x^{2} (e^{x} - 1) - x^{3} e^{x}}{{(e^{x} - 1)}^{2}} = 0 3 x^{2} (e^{x} - 1) - x^{3} e^{x} = 0 3 x^{2} e^{x} - 3 x^{2} - x^{3} e^{x} = 0 3 e^{x} - 3 - x e^{x} = 0$

Using matlab to solve the above equation: The code is:

Therefore, $x = 2.8214$

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Most popular questions from this chapter

At the center of the sun, the temperature is approximately $10^{7} K$ and the concentration of electrons is approximately $10^{32}$ per cubic meter. Would it be (approximately) valid to treat these electrons as a "classical" ideal gas (using Boltzmann statistics), or as a degenerate Fermi gas (with $T \approx 0$ ), or neither?

Evaluate the integrand in equation $7.112$ as a power series in x, keeping terms through $x^{4}$ • Then carry out the integral to find a more accurate expression for the energy in the high-temperature limit. Differentiate this expression to obtain the heat capacity, and use the result to estimate the percent deviation of $C_{v}$ from $3 N k a t T = T_{D} a n d T = 2 T_{D .}$

The planet Venus is different from the earth in several respects. First, it is only $70 %$ as far from the sun. Second, its thick clouds reflect $77 %$ of all incident sunlight. Finally, its atmosphere is much more opaque to infrared light.

(a) Calculate the solar constant at the location of Venus, and estimate what the average surface temperature of Venus would be if it had no atmosphere and did not reflect any sunlight.

(b) Estimate the surface temperature again, taking the reflectivity of the clouds into account.

(c) The opaqueness of Venus's atmosphere at infrared wavelengths is roughly $70$ times that of earth's atmosphere. You can therefore model the atmosphere of Venus as $70$ successive "blankets" of the type considered in the text, with each blanket at a different equilibrium temperature. Use this model to estimate the surface temperature of Venus. (Hint: The temperature of the top layer is what you found in part (b). The next layer down is warmer by a factor of $2^{1 / 4}$ . The next layer down is warmer by a smaller factor. Keep working your way down until you see the pattern.)

Consider the electromagnetic radiation inside a kiln, with a volume of V= I m³ and a temperature of 1500 K.

(a) What is the total energy of this radiation?

(b) Sketch the spectrum of the radiation as a function of photon energy.

(c) What fraction of all the energy is in the visible portion of the spectrum, with wavelengths between 400 nm and 700 nm?

$C o n s i d e r a c o l l e c t i o n o f 10, 000 a t o m s o f r u b i d i u m - 87, c o n f i n e d i n s i d e a b o x o f v o l u m e {(10^{- 5} m)}^{3} .$ $(a) C a l c u l a t e ε_{0}, t h e e n e r g y o f t h e g r o u n d s t a t e . (E x p r e s s y o u r a n s w e r i n b o t h j o u l e s a n d e l e c t r o n - v o l t s .)$ $(b) Calculate the condensation temperature, and compare k T_{c} to ϵ_{0} .$ $(c) S u p p o s e t h a t T = 0.9 T_{c} . H o w m a n y a t o m s a r e i n t h e g r o u n d s t a t e ? H o w c l o s e i s t h e c h e m i c a l p o t e n t i a l t o t h e g r o u n d - s t a t e e n e r g y ? H o w m a n y a t o m s a r e i n e a c h o f t h e (t h r e e f o l d - d e g e n e r a t e) f i r s t e x c i t e d s t a t e s ?$ $(d) R e p e a t p a r t s (b) a n d (c) f o r t h e c a s e o f 10^{6} a t o m s, c o n f i n e d t o t h e s a m e v o l u m e . D i s c u s s t h e c o n d i t i o n s u n d e r w h i c h t h e n u m b e r o f a t o m s i n t h e g r o u n d s t a t e w i l l b e m u c h g r e a t e r t h a n t h e n u m b e r i n t h e f i r s t e x c i t e d s t a t e .$

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