7.3 Heat capacities: extensive state variables (Hiroshi Matsuoka)

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1 7.3 Heat capacities: extensive state variables (Hiroshi Matsuoka) 1 Specific heats and molar heat capacities Heat capacity for 1 g of substance is called specific heat and is useful for practical applications. For example, the specific heat at constant pressure of liquid water at room temperature and under the atmospheric pressure is roughly 4 J ( g" K) while that of solid copper is roughly 0.4 J ( g" K). As mentioned above, heat capacities indicate how easily the temperature of the system can be changed by some heat flowing into the system. Having the larger specific heat at constant pressure, liquid water is therefore harder to warm up or cool down than a piece of solid copper with the same mass as the water. Molar heat capacity is actually more physical than specific heat because molar heat capacity corresponds to Avogadro s number of atoms or molecules and is therefore proportional to heat capacity per atom or molecule so that by comparing molar heat capacities of various systems, we can gain some insight into how the total energy of each of these systems is divided among its constituent atoms and molecules. More specifically, molar heat capacity at constant volume c v can tell us how the total energy is distributed among different types of kinetic and potential energies on the microscopic level. This is because c v is directly related to the molar internal energy u by c v = "u. v For example, look at the table of c v for various gases below. Molar heat capacity at constant volume c v for low-density gases at 15 C and 1 atm Substance c v R Ar 1.50 He 1.50 CO 2.49 H N CO

2 This table suggests that there is some connection between the value of c v and the internal structure of constituent molecules of gases as we find (we will address this issue in the next section). 2 Monatomic gases: c v ~ ( 3 2)R Diatomic gases: c v ~ ( 5 2)R Triatomic gases: c v ~ ( 7 2)R The table below also suggests that for elemental solids, c P ~ 3R = 25 J ( mol" K). Where does this constant of 3R comes from? We will discuss this question in the next chapter. Specific heats molar heat capacities for liquids solids at room temperature (~300 K) Substance Specific heat c P ( J ( g" K)) ( J ( mol" K)) Water Ethyl alcohol Copper Aluminum Gold Ice(-10 C) Heat capacity at constant volume, internal energy, entropy, and temperature We have thus far defined temperature as a label for equilibrium states so that two states with the same value of temperature are in thermal equilibrium with each other. We have also noted that energy flows as heat from a system with a higher temperature to another with a lower temperature so that we have effectively ordered equilibrium states according to their temperatures: if heat flows from a system to another, the we know the temperature of the former is higher than the latter. What is then temperature anyway, especially on the microscopic level? It is a difficult question to answer at this point because temperature is a concept closely coupled with entropy, a concept we have not defined yet. Also, in discussing the equations of state for gases, we have mentioned that as we increase the temperature of a gas, the average speed as well as the average kinetic energy of the molecules in the gas increase. How is the temperature of a system related to its kinetic energy after all?

3 To give you some idea, let s first look at a diatomic gas such as a nitrogen gas in the air at room temperature. As mentioned above, the heat capacity at constant volume of the nitrogen gas at room temperature is 3 C V = 5 2 nr. As the internal energy U is related to C V by "U = C V, we obtain U = 5 2 nrt, where we have set U( T = 0 K,V,n) = 0. It turns out that this internal energy is a sum of two terms, one of which comes from the kinetic energy for the translational motion of each molecule as a whole and the other of which comes from the kinetic energy for the rotational motion of each molecule: U = U trans + U rot = 3 2 nrt + nrt = 3 2 Nk B T + Nk B T, where N is the total number of the molecules and satisfies nr = Nk B. Clearly, the average kinetic energy per molecule ( 3 2)k B T is proportional to the temperature of the gas. As we will see in Ses.8.1, the entropy of the gas is given by (" S( T,V,n) = S ( T 0,V 0,n ) + nrln * $ T * T 0 ) ' 5 2 " V + $ '- V 0,-,

4 where a reference state is chosen to be at ( T 0,V 0,n). We can then express the entropy as a function of the internal energy U: 4 (" S( U,V,n) = S ( U 0,V 0,n ) + nrln * $ U * U 0 ) ' 5 2 " V + $ '- V 0,-, where U = ( 5 2)nRT and U 0 = ( 5 2)nRT 0. We the find "S = 5 2 nr 1 U = 1 T, which is also a general relation between entropy and temperature: a rate of change of entropy with respect to a change in internal energy is always the inverse temperature, 1 T. As we will see later that the entropy of a system is, on the microscopic level, related to the number W of energy eigenstates whose energy eigenvalues are near U or in an interval [ U,U + du], where du << U, by S( U,V,n) = k B lnw ( U,V,n). W basically counts the number of ways of dividing the energy U among the constituent molecules. The above relation between S and T then becomes a relation between W and T: 1 W "W = " lnw = 1 "S k B = 1 k B T, which means that the fractional increase of W is proportional to 1 T : the higher the temperature is, the smaller the fractional increase of the number of eigenstates degenerate in energy becomes. W is analogous to the surface area A = 4"r 2 of a sphere while U is analogous to its diameter D and k B T is analogous to its radius r as 1 da A dd = 1 d 4"r 2 $ 2dr 4"r2 ( = 1 ' r.

5 As another example, consider an insulator solid whose heat capacity at constant volume of the nitrogen gas at low temperatures much lower than its Debye temperature " is found (see Sec7.4.2) to be 5 C V = anr T 3, where a is a positive constant. The internal energy U is then U = 1 4 anr" T 4, where we have set U( T = 0 K,V,n) = 0 and " is a function of the molar volume v of the solid. It turns out that this internal energy is a sum of two terms, one of which comes from the vibrational kinetic energy of each atom and the other of which comes from the potential energy among the atoms due to inter-atomic forces: U = U kin + U pot = 1 8 anr" T anr" T 4 = 1 8 ank B" T ank B" T 4, where N is the total number of the atoms and satisfies nr = Nk B. In this case, the average kinetic energy per atom is not simply proportional to the temperature of the solid. The entropy of the solid is then given by ( ) = 1 3 anr T $ " S T,V,n 3 (, ' where we have used the third law of thermodynamic: S( T = 0 K,V,n) = 0. We can then express the entropy as a function of the internal energy U: S( U,V,n) = 1 3 4U anr ( $ anr" ' 3 4

6 so that we obtain the general relation between S and T as we have found for the nitrogen gas at room temperature: 6 "S = 1 anr) = 1 ) $ 4U ' T. At temperatures higher than the Debye temperature ", C V is found (see Sec7.4.2) to be C V " 3nR, so that the internal energy U is roughly U " 3nRT, where we have set U( T 0,V,n) = 3nRT 0 and T 0 is a reference temperature that satisfies T 0 >> ". This internal energy is also a sum of the total vibrational kinetic energy of the atoms and the total potential energy among the atoms due to inter-atomic forces: U = U kin + U pot = 3 2 nrt nrt = 3 2 Nk B T Nk B T, Note that the average kinetic energy per atom is proportional to the temperature of the solid. The entropy of the solid is then given by S( T,V,n) = S ( T 0,V,n ) + 3nRln $ T T 0 " ', so that we can express the entropy as a function of the internal energy U: and S( U,V,n) = S ( U 0,V,n ) + 3nRln $ U U 0 "S = 3nR U = 1 T. " '

7 7 Appendix: A more careful derivation of C P = C V + TV" 2 T To derive this equation, we use the first law of thermodynamics or the law of conservation of energy. Suppose that our system is initially at an equilibrium state ( T,V,n), where the pressure is determined through the equation of state P = P( T,V,n). If we slowly change the temperature and volume by infinitesimal amounts dt and dv while keeping the pressure constant, the system ends up in an equilibrium state ( T + dt,v + dv,n) for which the pressure remains at P. The volume change dv is related by dv = "V dt = V 1 "V P.n V P.n dt = V)dT. According to the first law, in this infinitesimal quasi-static isobaric (i.e., constant-pressure) process, the internal energy changes by du P,n = "Q qs P,n +"W qs P,n = C P dt PdV. We can also start from the same initial state and end with the same final state through a different route. We first slowly change the temperature by dt while keeping the volume constant and then slowly change the volume from V to V + dv while keeping the temperature at T + dt. In the first leg of this roundabout process, the internal energy changes by du = "Q qs +"W qs = C V dt, where "W qs = PdV = 0 as dv = 0. In the second leg of this roundabout process, the internal energy changes by du T +dt,n = "Q qs T +dt,n +"W qs T +dt,n = T + dt T $ T dv PdV = T $ T ( ) ( T + dt) $ T ( T + dt) dv P ( T + dt,v,n )dv VdT ( ) PdV = TV 2 $ T dv PdV

8 where and ( ) ( ) ( T + dt) " T + dt T T + dt " 1+ ( ) - dv = T + dt + 1 $" (, ' * " $T ) + T 1+ 1 $ T (, ' * - T $T ) 1 T " dv + O( dtdv) T P( T + dt,v,n)dv = P( T,V,n) + O( dtdv ). dt/ 0 dv. dt/ 0 8 so that we have neglected terms on the order of or smaller than dtdv. Finally, as the internal energy is a state variable, its change du P,n must be equal to the difference between its final value U( T + dt,v + dv,n) and initial value U( T,V,n) so that ( ) "U( T,V,n) { ( ) "U( T + dt,v,n) } + { U( T + dt,v,n) "U( T,V,n) } du P,n = U T + dt,v + dv,n = U T + dt,v + dv,n = du T +dt,n + du from which we get C P dt " PdV = TV 2 ( ' dt " PdV * + C V dt = C V + TV 2 ( ' * dt " PdV $ T ) ) and C P = C V + TV " 2. T $ T This derivation shows that the second term TV" 2 qs T comes from "Q T +dt,n because "Q qs T +dt,n is positive as long as dt is positive. and that C P " C V

9 SUMMARY FOR SEC The heat capacity at constant volume C V of a system is an extensive state variable related to the internal energy U of the system by C V = "U = nc v, where c v is the molar heat capacity at constant volume and is related to the molar internal energy u by c v = "u. v 2. imply "U > 0 and "u C V > 0 and c v > 0. v > 0 3. The heat capacity at constant pressure C P of a system is an extensive state variable related to the enthalpy H " U + PV of the system by C P = "H P,n = nc P, where c P is the molar heat capacity at constant pressure and is related to the molar enthalpy h by c P = "h. 4. The enthalpy H of a system is an extensive state variable defined by P H " U + PV = n( u + Pv) = nh( T,P). 5. C P and C V are related with each other by C P = C V + TV " 2 6. c P and c v are related with each other by c P = c v + Tv " 2 T $ C V. T $ c v.

10 For a low-density gas, for which the ideal gas law, Pv = RT, is a good approximation for its equation of state, we find c P = c v + R. 7. As C P " C V > 0 and c P " c v > 0, 10 "H P,n > 0 and "h P > 0, which imply that H as a function of T, P, and n is an increasing function of T and that h as a function of T and P is an increasing function of T. 8. We can calculate the molar enthalpy h of a system by ( T,P)d T ". h( T,P) = h( T 0,P) + c P " 9. We can calculate the molar internal energy U of a system by u( T,P) = h( T,P) " Pv( T,P). 10. At room temperature and under the atmospheric pressure Monatomic gases: c v ~ ( 3 2)R Diatomic gases: c v ~ ( 5 2)R Triatomic gases: c v ~ ( 7 2)R T T 0 Elemental solids: c P ~ 3R Answer for the homework question in Sec.7.3 HW7.3.1 For low-density gases: Pv = RT, which leads to so that! = 1 and! T T = 1 P c v = c P! Tv " 2 ( )2 = c P! Tv 1 T T 1 P ( ) = c P! Pv T = c P! R.

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