Lecture Ch. a Energy and heat capacity State functions or exact differentials Internal energy vs. enthalpy st Law of thermodynamics Relate heat, work, energy Heat/work cycles (and path integrals) Energy vs. heat/work? Adiabatic processes Reversible P-V work define entropy Curry and Webster, Ch. pp. 35-47 Van Ness, Ch. Key Combined st + nd Law Results st Law: du=dq+dw; u is exact Eq..8 du=dq rev -pdv (expansion only) p. 56 Define Enthalpy: H=U+pV Eq.. dh=du+pdv+vdp nd Law: [dq rev /] int.cycle =0 Eq..7 Define Entropy: d=dq rev / Eq..5a d=dq rev du=d-pdv Define Gibbs Energy: G=H- Eq..33 dg=dh-d-=(du+pdv+vdp)-d- dg=du-(d-pdv)+vdp-=vdp- p. 58 (p/t) g =/v Eq..40 Lord Kelvin (a.k.a William homson) James P. Joule he First Law of hermodynamics Consequences Uniqueness of work values W rev = pdv Reversible Definition of energy Conservation of energy Q = 0 E = W Adiabatic Q = 0,W = 0 E = 0 E = E State function Impossibility of perpetual motion machine Q = 0,E = 0 W = 0 See also nd law (Relativity) E = mc Proof for hmwk Other Kinds of Energy What is the difference between E and U? In addition to changes in internal energy, a system may change Potential energy for height change z Kinetic energy for velocity change v Nuclear energy for mass change m E = U( p,v,) + mgz + mv c m = Q + W Exact Differentials State functions are exact differentials State functions have no memory if E $ U( p,v,), then U( p,v,) = Q + W Van Ness, p. 3 State functions result from two non-interacting contributions.
Heat Capacity For all gases: Heat Capacity Difference b/w U and H U depends on v H depends on p From st Law (i.e. U is exact differential) Defined above For ideal gases: Specific heats [a.k.a. heat capacity] c v is constant v c p is constant p Work Expansion work W=-pdV or w=-pdv Lifting/rising Mixing Convergence Other kinds of work? Electrochemical (e.g. batteries) Cycles Work and heat are path-dependent transfers W work Q heat State functions are unique states U internal energy H enthalpy (also S) entropy A Helmholtz free energy G Gibbs energy
Reversible-Adiabatic-Work Reversible-Adiabatic-Work Reversible mass is conserved Ideal Gas Adiabatic thick walls First Law Internal Energy Reversible, Adiabatic u = Q + W u = c v Frictionless Low P, High = P % $ ' & P R c p dw = pdv p v = R = p v Q = 0 Reversible mass is conserved Ideal Gas Adiabatic thick walls First Law Internal Energy Reversible, Adiabatic u = Q + W u = c v Frictionless Low P, High = P % $ ' & P R c p dw = pdv p v = R = p v Q = 0 pdv = c v R & dv = c $ v ' v dv ) R v = ) c v Reversible, Adiabatic Expansion of an Ideal Gas pdv = c v R & dv = c $ v ' v dv ) R v = c ) v R( lnv lnv ) = c v ( ln ln ) ln v R & = ln & $ v ' $ ' & = v & $ ' $ v ' R c v c v R & & & %% ( ( p = % $ ' ( $ ' % R &( % $ $ p ( '' +R & cv = p & $ ' $ p ' R cv R cv & = p R & R +cv = p & $ ' $ p ' $ p ' = & $ ' R c p & $ p ' R cv p R cv Reversible Processes Reversible mass is conserved Frictionless grain of sand dw rev = pdv Always at or infinitesimally close to equilibrium Infinitesimally small steps Infinite number of steps Each step can be reversed with infinitesimal force Lecture Ch. b Entropy Second law of thermodynamics Maxwell s equations Heat capacity Meteorologist s entropy Entropy Is there a way to quantify useful energy? Need a measure that is conserved, exact, unique While Q is not exact, Q rev is exact Reversible heat is limit of maximum work done Since path is specified, cyclic integral is 0 Curry and Webster, Ch. pp. 47-6 Van Ness, Ch. 5-7 Curry and Webster, Ch. pp. 47-6 Van Ness, Ch. 5-7 3
Entropy st Law Second Law of hermodynamics Heat cannot pass of itself from a colder body to a hotter body. Definition 9 o C o C not 0 o C 30 o C possible A system left to itself cannot move from a less ordered state to a more ordered state. room containing air not possible O here N here he entropy of an isolated system cannot decrease. S system 0 S system = state dq rev state system Clausius Inequality Maxwell s Equations At dg=0, we get n=vdp or (dp/) g =n/v or Energy spontaneously tends to flow only from being concentrated in one place to becoming diffused or dispersed and spread out. typo in book Maxwell s Equations How to Derive a Maxwell Eqn. Start with first law and definition of entropy du = dq + dw du = dq rev + dw rev = [ d ] + [pdv] du = d pdv Assumed reversible, but result is path independent Use properties of exact differentials df = F & dx + F & dy ) Mdx + Ndy * M & = N & $ x ' y $ y ' y x $ ' $ x ' x y o give $ ' $ & ) = + p ' & ) % v ( %*( * v 4
What is an energy/work cycle? Visualize on a P-V diagrams Specific pathways, multiple steps Work is determined by pathway Heat/Work Cycles he efficiency with which work is accomplished in a reversible cyclic process depends only on the temperature of the reservoirs to which heat is transferred Q FLUID W HE CARNO CYCLE SEP : Expand isothermally and reversibly at W = Q = R ln P A P B SEP : Expand adiabatically and reversibly W = C v ( ) SEP 3: Compress isothermally and reversibly at Carnot was an engineer in Napoleon s defeated army with an interest in engines. Q W = Q = R ln P C P D SEP 4: Compress adiabatically and reversibly W = C v ( ) Efficiency: = Cold Hot Nikolaus Otto developed the Otto cycle in 876. Other Work Cycles Rudolf Diesel developed the Diesel cycle in 89. 4 Steps of Carnot Engine he Otto Cycle works by compressing a mixture of air and fuel in a piston and then igniting the mixture with a spark. he Diesel Cycle works by compressing air and then adding fuel directly to the piston. he compressed air then combusts the mixture. Efficiency: B C = A D he compression ratio of the Diesel Cycle ranges from 4: to 5:, while the Otto Cycle range is significantly lower, from 8: to :. he defining feature of the Diesel engine is the use of compression ignition to burn the fuel, which is injected into the combustion chamber during the final stage of Efficiency: compression. his is in contrast to a gasoline engine, & which utilizes the Otto cycle, in which ignition is B ' C ) = ' initiated by a spark plug following the aspiration and $ ( % A ' D compression of a fuel/air mixture. = 5 3 for monatomic ideal gas :Add Heat (isothermally) :Adiabatic 3:Lose Heat (isothermally) 4:Adiabatic 5
Virtual emperature Potential emperature v, v + [0 $ 3K] Virtual Potential emperature Potential emperature (for dry & moist air) Dry (.6): Moist (.67a): = p $ cpd 0 &Rd = p $ 0 p & % p % Virtual Potential emperature R d p v = 0 $ cpd v & p % v = ( + 0.608q v ) p & 0 $ p ' R d c pd R d ('.6q v ) cpd Moist (.67b) Meteorologists Entropy c p = ln $ $ $ % = exp ( % ' * = exp + ( ' * $ & c p ) & c p ) rajectories Example: NOAA HYSPLI Model Example: NOAA HYSPLI * Single or multiple (space or time) simultaneous trajectories * Optional grid of initial starting locations * Computations forward or backward in time * Default vertical motion using omega field * Other motion options: isentropic, isosigma, isobaric, isopycnic * rajectory ensemble option using meteorological variations * Output of meteorological variables along a trajectory http://www.arl.noaa.gov/ready/hysplit4.html http://www.arl.noaa.gov/ready/hysplit4.html 6
Ideal Gases For an ideal gas Simplify Eqn..5a and.5b to get or pv=nr where v=v/n c p h % $ ' & p = dh [ypes of processes] Constant pressure Constant volume 7