class12lecnotes

class12lecnotes - Class 12, W~sday, February 3,2010...

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Class 12, W~sday, February 3,2010 Reading: 1'1, / ,; / cr. z.. A microscopic view of the Venturi tube: discussion Last class we talked about the necessity for gases to have different specific heats when they undergo transformations under constant volume or under constant pressure. The table below lists heat capacities for different substances. Do you notice any interesting patterns? We can divide them up into groups. Why are the heat capacities for group A the same? Why for group B and C? We will address these questions below. Substance C p e v c c p Cv He 2O.S 125 I 5200 3125 I 10S0 20.S 12.5 I 520 -.----. ...-~--------- 2S.7 20.4 I 14,350 29.1 20.S I 1,039 29.2 20.9 I 9,125 --,-"¥-----~ ~--,---~.- -- -~- "-- j ., AI I 24.3 900 tifc/nWJ CJ (;< /McIJ!/IIJ Fe 25.1 449 dl th (MJ?1J fr-i.M-Y' "ll' Gold 25.4 129 Ice 37.6 2,090 Water 75.4 4,190 Microscopic view: !J [ .:= 1- n ~. b T (1M (J.£)rk Macroscopic view: At:; n C ~,/j T , , ~ .3 ! c: ) j/ R·¥ = J'1 Cf/·t/ =-> C v ;- T £. . For an ideal gas at constant volume, we can determine the mo ar specific heat, C v , 3 £J _ .1, tJ JI ,L - - ," - 2 o. J II 2 /11rr'K JYV~ Check this with the table ostdilolu'sujmw i §e. It's a pretty good agreement!
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How is the energy 3/2-nkT. distributed among the atoms? The Equipartition Theorem The thermal energy of a system is equally divided between all possible degrees of freedom of its constituent particles. For a system of N particles at temperature T. the energy per degree of freedom is -1 IIi T t)y J kif. T ()r' ~ leT jUi" /,tJwhdlJjVr 2 Z. 2 v/tYeL ~ /rt1'/tJWf The proof for this is based on statistical mechanics and is too advanced for the purpose of this class. In a monoatomic gas. atoms have three independent modes of storing energy. This corresponds to their three degrees of freedom associated with their translational motion along each of the three axes in space. 3x~NJ.T r .2 => r= If/if) ["iN;'! 2 For solids. there are 6 degrees of freedom. You can figure these out by thinking about the atoms in solids as connected by elastic springs. They are allowed to oscillate along all three spatial directions around their points of equilibrium. These modes are analogous to the translational modes in a monoatomic gas_ In a solid. however. one cannot neglect the potential energy of the system resulting from the electrostatic interaction between the particles. These are like the potential energy stored in a compressed spring. The spring can be stretched/compressed in all three directions as well. Thus additional three degrees of freedom result from this. The intf?t/lJ. energy of a solid thus is: Fe 6 x-INk T ~)g 3A1kTJ Z = JJI£T We can now find out the molar specific heat for solids: JE=- 317 R .1:J T 3//R jif:p{ I;YT Q: I/! c!J T= (J {6. T At:: Q ~>f(= 3 12 1 If we compare this with the table at the end of the notes. we see that there are small differences in the molar specific heats of solids. These result from relaxing simplifying 12,2
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assumptions made, as for example treating the bonds as elastic springs, and from quantum effects.
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class12lecnotes - Class 12, W~sday, February 3,2010...

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