1 st Law: du=dq+dw; u is exact Eq. 2.8 du=dq rev -pdv (expansion only) p. 56. (δp/δt) g =η/v Eq. 2.40

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1 Lecture Ch. a Energy and heat capacity State functions or exact differentials Internal energy s. enthalpy st Law of thermodynamics Relate heat, work, energy Heat/work cycles (and path integrals) Energy s. heat/work? Adiabatic processes Reersible P-V work define entropy Curry and Webster, Ch. pp Van Ness, Ch. Key Combined st + nd Law Results st Law: du=dq+dw; u is exact Eq..8 du=dq re -pd (expansion only) p. 56 Define Enthalpy: H=U+PV Eq.. dh=du+pd+dp nd Law: [dq re /] int.cycle =0 Eq..7 Define Entropy: dη=dq re / Eq..5a dη=dq re du=dη-pd Define Gibbs: G=H-η Eq..33 dg=dh-dη-η=(du+pd+dp)-dη-η dg=du-(dη-pd)+dp-η=dp-η p. 58 (δp/δt) g =η/ Eq..40 Lord Kelin (a.k.a William homson) James P. Joule he First Law of hermodynamics Consequences Uniqueness of work alues Definition of energy Conseration of energy Impossibility of perpetual motion machine (Relatiity) W re = pd Q = 0 ΔE = W Q = 0,W = 0 ΔE = 0 E = E Q = 0,ΔE = 0 W = 0 ΔE = mc Reersible Adiabatic State function See also nd law! 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 Δz Kinetic energy for elocity change Δ Nuclear energy for mass change Δm Exact Differentials State functions are exact differentials ΔE = ΔU( p,v,) + mgδz + mδ c Δm = Q + W if ΔE ΔU( p,v,), then ΔU( p,v,) = Q + W Van Ness, p. 3

2 Heat Capacity For all gases: Heat Capacity Difference b/w U and H U depends on H depends on p From st Law (i.e. U is exact differential Defined aboe For ideal gases: Specific heats [a.k.a. heat capacity] c is constant c p is constant p Work Expansion work W=-pdV or w=-pd Lifting/rising Mixing Conergence 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 η (also S) entropy A Helmholtz free energy G Gibbs energy

3 Reersible" mass is consered" Ideal Gas" Adiabatic" thick walls" First Law" Internal Energy" Reersible, Adiabatic" Δu = Q + W Δu = c Frictionless Low P, High = P P R c p dw = pd p = R = p Q = 0 Reersible" mass is consered" Ideal Gas" Adiabatic" thick walls" First Law" Internal Energy" Reersible, Adiabatic" Δu = Q + W Δu = c Frictionless Low P, High = P P R c p dw = pd p = R = p Q = 0 pd = c R d = c d R = c Reersible, Adiabatic Expansion of an Ideal Gas pd = c R d = c d R = c R( ln ln ) = c ( ln ln ) ln R = ln = R c c R p = R p + R c = p p R c R c = p R R +c = p p p = R c p R c p p R c Reersible" mass is consered" Frictionless grain of sand dw re = pd Always at or infinitesimally close to equilibrium Infinitesimally small steps Infinite number of steps Each step can be reersed 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 consered, exact, unique While Q is not exact, Q re is exact Reersible heat is limit of maximum work done Since path is specified, cyclic integral is 0 Curry and Webster, Ch. pp Van Ness, Ch. 5-7 Curry and Webster, Ch. pp Van Ness, Ch

4 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 moe from a less ordered state to a more ordered state. room containing air not possible O here he entropy of an isolated system cannot decrease. ΔS system 0 ΔS system = N here state dq re state system Clausius Inequality Maxwell s Equations At dg=0, we get n=dp or (dp/) g =n/ typo in book Maxwell s Equations Sample Deriation Start with first law and definition of entropy du = dq + dw du = dq re + dw re = [ dη] + [ pd] du = dη pd Assumed reersible, 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 o gie = p η η y 4

5 Heat/Work Cycles Carnot was an engineer in Napoleon s defeated army with an interest in engines. he efficiency with which work is accomplished in a reersible cyclic process depends only on the temperature of the reseroirs to which heat is transferred Q HE CARNO CYCLE SEP : Expand isothermally and reersibly at W = Q = R ln P A P B FLUID Q W SEP : Expand adiabatically and reersibly W = C ( ) SEP 3: Compress isothermally and reersibly at W = Q = R ln P C P D SEP 4: Compress adiabatically and reersibly W = C ( ) Efficiency: η = Cold Hot P-V diagrams of work Work is determined by pathway Nikolaus Otto deeloped the Otto cycle in 876. Other Work Cycles Rudolf Diesel deeloped the Diesel cycle in 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 η = 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 compression. his is in contrast to a gasoline engine, which utilizes the Otto cycle, in which ignition is initiated by a spark plug following the aspiration and compression of a fuel/air mixture. γ Efficiency: B C η = γ A D = 5 3 for monatomic ideal gas :Add Heat (isothermally) :Adiabatic 3:Lose Heat (isothermally) 4:Adiabatic 5

6 Potential emperature Virtual Potential emperature Potential emperature (for moist air) Virtual Potential emperature θ = ( q ) p 0 p R d c pd Virtual emperature Meteorologists Entropy, + [0 3K] η η c p = ln θ θ θ = exp η η = exp Δη θ 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 ertical motion using omega field * Other motion options: isentropic, isosigma, isobaric, isopycnic * rajectory ensemble option using meteorological ariations * Output of meteorological ariables along a trajectory " " 6

7 Ideal Gases For an ideal gas Simplify Eqn..5a and.5b to get c p h = dh = h p [ypes of processes] Constant pressure Constant olume he nd Law Energy spontaneously tends to flow only from being concentrated in one place to becoming diffused or dispersed and spread out. Hurricane as Carnot Cycle :Adiabatic :Add Heat (isothermally) 4:Adiabatic 3:Lose Heat (isothermally) Hurricane as Carnot Cycle Step (A-B): air spirals in, acquiring heat isothermally expanding Step (B-C): air ascends and expands adiabatically in eye Step 3 (C-D): air loses heat and begins to compress isothermally Step 4 (D-A): air compresses adiabatically and returns to surface Caeats Not open: rain, mixing 7