Chapter 3- Answers to selected exercises

Transcription

1 Chater 3- Anwer to elected exercie. he chemical otential of a imle uid of a ingle comonent i gien by the exreion o ( ) + k B ln o ( ) ; where i the temerature, i the reure, k B i the Boltzmann contant, and the function o ( ) and o ( ) are well behaed. Show that thi ytem obey Boyle law, V Nk B. Obtain an exreion for the eci c heat at contant reure. What are the exreion for the thermal comreibility, the eci c heat at contant olume, and the thermal exanion coe cient? Obtain the denity of Helmholtz free energy, f f (; ). **** Note that (; ) g (; ), where g (; ) i the Gibb free energy er article. hu, k B ; which i the exreion of Boyle law, and d o d + k k B d o B ln o ( ) o ( ) d ; from which we obtain the eci c heat at contant reure. All other exreion are traightforward. In articular, f g You hould gie f in term of and, f f (; ).. Conider a ure uid of one comonent. Show that cv Ue thi reult to how that the eci c heat of an ideal ga doe not deend on olume. Show that ;N ;N

2 *** From the de nition of the eci c heat, we hae cv c V ) Note that (; ) i an equation of tate in the Helmholtz rereentation. hen, we write f f df d d ) ; ) ; which lead to the rt identity. he roof of the econd identity require imilar trick. 3. Conider a ure uid characterized by the grand thermodynamic otential V f o ( ) ex ; k B where f o ( ) i a well-behaed function. Write the equation of tate in thi thermodynamic rereentation. Obtain an exreion for the internal energy a a function of, V, and N. Obtain an exreion for the Helmholtz free energy of thi ytem. Calculate the thermodynamic deriatie and a a function of temerature and reure. *** From Euler relation, we hae hu, we can write V f o ( ) ex k B ln f o ( ) ; k B which i identical to the exreion for the chemical otential in the rt exercie, if we make o 0 and o ( ) f o ( ). herefore, we hae Boyle law and the uual exreion for and. 4. Obtain an exreion for the Helmholtz free energy er article, f f (; ), of a ure ytem gien by the equation of tate u 3 and a 4 ;

3 where a i a contant. *** hee equation of tate can be exlicitly written in the entroy rereentation, 4 3a u 4 and 3 4 3a u 34 ; from which we obtain the fundamental equation a u 34 + c; where c i a contant. he Helmholtz free energy er article i gien by " f (; ) u 3a a 3a c# *** Let u conider a imilar roblem, with a light modi cation in one of the equation of tate, u 3 and a n Note that, intead of 4, we are writing n, where n i an arbitrary integer. I thi a bona de thermodynamic ytem? I it oible to hae n 6 4? Again, we rewrite the equation of tate in the entroy rereentation, n 3a n u n and 3 n 3a +n u n In thi rereentation we hae ; u from which we obtain 3a n n +n u n 3 u ) u u n 3a +n 3 ; u n ; n

4 leading to the only thermodynamic bona de olution, n Obtain an exreion for the Gibb free energy er article, g g (; ), for a ure ytem gien by the fundamental equation S N 4 c a V U N ; 3 where a and c are contant. *** From the fundamental equation a 4 4 u + c; we write the equation of tate u a4 4 u and u 4 a4 34 u he Gibb free energy er article i gien by the Legendre tranformation g u ; where u and come from the equation of tate. Note that g ha to be gien in term of and. 6. Conider an elatic ribbon of length L under a tenion f. In a quaitatic roce, we can write du ds + fdl + dn Suoe that the tenion i increaed ery quickly, from f to f +f, keeing the temerature xed. Obtain an exreion for the change of entroy jut after reaching equilibrium. What i the change of entroy er mole for an elatic ribbon that behae according to the equation of tate LN cf, where c i a contant? *** Uing the Gibb rereentation, we hae the Maxwell relation S L f 4 f

5 From the equation of tate, LN cf, we hae S N cf f 7. A magnetic comound behae according to the Curie law, m CH, where C i a contant, H i the alied magnetic eld, m i the magnetization er article (with correction due to reumed urface e ect), and i temerature. In a quai-tatic roce, we hae du d + Hdm; where u u (; m) lay the role of an internal energy. For an in niteimal adiabatic roce, how that we can write CH c H H; where c H i the eci c heat at contant magnetic eld. *** We hae to calculate the artial deriatie ( ) at xed entroy. Uing Jacobian, it i eay to write (; ) (; ) (; H) (H; ) (; H) (H; ) All deriatie are written in term of the indeendent ariable and H. We then introduce the Legendre tranformation from which we hae g g u Hm ) dg d mdh; H ; m g ) H m H Inerting the equation of tate in thi Maxwell relation, it i eay to comlete the roof. 8. From tability argument, how that the enthaly of a ure uid i a conex function of entroy and a concae function of reure. 5

6 *** he entaly er article i gien by from which we hae dh d + d ) It i eay to how that h h u + ; h and > 0 c h Alo, we hae h < 0 It i traightforward to ue tandard trick (Jacobian, for examle) to write an exreion for the adiabatic modulu of comreibility, in term of oitie quantitie. *9. Show that the entroy er mole of a ure uid, (u; ), i a concae function of it ariable. Note that we hae to analyze the ign of the quadratic form d u (du) + u dud + (d) ; *** hi quadratic form can be written in the matrix notation du d u du d d u u he eigenalue of the matrix are the root of the quadratic equation u u u 6

7 For a concae function, the eigenalue are negatie, that i, u > 0 u and u + > 0 Now it i traightforward to relate thee deriatie of the entroy with oitie hyical quantitie (a the comreibilitie and the eci c heat). Feb. 0, 00 7