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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition · 356 pages

ISBN: 9780201380279

Chapter 7: Q. 7.66 (page 323)

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

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Most popular questions from this chapter

Each atom in a chunk of copper contributes one conduction electron. Look up the density and atomic mass of copper, and calculate the Fermi energy, the Fermi temperature, the degeneracy pressure, and the contribution of the degeneracy pressure to the bulk modulus. Is room temperature sufficiently low to treat this system as a degenerate electron gas?

(a) As usual when solving a problem on a computer, it's best to start by putting everything in terms of dimensionless variables. So define $t = T / T_{c}$ $c = μ / k T_{c}, and x = ϵ / k T_{c}$ . Express the integral that defines $μ$ , equation 7.122, in terms of these variables. You should obtain the equation

(b) According to Figure 7.33, the correct value of $c$ when $T = 2 T_{c}$ is approximately $- 0.8$ . Plug in these values and check that the equation above is approximately satisfied.

(c) Now vary $μ$ , holding $T$ fixed, to find the precise value of $μ$ for . Repeat for values of $T / T_{c}$ ranging from $1.2$ up to $3.0$ , in increments of $0.2$ . Plot a graph of $μ$ as a function of temperature.

In Section 6.5 I derived the useful relation $F = - k T \ln (Z)$ between the Helmholtz free energy and the ordinary partition function. Use analogous argument to prove that $ϕ = - k T \times \ln (\hat{Z})$ , where $\hat{Z}$ is the grand partition function and $ϕ$ is the grand free energy introduced in Problem 5.23.

For a system of particles at room temperature, how large must $ϵ - μ$ be before the Fermi-Dirac, Bose-Einstein, and Boltzmann distributions agree within $1 %$ ? Is this condition ever violated for the gases in our atmosphere? Explain.

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?

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