Specific Heat Capacity
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1 Specific Heat Capacity Equation of state describes all material properties. here are many relationships between thermodynamic quantities. Measuring one quantity makes it possible to calculate others Specific heat capacity (temperature is easy to measure olume (for crystals is easily measured October 20, / 17
2 he difference in heat capacities, C C What is the general relation between C and C? C C = nr is valid only for an Ideal gas Constant volume = no work. du = ds d = ds ( ( U S C = d Q /d = = his suggests working with the entropy : S = S(, ( ( S S ds = d + d o relate to C p = ( H/ p = ( S/ p, differentiate the equation wrt at constant. ( ( ( ( ( S S S C = = + October 20, / 17
3 Identify terms which are materials properties ( ( ( S S C = + Introducing heat capacity ( S C = and isobaric thermal expansivity: β = 1 ( ( S C C = β October 20, / 17
4 Eliminate the unmeasurable ( S C C = β Recall S is not readily measurable, so use Maxwell to eliminate S. ( S = ( = ( ( and introduce the isothermal bulk modulus and thermal expansivity (again ( ( 1 = = 1 K β = Kβ K = ( and β = 1 ( C C = β 2 K = β 2 /κ where κ = 1/K is the isothermal compressibility. October 20, / 17
5 Deconstruct C C = β 2 K = β 2 /κ C C = extensive quantity. K is positive for all known substances. β 2 is positive. C > C, even for negative thermal expansion. C = C at the density maximum in water. β is small except for gases. C C : for solids and liquids often lazily just give heat capacity Difference is NO due to work done expanding the material. October 20, / 17
6 ariations in C and C What is ( C ( C? ( ( S = ( ( = ( ( S = ( 2 = 2 using a Maxwell relation. An analogous analysis for C yields ( ( C 2 = 2 We can calculate these directly from equation of state. is Zero for Ideal Gas =R/. e.g.( C formally, infinite for a phase transition (volume changes at constant,.. October 20, / 17
7 Aside: Latent heats and the Lambda Function Heat capacity of liquid He becomes infinite at a phase transition. L = +δt δt c v d October 20, / 17
8 he energy equation. Why demonstrations don t work. Relate the volume and pressure derivatives of the internal energy to material properties and gradients of equation of state. Differentiate du = ds d, wrt d and eliminate S using a Maxwell relation: ( U = ( ( S = = β κ Is force the derivative of energy? Compare F = U, = ( U S October 20, / 17
9 Increasing energy under pressure Similarly, differentiate du = ds d, wrt d and eliminate S using a Maxwell relation: ( ( ( U = = β + κ October 20, / 17
10 he ratio of heat capacities C /C It can be shown using Maxwell relations that C C = κ κ S where κ and κ S are, respectively, the isothermal and adiabatic compressibilities: κ = 1 ( and κ S = 1 ( S his provides another useful link between thermal and mechanical properties of materials. October 20, / 17
11 he entropy of an ideal gas, again For entropy per mole s = s(,, we can always write ( ( s s d ds = d + d = c v + βkd his equation applies to any fluid. β is thermal expansivity, K bulk modulus. For an Ideal gas βk = R/v and c v is a constant. Integration then gives s = c v ln + R ln v + s 0 Similarly s = c ln R ln + s 0. Again, we relate changes in entropy to measurable quantities via the equation of state. October 20, / 17
12 Availability 1904 rince iero Ginori Conti. generating electric energy from geothermal steam How much work can be extracted from a system? Depends on surroundings, but how? Availability is the thermodynamic quantity to calculate. October 20, / 17
13 Availability - non Infinite reservoirs Second-law for system with surrounding reservoir at 0, 0 S + S surr 0 S Q 0 0 Q is heat transferred from the reservoir into the system. First law for system U = Q 0, allow us to eliminate Q: U S 0 Define a new function called the Availability, A, A = U 0 S + 0 October 20, / 17
14 Availability A = U 0 S + 0 Availability is not Gibbs free energy. Depends on both the system and surroundings. A(S,, 0, 0, Spontaneous changes in availability are always negative A = U 0 S Availability is minimised when system and surroundings are in equilibrium ( = 0, = 0. da = du ds + d = 0 October 20, / 17
15 Availability incorporates all potentials If = 0 & = 0 A = U S + G is minimum If = 0 & = const A = U S + const F is minimum If S is const. & = const A = U + const U is minimum If U is const. & = const A = const 0 S S is maximum Minimising Availability maximises entropy of the Universe. Availability is minimised when system and surroundings are in equilibrium. Availability tells us how far from equilibrium we are October 20, / 17
16 Useful Work here was a subtle assumption in the previous section that the only work done by the system was in expanding against the environment 0. In general, system could produce other useful work. In that case the First Law becomes U = Q W = Q W useful 0 (1 where Q is the heat transported from the surroundings to the system, U S + W useful = W useful A 0 Maximum useful work is W max = A. (iff all changes are reversible. his is also clearly seen if we look at a small differential change of A: dw useful da = du + 0 ds 0 d = ( 0 ds + ( 0 d October 20, / 17
17 Available for work dw useful ( 0 ds + ( 0 d Useful work could come from moving entropy (heat from hot body ( to cold body ( 0. pushing a piston against a pressure ( 0 Once the availability is used up - no more work. n.b. Idealised engines had two infinite reservoirs, hot and cold. Real engines use up energy to maintain the temperatures. October 20, / 17
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