{"id":38,"date":"2022-05-23T14:04:47","date_gmt":"2022-05-23T14:04:47","guid":{"rendered":"https:\/\/academic.csuohio.edu\/kaufman-miron\/?page_id=38"},"modified":"2022-05-23T15:30:21","modified_gmt":"2022-05-23T15:30:21","slug":"homework","status":"publish","type":"page","link":"https:\/\/academic.csuohio.edu\/kaufman-miron\/phy475-statistical-physics-spring-2017\/homework\/","title":{"rendered":"Homework"},"content":{"rendered":"\n

1) For a simple thermodynamic system show that:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

 2) Compute the slope of the coexistence line dp\/dT in Pa\/K for water in the following cases: (a) boiling under atmospheric pressure: T = 100 C, latent heat l = 540cal\/g, vgas<\/sub> = 1.6729 l\/g, vliq<\/sub> = 1.044*10-3<\/sup> l\/g; (b) freezing under atmospheric pressure: T = 0 C, latent heat l = 80 cal\/g, vice<\/sub> = 1.25 cm3<\/sup>\/g, vliq<\/sub> = 1.0 cm3<\/sup>\/g.<\/p>\n\n\n\n

 DUE:<\/strong><\/p>\n\n\n\n

\n\n\n\n

HOMEWORK #2<\/h2>\n\n\n\n

1) Study the accuracy of the Stirling approximation by computing the fractional error:<\/p>\n\n\n\n

e (N) = 1 – (NlnN – N)\/ln(N!)<\/p>\n\n\n\n

for N = 1, 2, …10. Sketch e (N).<\/p>\n\n\n\n

2) (a) Consider four objects {a, b, c, d}. List all permutations of these 4 objects. How many permutations did you find?<\/p>\n\n\n\n

(b) You have four identical (indistinguishable) marbles and three boxes. List all possible ways of distributing the 4 marbles in the 3 boxes. How many ways there are?<\/p>\n\n\n\n

DUE:<\/strong><\/p>\n\n\n\n

\n\n\n\n

HOMEWORK #3<\/h2>\n\n\n\n

CALLEN: CH.15, SECT.2, PR.2<\/p>\n\n\n\n

For lead U \ufffd 3NkB<\/sub>T for temperatures above about 90K, while for diamond this holds for temperatures above about 2000K. By using the Einstein model estimate the angular frequency w , and the elastic constant k for Pb and C (diamond).<\/p>\n\n\n\n

DUE:<\/strong><\/p>\n\n\n\n

\n\n\n\n

HOMEWORK #4<\/h2>\n\n\n\n

For the two-state model graph: (a) the fundamental equation s(u); (b) T(u). Discuss the negative temperature phenomenon.<\/p>\n\n\n\n

DUE:<\/strong><\/p>\n\n\n\n

\n\n\n\n

HOMEWORK #5<\/h2>\n\n\n\n

CALLEN: CH.16, SECT.2, PR.6<\/p>\n\n\n\n

Calculate the fractional energy fluctuation for the Einstein crystal and for an ideal gas. Hint: use Gibbs – Einstein formula.<\/p>\n\n\n\n

DUE:<\/strong><\/p>\n\n\n\n

\n\n\n\n

HOMEWORK #6<\/h2>\n\n\n\n

Calculate the partition function Z and the Helmholtz free energy F for the classical <\/em>crystal by using the canonical ensemble. (b) Calculate the energy, the entropy and the heat capacity.<\/p>\n\n\n\n

DUE:<\/strong><\/p>\n\n\n\n

\n\n\n\n

Homework #7<\/h2>\n\n\n\n

(a) Prove the Wien displacement law: l m<\/sub>*T = 2.898*10-3<\/sup> m*K, where l m <\/sub>is the wavelength for which the spectral energy density dU\/dl is maximum.<\/p>\n\n\n\n

(b) Compute l m <\/sub>at room temperature. Check it is in the infrared part of the spectrum.<\/p>\n\n\n\n

(c) The maximum spectral energy density dU\/dl in the sun\ufffds spectrum occurs at a wavelength of 4700A. What is the surface temperature of the sun?<\/p>\n\n\n\n

(d) Compute the heat capacity of radiation inside a container of volume 1liter at room temperature.<\/p>\n\n\n\n

DUE:<\/p>\n\n\n\n

\n\n\n\n

HOMEWORK #8<\/h2>\n\n\n\n

For each statistical mechanics ensemble (microcanonical, canonical, grandcanonical) prove the following formula for the entropy:<\/p>\n\n\n\n

S = -kB<\/sub> S Pj<\/sub> ln(Pj<\/sub>)<\/p>\n\n\n\n

where Pj<\/sub> is the probability of microstate j.<\/p>\n\n\n\n

2) The grand canonical potential W for an ideal gas is: W = -kB<\/sub>T*V*exp(m \/kB<\/sub>T)*1\/l T<\/sub>3<\/sup><\/p>\n\n\n\n

Show: (a) U = 3pV\/2; (b) pV = NkB<\/sub>T. (c) Determine the fundamental equation: m = m (T,p).<\/p>\n\n\n\n

DUE:<\/strong><\/p>\n\n\n\n

\n\n\n\n

HOMEWORK #9<\/h2>\n\n\n\n

Consider the ideal Fermi gas. Prove the following formula for the entropy:<\/p>\n\n\n\n

S = -kB<\/sub> S [nj <\/sub>ln(nj <\/sub>) + (1 – nj <\/sub>)ln(1 – nj<\/sub>)]<\/p>\n\n\n\n

where nj<\/sub> is the average number of fermions occupying quantum state #j.<\/p>\n\n\n\n

By using the free-electron approximation for metals compute: the Fermi energy in eV, the Fermi temperature in K, the pressure in Pa, the bulk modulus in Pa, for gold. The number of free electrons per volume is N\/V = 5.9*1028<\/sup>m-3<\/sup>.<\/p>\n\n\n\n

DUE:<\/strong><\/p>\n\n\n\n

\n\n\n\n

HOMEWORK #10<\/h2>\n\n\n\n

1. Consider the ideal Bose-Einstein gas. Prove the following formula for the entropy:<\/p>\n\n\n\n

S = -kB<\/sub> S [nj <\/sub>ln(nj <\/sub>) – (1 + nj <\/sub>)ln(1 + nj<\/sub>)]<\/p>\n\n\n\n

 where nj<\/sub> is the average number of bosons occupying quantum state #j.<\/p>\n\n\n\n

2. Compute the Bose condensation temperature for He4 if N\/(gV) = 2.2*1028<\/sup>m-3<\/sup> and m = 6.65*10-27<\/sup> Kg.<\/p>\n\n\n\n

3. Consider the ideal Bose-Einstein gas of ultrarelativistic particles. The energy and momentum are related through the relativistic formula e = pc. (a) Determine the grand-canonical potential W , the energy U, and the number of bosons N as functions of T, V, and m . (b) Determine the condensation temperature TC <\/sub>as a function of N\/V.<\/p>\n\n\n\n

DUE:<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

1) For a simple thermodynamic system show that:  2) Compute the slope of the coexistence line dp\/dT in Pa\/K for water in the following cases: (a) boiling under atmospheric pressure: T = 100 C, latent heat l = 540cal\/g, vgas = 1.6729 l\/g, vliq = 1.044*10-3 l\/g; (b) freezing under atmospheric pressure: T = 0 C, latent heat…<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":36,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_relevanssi_hide_post":"","_relevanssi_hide_content":"","_relevanssi_pin_for_all":"","_relevanssi_pin_keywords":"","_relevanssi_unpin_keywords":"","_relevanssi_related_keywords":"","_relevanssi_related_include_ids":"","_relevanssi_related_exclude_ids":"","_relevanssi_related_no_append":"","_relevanssi_related_not_related":"","_relevanssi_related_posts":"1","_relevanssi_noindex_reason":"","footnotes":""},"class_list":["post-38","page","type-page","status-publish","hentry"],"featured_image_src":null,"_links":{"self":[{"href":"https:\/\/academic.csuohio.edu\/kaufman-miron\/wp-json\/wp\/v2\/pages\/38","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/academic.csuohio.edu\/kaufman-miron\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/academic.csuohio.edu\/kaufman-miron\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/academic.csuohio.edu\/kaufman-miron\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/academic.csuohio.edu\/kaufman-miron\/wp-json\/wp\/v2\/comments?post=38"}],"version-history":[{"count":2,"href":"https:\/\/academic.csuohio.edu\/kaufman-miron\/wp-json\/wp\/v2\/pages\/38\/revisions"}],"predecessor-version":[{"id":59,"href":"https:\/\/academic.csuohio.edu\/kaufman-miron\/wp-json\/wp\/v2\/pages\/38\/revisions\/59"}],"up":[{"embeddable":true,"href":"https:\/\/academic.csuohio.edu\/kaufman-miron\/wp-json\/wp\/v2\/pages\/36"}],"wp:attachment":[{"href":"https:\/\/academic.csuohio.edu\/kaufman-miron\/wp-json\/wp\/v2\/media?parent=38"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}