1. Lecture 4 – The First Law (Ch. 1) Monday January 14 th Finish previous class: functions of state Reversible work Enthalpy and specific heat Adiabatic processes Reading: All of chapter 1 (pages 1 - 23) 1st homework set due next Friday (18th). Homework assignment available on web page. Assigned problems: 2, 6, 8, 10, 12

2. How to know if quantity is a function of state There is a mathematical basis..... Consider the function F = f(x,y) : z y x dS dr

3. How to know if quantity is a function of state U1U1 U2U2 đW is path dependent đQ + đW does not depend on path

4. dF ( = Adx  Bdy In general, F is a state function if the differential dF is ‘exact’. dF ( = Adx  Bdy ) is exact if: See also: Appendix EAppendix E PHY3513 notesPHY3513 notes Appendix A in Carter bookAppendix A in Carter book In thermodynamics, all state variables are by definition exact. However, differential work and heat are not. How to know if quantity is a function of state There is a mathematical basis..... Consider the function F = f(x,y) :

5. Differentials satisfying the following condition are said to be ‘exact’: This condition also guarantees that any integration of dF will not depend on the path of integration, i.e. only the limits of integration matter. This is by no means true for any function! If integration does depend on path, then the differential is said to be ‘inexact’, i.e. it cannot be integrated unless a path is also specified. An example is the following: đF = ydx  xdy. Note: is a differential đ F is inexact, this implies that it cannot be integrated to yield a function F. How to know if quantity is a function of state

6. Calculation of work for a reversible process đQ + đW P V (1) (2) (3) (4) 1.Isobaric (P = const) 2.Isothermal (PV = const) 3.Adiabatic (PV  = const) 4.Isochoric (V = const) For a given reversible path, there is some associated physics.For a given reversible path, there is some associated physics.

7. Heat Capacity The heat capacity C of a system is defined as the limiting ratio of the heat Q added to a system (causing it to change from one equilibrium state to another) divided by the accompanying temperature increase: Note that this is a rather awkward definition, because the differential đ Q is inexact. The specific heat capacity c of a system, often abbreviated to “specific heat”, is the heat capacity per unit mass (or per mole, or per kilomole)

8. Heat Capacity Because the differential đ Q is inexact, we have to specify under what conditions heat is added. Or, more precisely, which parameters are held constant. This leads to two important cases: the heat capacity at constant volume, C V the heat capacity at constant pressure, C p

9. More on heat capacity Using the first law, it is easily shown that: Finding a similarly straightforward expression for C P is not as easy, and requires knowledge of the state equation. U is a function of state, so it does not actually matter how we add the heat! For an idea gas, it can be shown that the internal energy depends only on the temperature of the gas . Therefore, Always true

10. Enthalpy and heat capacity Enthalpy, H = U + PV, turns out to be a useful quantity for calculating the heat capacity at constant pressure Always true For an idea gas, it can be shown that the enthalpy depends only on the temperature of the gas . Therefore, dH = dU + PdV + VdP = đQ + VdP

11. Configuration Work and ideal gases Note: for an ideal gas, U = U(   ), so W =  Q for isothermal processes. It is also always true that, for an ideal gas, Adiabatic processes: đQ = 0, so W =  U, also PV  = constant.