Introduction First Law of Thermodynamics

1. h Introduction First Law of Thermodynamics Thermodynamic Property Measurable quantity characterizing the state of a system • Temperature • Volume • Pressure • Voltage • Magnetic field strength • Energy, etc. Notable exceptions to this are Work and Heat – It doesn’t make any sense to say that some object has a certain amount of work or heat. These quantities can’t always easily be distinguished!

2. This reduces the number of independent parameters characterizing the system from three to two. Introduction First Law of Thermodynamics • Thermodynamic equilibrium • mechanical equilibrium • thermal equilibrium • chemical equilibrium The state of a system refers to the condition of the system as described by the thermodynamic parameters characterizing the system. Equation of state A functional relationship among the thermodynamic parameters characterizing the system.

3. Mechanical Work Terms General form dW = Intensive x d (Extensive) Intensive variable - mass independent Extensive variable- mass dependent dW = pdV; hydrostatic work = -dZ; electrical work = -HdM; magnetic work = -γdA; surface work = -Fds dW = ΣXidxi = pdV - dZ – HdM +…

4. Most materials science texts on thermodynamics If work is done on the “system” then W < 0. For example, compression of the gas in a piston cylinder arrangement. Sign Convention on Work Atkins and de Paula (and other chemistry texts): If work is done on the “system” so as to add energy to it then W > 0. For example, compression of the gas in a piston cylinder arrangement. Conceptual example: Consider the stretching a linear spring: The spring is our “system”, and work is done ON the system to stretch it.

5. F = mg vacuum dW = -Fdz F = pexA W = - pex(Vf-Vi ) Pext vacuum dW = -Fdz F = 0 W = 0 Free expansion expansion against mg expansion against a constant atmospheric pressure, Pext Work If F and s are in the same direction, W > 0. If F and s are in the opposite direction, W < 0. Massless, frictionless piston cylinder arrangements Hydrostatic Work

6. Heat Historically, the concept of heat was difficult to quantify because it is an energy flow based on microscopic actions rather than correlated macroscopic action. If energy “enters a system” by some correlated motion, i.e., the movement of a piston, we consider that as work. If energy enters a system owing to some microscopic transport, i.e., hot piston uncorrelated motion of molecules, we consider that as heat. If, ΔE + *W ≠ 0, the remainder of the energy change is due to heat. Sign convention ΔE + W = Q

7. In the case of heat transport the heat flux is given by, jheat = -κ grad T or The minus sign is there because the flux is always in a direction of decreasing temperature. Heat Transport Quite generally all transport processes can be described by an equation of the form, v = mx F, where v is a generalized velocity or flux of transport and F is a generalized force. This works for thermal transport, mass transport, electrical transport, etc. All transport equations take the following form, j = -k grad Φ.

8. For example - In a conservative field the quantity is path independent, i.e., Note that leads directly to Maxwell the reciprocal relations. First Law of Thermodynamics First Law - Conservation of Energy dE d(KE) + d(PE) + dU dU change in internal energy δQ heat added to the system δW work done by the system dU is an exact differential whereas δQ and δW are not. *What’s and exact differential? Is it possible to somehow turn a differential quantity that isn’t exact in to an exact differential? * Q is positive for heat flow in to the system W is positive for work done by the system.

9. p p = constant/V γ V What’s the physical manifestation of the increase in internal energy? Vi Vf Sign convention: The work is taken negative if it increases the energy in the system. If the volume of the system is decreased work is done on the system, increasing its energy; hence the positive sign in the equation . This unusual convention was established to fit the behavior of heat engines whose normal operation involves the input of heat and the output of work. So according to this unfortunate convention work is positive if the engine is doing its job. Note that some textbooks e.g., Equilibrium Thermodynamics by Adkins will write the first law as which is in line with results in the opposite sign convention for work. In this convention positive work is work going in to a system raising its energy. So, the manner in which you write the first law establishes the sign convention that you use. Adiabatic Work Adiabatic walls P1, V1 P2, V2 Quasistatic compression of an ideal gas.

10. Composite system 2 P1, V1 A P2, V2 B C 1 V Adiabatic Work p If a system is caused to change to an initial state to a final state by adiabatic processes, the work done is the same for all adiabatic paths connecting the two states. Different adiabatic paths between two states of a fluid. • 1A2. An adiabatic compression followed by electrical work at constant volume performed via a “heater” of negligible thermal capacity immersed in the fluid. • 1B2. The same process but in reverse order • 1C2. A complex route requiring simultaneous electrical and mechanical work.

11. p T p = nRT/V V Vi Vf Heat The heatflux to a system in any process is simply the difference in internal energy between the final and initial states plus the work done in that process. Non adiabatic Work Heat reservoir at fixed T Diathermal walls P1, V1 Heat reservoir at fixed T P2, V2 Q Quasistatic isothermal compression of an ideal gas.

12. Simple thermodynamic system A system that is macroscopically homo- geneous, isotropic, and uncharged, that is large enough so that surface effects can be neglected, and that is not acted on by external electric, magnetic or gravitational fields. All thermodynamic systems have an enor- mous number of atomic coordinates. Only a few of of these survive the statistical averaging associated with a transition to a macroscopic description. Certain of these coordinates are “mechanical” in nature. Thermodynamics is concerned with consequences of atomic coordinates that, by virtue of the coarseness of observation, do not appear in a macroscopic description of the system. Thermodynamic Equilibrium In all systems there is a tendency to evolve toward states in which the properties are determined by intrinsic factors and not by previously applied external influences. Such simple terminal states are time independent and are called equilibrium states. Equilibrium states of simple systems are characterized completely by the internal energy U, the volume V, and the mole numbers N1, N2, … of the chemical components.

13. Ideal Gas Temperature Scale pV/nk pV = nkT Equation of State Functional relationship among the thermodynamic parameters for a system in equilibrium. If p,V and T are the parameters, f (p, V, T) = 0, This reduces the number of independent parameters. Any point lying on this surface represents a state on equilibrium. 100 divisions T 0 K Fp H20 Bp H20 Surface representing the equation of state. T Equilibrium point V p Thermodynamic Parameters Measurable macroscopic quantities associated with the system, e.g., p, V, T, H, etc.

14. Mathematical digression Suppose that three variables are related as in an equation of state: This expression can be rearranged to yield any one variable in Terms of the other two independent variables, Differentiating this expression, (A)

15. Mathematical digression The terms in brackets are partial derivatives which are defined In a manor analogous to normal derivatives Also since, z =z (x, y) Substituting in equation (A) for dz, (B)

16. Mathematical digression This result is true whatever pair of variables we choose as independent. In expression (B) suppose, dy = 0, then This is the reciprocal theorem which allows us to replace any partial derivative by the reciprocal of the inverted derivative with the same variable(s) held constant.

17. f(xo) Δx = xA - xo x xo Mathematical digression In expression (B) suppose, dx = 0, and dy ≠ 0, then, by repeated application of the reciprocity theorem. Taylor series expansion in 1 variable.

18. B z 2 Δz A 1 Δy y Mathematical digression Taylor series expansion in 2 variables, x = x(y,z). Given x at y1, z1, we want to evaluate the value of the function x at point 2. Proceeding from 1 to A, And then from point A to 2, Substitution for xA from the 1st relation,

19. Mathematical digression If we had proceed from 1 to B to 2, Since the result must be the same, the last 2 terms in each of the Expressions must be equal,

20. Mathematical digression (C) Exact differentials If x is a function of y and z, we can write an expression for an infinitesimal change in x owing to infinitesimal changes in y and z, Here Y and Z correspond to the partial derivatives. In principle, dx can always be integrated since all we need are the initial and final states. If a quantity is not an exact differential, δW, then in order to integrate To get the work, we need to know the path.

21. Mathematical digression Since dx is an exact differential, Which just expresses the condition (C) on the previous page.

22. Internal Energy Macroscopic systems have a definite and precise energy energy subject to a conservation principle. The energy of the universe is the same today as it was eons ago. Thermodynamics is concerned with measuring energy differences of a system owing to external influences. We can easily determine the energy difference between two equilibrium states of a system by enclosing the system with adiabatic walls so that energy change can only occur by doing some form of mechanical work on the system. For a simple hydrostatic system (pdV work term) the change in internal energy is describable in terms of any two of the variables p, V and T since the third is by an equation of state (for an ideal gas, pV = nRT):

23. Heat The heatflux to a system in any process is simply the difference in internal energy between the final and initial states plus the work done in that process, . Heat is what is adsorbed by a system if its temperature changes while no work is done. If δδQ is the small amount of heat adsorbed in a system causing a temperature change of dT the ratio δQ /dT is called the heat capacity C, of that system. The specific heat c, is the heat capacity per unit mass. For example the heat capacity per mole of substance or the molar heat capacity is defined as; Specific heat In the case of a hydrostatic system the heat capacity is unique when constraints of either constant pressure or volume are imposed so that

24. Here the symbol His a thermodynamic potential or “energy” called the enthalpy which I will define shortly. The second of the set of equalities is easily demonstrated just by writing out the first law, and substituting for dU obtaining At constant volume,

25. If we try to do the same thing for Cpwe run in to a problem because of the form of the internal energy function, At constant pressure You can see that this is messy compared to the simple equation for CV. It is convenient to define a new function H, such that dH = d(U+pV). Then Introduction of the enthalpy function allows for a simple relation for Cp.