Outline. 1. Work. A. First Law of Thermo. 2. Internal Energy. 1. Work continued. Category: Thermodynamics. III. The Laws of Thermodynamics.

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1 ategory: hermodynamics Outline III. he Laws of hermodynamics A. First Law of hermo B. Second Law of hermo (Entropy). Statistical Mechanics D. References Updated: 04jan A. First Law of hermo. Work 4 Stored eat (a) Recall Definition W=Fx eat into a system is either stored as internal energy, or performs work on the universe EA IN WORK OU (b) (c) For a gas expanding against a piston the work done is: W=Fx=(PA)x=P(Ax) Work done by gas on environment is pressure time change in volume: W=+PV he above formula is only really valid for isobaric process (P=0 ). Generally it s the area under the curve of a PV diagram. (example to right is isothermal process =0 ). Work continued (a) Isobaric Process: P=0 W is the work done BY the system Exact Form: W=+PV (b) Isothermal Process (=0 ): dv W PdV nr V V W nr Ln V. Internal Energy Adding heat to a gas (at constant volume isovolumetric ) increases its internal energy U For monatomic gas (noble gasses, e, Ne) all the internal energy is in the kinetic energy of molecules For diatomic gas (e.g. O or ), there is more energy stored in rotation and vibration of molecule U mv U N k k U nr Note: N=number of molecules, n=number of moles, k=boltzmann s constant, R=gas constant, in Kelvin, & U in Joules nr 6

2 b Specific eat and gamma 7.c Adiabatic Process 8 (a) onstant Volume Molar Specific eat v Monatomic Ideal Gas U nr More generally: U nv for monatomic v R for diatomic v R (b) onstant Pressure Molar Specific eat p = v +R = v = p / v, / for monatomic (7/ diatomic) Definition: =0 (no heat in) A gas that expands adiabatically will cool (by the first law): =0=U+W U W nr PV V V onst PV onst V. First Law of hermodynamics (a) he First Law =U+W is heat INO system U is change of internal energy of system W is the work done by system on environment So its based on the law of ONSERVAION of ENERGY BEWARE: Some authors define work as done ON the system by environment and heat flowing OU of system, so equation will have minus signs. 9 Definition: =U+PV.b Enthalpy hange in Enthalpy: =U+PV+ VP Now insert the nd law to get =+ VP ence Enthalpy will be conserved in a process which is both isobaric (P=0) and adiabatic (=0) Or, for chemistry in an open beaker: = 0.c Gibbs Free Energy B. Second Law of hermo Definition: G=U+PV-S = -S hange in Gibbs: G=VP - S ence Gibbs Energy will be conserved in a process which is both isobaric (P=0) and isothermal (=0) [i.e. hemistry in an open beaker in an ice bath] Spontaneous reactions occur when: G < 0 Why does heat flow from hot to cold (instead of the other way around?). Definition of Entropy: S=/. S Diagrams. Entropy of Ideal Gas

3 . Entropy.b Entropy and First Law (details) 4 (a) Definition Rudolf lausius (86) Extrinsic uantity: Units: Joule/Kelvin S For example, it takes 4 Joules to melt gram of ice. he entropy of the liquid water is bigger than that of ice: 4 J S. 7 K J K From Definition: =S Isentropic Process (S=0) is equivalent to saying process is adiabatic (if reversible) First Law Restated in terms of Entropy: S = U + PV.c Entropy and Gas Volume.d Entropy of Ideal Gas 6 First Law: S = U + PV For ideal gas, isothermal process (=0, or U=0) reduces to: S=PV First Law of hermo: U=S-PV For monatomic ideal gas: U=(/)nR For ideal gas: P=(nR)/V Substitute into first law and solve for entropy: From this we can deduce change in entropy in an isothermal gas expansion is equal to isothermal work done by gas S V NR NR V W S V nr Ln V Integrate to get entropy of monatomic ideal gas S( n,, V ) nr Ln V. Reversibility 7 b. lausius Inequality 8 (a) Second Law of hermodynamics hange in (total) entropy of a closed system tends to increase in time: S0 Explains why heat flows from hot to cold, but not the other way around Example: eat removed from hot water and given to cold water ( > ) S 0 Any process obeys the inequality: For a system to be reversible, all the processes must obey the equality S S Example: Free expansion of gas in vacuum. No heat was added (=0), but the entropy has increased due to increased volume: V S nr Ln V 0

4 c. Irreversibility 9. Efficiency 0 otal entropy of universe increases (this tells us the direction of time!) A open system may decrease in entropy (e.g. freeze water), but the heat exhausted from it increases the entropy of the environment such that the total entropy increases S universe S system S environment 0 (a) arnot Energy Diagram Sadi arnot (84) =eat from fuel W=useful work by engine =waste heat exhausted onserve Energy W Efficiency W b. Engines MUS waste heat From second Law, Entropy must increase S 0 c. arnot ycle (84) he arnot ycle is the best efficient engine that can be made (reversible processes) W Must waste heat to environment. he colder the environment the less heat must be wasted. h h q S S B A his puts a limit on efficiency (arnot 84) W c c q S S D. Statistical Mechanics. he rd law of thermodynamics 4 Aka Nernst s heorem (906). he rd Law of hermodynamics a) Impossible to reach absolute zero. Probability and Equilibrium b) As 0 the heat capacity approaches zero. Boltzmann s Definition of Entropy c) As 0 all thermodynamic processes stop d) At absolute zero the entropy of a system would be minimum (S=0) 4

5 . Probability and Equilibrium c. Large N systems 6 (a) Equilibrium is the most probable situation (b) onsider 4 particles in a box. Each has a 0/0 chance of being on the Left half ( L ) of the box. here are 4 =6 possible microstates he most probable situation is that will be on either side. For example, consider N=40 atoms otal number of microstates is N =,099,,67,776 Macrostate Microstates Frequency Probability 0 LLLL 6.% Only possible for all on same side: Probability 9x0 - % LLLR, LLRL, LRLL, 4 % RLLL LLRR, LRLR, RLLR, 6 7.% RRLL, RLRL, LRRL RRRL, RRLR< 4 % RLRR, LRRR 4 RRRR 6.% For N=0 4 atoms deviations from the average situation are only in the th decimal place!. Boltzmann Entropy Entropy is a measure of the randomness (disorder) of a system Details: S=k Ln() k=boltzmann s onstant =Number of Microstates (for that macrostate), i.e. a measure of disorder. 7 References Good One 8 Maximum order: =, so S=0 Maximum disorder is the average equilibrium situation of half on each side of box N! N! S knln( ) Notes 9 Discuss elmholtz contraction of a star What about Refrigerators? Example: change in entropy by melting ice Demos: Stirling Engine

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