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Lecture 7 The second law of thermodynamics. Heat engines and refrigerators. The Carnot cycle. Entropy.
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Irreversibility of processes There exist many processes that are irreversible: the net transfer of energy by heat is always from the warmer object to the cooler object, never from the cooler to the …
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Heat Engines A heat engine is a device that takes in energy by heat and, operating in a cyclic process, expels a fraction of that energy by means of work. Weng – work done by the heat engine Qh – …
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Thermal Efficiency of a Heat Engine Good Automobile engine efficiency is about 20% Diesel engine efficiency is about 35%-40%
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Heat Pumps or Refrigerators In a heat engine a fraction of heat from the hot reservoir is used to perform work. In a refrigerator or a heat pump work is used to take heat from the cold reservoir and …
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Refrigerator W – work done on the heat pump Qh – heat, put into the hot reservoir. Qc - heat, taken from the cold reservoir.
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Coefficient of performance of a refrigerator The effectiveness of a refrigerator is described in terms of a number called the coefficient of performance (COP). COP = Qc /(Qh - Qc) = Qc /W Good …
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The Second Law of Thermodynamics The Kelvin form: It is impossible to construct a cyclic engine that converts thermal energy from a body into an equivalent amount of mechanical work without a further …
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The Second Law of Thermodynamics The Clausius form: It is impossible to construct a cyclic engine which only effect is to transfer thermal energy from a colder body to a hotter body. Thus for …
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Carnot cycle 1. A-B isothermal expansion B-C adiabatic expansion 3. C-D isothermal compression 4. D-A adiabatic compression
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Carnot Efficiency Using the equation of state and the first law of thermodynamics we can easily find that (look Servay p. 678; Fishbane p. 581): Let’s prove it: During the isothermal expansion …
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So, the work done on a gas during an isothermal process A → B is: So, the work done on a gas during an isothermal process A → B is: (1) Similarly, for isothermal C → D: (2) Deviding (2) over (1): (3)
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For adiabatic processes: For adiabatic processes: So, statement (3) gives us:
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So, using the last expression and the expression for efficiency: So, using the last expression and the expression for efficiency: Thus we have proved that the Carnot Efficiency equals Carnot Engine …
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Carnot theorem The Carnot engine is the most efficient engine possible that operates between any two given temperatures. (look Servay p. 675; Fishbane p. 584)
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Carnot Theorem Proof
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Entropy Measures the amount of disorder in thermal system. It is a function of state, and only changes in entropy have physical significance. Entropy changes are path independent. Another statement …
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Entropy change calculations Entropy is a state variable, the change in entropy during a process depends only on the end points and therefore is independent of the actual path followed. Consequently …
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So for infinitesimal changes: So for infinitesimal changes: The subscript r on the quantity dQr means that the transferred energy is to be measured along a reversible path, even though the system may …
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Change of Entropy in a Carnot Cycle Carnot engine operates between the temperatures Tc and Th. In one cycle, the engine takes in energy Qh from the hot reservoir and expels energy Qc to the cold …
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Reversibility of Carno Cycle Using equality, proved for the Carnot Cycle (slide N13): We eventually find that in Carno Cycle: S=0
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Reversible Cycle Now consider a system taken through an arbitrary (non-Carnot) reversible cycle. Because entropy is a state variable —and hence depends only on the properties of a given equilibrium …
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Ideal Gas Reversible Process Suppose that an ideal gas undergoes a quasi-static, reversible process from an initial state Ti, Vi to a final state Tf, Vf . 1st law of thermodynamics: dQr = ΔU + W, …
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- This expression demonstrates that S depends only on the initial and final states and is independent of the path between the states. The only claim is for the path to be reversible. -S can be …
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The Second Law of Thermodynamics The total entropy of an isolated system that undergoes a change cannot decrease. If the process is irreversible, then the total entropy of an isolated system always …
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Microscopic States Every macrostate can be realized by a number of microstates. Each molecule occupies some microscopic volume Vm. The total number of possible locations of a single molecule in a …
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Entropy on a Microscopic Scale Let’s have an ideal gas expanding from Vi to Vf. Then the numbers of microscopic states are: For initial state: Wi = wiN = (Vi /Vm)N . For final state: Wf = wfN = (Vf …
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After further transformations: After further transformations: n – number of moles, R=kbNa. Then we use the equation for isothermal expansion (look Servay, p. 688): Using the expression from the …
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Entropy is a measure of Disorder The more microstates there are that correspond to a given macrostate, the greater is the entropy of that macrostate. Thus, this equation indicates mathematically that …
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Independent Study Reynold’s number, Poiseuille flow, viscosity, turbulence (Fishbane p. 481, Lecture on physics Summary by Umarov). Entropy Change in a Free Expansion. (Servay p. 688). Entropy Change …
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