Lecture 4 – The First Law (Ch. 1) Monday January 14 th

1. Lecture 4 – The First Law (Ch. 1) Monday January 14th • 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 z dS y x dr There is a mathematical basis..... Consider the function F = f(x,y):

3. How to know if quantity is a function of state đQ + đW does not depend on path đW is path dependent U1 U2

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

5. How to know if quantity is a function of state 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 đF is inexact, this implies that it cannot be integrated to yield a function F. 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.

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

7. Heat Capacity 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) 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 đQ is inexact.

8. Heat Capacity Because the differential đQ 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, CV • the heat capacity at constant pressure, Cp

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

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

11. Configuration Work and ideal gases Note: for an ideal gas, U = U(q), so W = -Q for isothermal processes. It is also always true that, for an ideal gas, Adiabatic processes: đQ = 0, so W = DU, also PVg = constant.