Chapter 4: Applications of the First Law

1. Chapter 4: Applications of the First Law Different types of work: • Configuration work: (reversible process) • Dissipative work: (irreversible process) • Adiabatic work: (independent of path) • Work: not a system property, i.e. not a state variable • Conventional sign of work: on (-) or by (+) the system • Expansivity and isothermal compressibility

2. 4.2 Mayer’s equation • Heat capacity: limiting ratio of …… ( not an exact differential !!) • Heat capacity depends on the conditions at which heat transfer takes place • Specific heat capacity cv and cp • For the ideal gas system: cp – cv = R • Calculate the internal energy of an ideal gas system based on the definition of cv

3. If cv is independent of temperature, then u = u0 + cv(T – T0)

4. 4.3 Enthalpy and Heats of Transformation • The heat of transformation is the heat transfer accompanying a phase change. • Phase change is an isothermal and isobaric process. • Phase change only entails a change of volume. thus the work (only configuration work) equals: w = P(v2 – v1)

5. Variation in the internal energy is expressed as: du = dq - Pdv for a finite change: u2 – u1 = l– P(v2-v1) where l represents the latent heat of transformation. thus l = (u2 + Pv2) – (u1 + Pv1) At this point, introducing h = u + Pv (the small h denotes the specific enthalpy) • Therefore, the latent heat of transformation is equal to the difference in enthalpies of the two phases.

6. Conventional notation: 1 denotes a solid, 2 a liquid and 3 a vapor, i.e. h’ represents the enthalpy of solid, h’’ is the enthalpy of liquid, …. • l12 = h’’ – h’ represents solid to liquid transformation (fusion). l23 = h’’’ – h’’ represents liquid to vapor transformation (evaporation). • Enthalpy is a state function, i.e. integration around a closed cycle produces 0!! (see Fig. 4.2)

7. 4.4 Relationships involving enthalpy • The natural choice in the variable h is h = h(T, P) • The analysis can be proceeded in the same way as to the internal energy u

8. As Thus: Since: then

9. For an ideal gas • Then • Since for ideal gas h depends on T only,

10. 4.5 Comparison of u And h • See Table 4.2 in the textbook

11. 4.6 Work done in an adiabatic process • In adiabatic process: dq = 0. • The equation dq = cpdT – vdP can be rearranged into vdP = cpdT • Similarly, one gets Pdv = -cvdT • Dividing the above two equation:

12. Assuming • The integration of the above equation leads to where K is a constant • Similarly, one gets

13. The work done in the adiabatic process is • For a reversible adiabatic process: w = u1 – u2 = cv(T1 – T2)

14. Example: An ideal monatomic gas is enclosed in an insulated chamber with a movable piston. The initial values of the state variables are P1 = 8atm, V1 = 4 m3 and T1 = 400K. The final pressure after the expansion is P2 = 1 atm. Calculate V2, T2, W and ∆U. • Solution: For an ideal monatomic gas, the ratio of specific heats γ = 5/3 since P1V1γ = P2V2γ V2 = V1(P1/P2)1/γ thus V2 = 13.9m3 According to ideal gas law: PV = nRT T2 can be easily calculated as 174K The work done by the system is w = 2.74 x 106 J For the adiabatic process: ∆U is equal to the work done on the system thus is -2.74 x 106 J

15. Chapter 5: Consequences of the First Law

16. 5.1 The Gay-Lussac-Joule Experiments