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Real Gases: Degrees of Freedom

Real Gases: Degrees of Freedom. An ideal gas is monatomic , and stores internal energy in translational motion only. Real gases are mostly polyatomic , and store internal energy in translational motion vibrational motion Degrees of Freedom rotational motion Equipartition Theorem:

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Real Gases: Degrees of Freedom

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  1. Real Gases: Degrees of Freedom • An ideal gas is monatomic, and stores internal energy in translational motion only. • Real gases are mostly polyatomic, and store internal energy in • translational motion • vibrational motion Degrees of Freedom • rotational motion Equipartition Theorem: • Internal energy is divided equally among the degrees of freedom: • ½kBT per degree of freedom per molecule. • ½RT per degree of freedom per mole.

  2. Example: Diatomic Molecule A diatomic molecule has • 3 translational degrees of freedom. • 2 vibrational degrees of freedom. • 2 rotational degrees of freedom. • 7 total degrees of freedom Internal energy of n moles of diatomic molecules: U = 3n(½RT) + 2n(½RT) + 2n(½RT)= (7/2)nRT translation vibration rotation total Molar Specific Heat At Constant Volume

  3. Heat-Transfer Mechanisms Conduction: • is heat transfer through one material or between materials in contact. • requires a temperature gradient dT/dx. Heat Transfer Rate: (Power) k = thermal conductivity (property of material) A = cross-sectional area dT/dX = temperature gradient

  4. Heat Transfer Rate: Example of Heat TransferUniform Rod: Thermal Insulation “R-Value” =l/k

  5. Thermal Conductivities of Some Materials

  6. Heat-Transfer Mechanisms:Convection • Convection is the transfer of heat due to the net movement of the medium by gravitational forces. • e.g. warm air is less dense than cold air and rises under the influence of gravity. Convection Heating System for a Home

  7. Heat-Transfer Mechanisms:Radiation • All objects radiate energy because of microscopic movements (accelerations) of charges, which increase with temperature. Radiated Power: σ = 5.67 x 10-8 W/(m2 K) A = surface area ε = emissivity of material (number between 0 and 1) ε = 0 for perfect reflector ε = 1 for perfect absorber T = absolute temperature in Kelvin

  8. Heat Transfer by Radiation • If an object is at temperature T1 and its surroundings are at temperature T2, the net flow of heat radiation between the object and its surroundings is • Heat energy will always flow in the direction from warmer to colder.

  9. Second Law of Thermodynamics • Heat flows spontaneously from a hot object to a cold object, but will not flow spontaneously from a cold object to a hot object. Q heat Hot Object Cold Object It is relatively easy to produce thermal energy by doing work (e.g. against the force of friction). It is also possible to convert internal (heat) energy to work.

  10. Heat Engine A HEAT ENGINE is any device that converts thermal energy to mechanical work. FIRST LAW: Energy Conservation Input = Output QH = QL + W Efficiency =ε

  11. Steam Engines(External Combustion)

  12. Carnot (Ideal) Heat Engine Ideal (Carnot) Efficiency Operates in a reversible cycle: a  b isothermal expansion (ΔT=0) b  c adiabatic expansion (Q=0) c  d isothermal compression (ΔT=0) d  a adiabatic compression (Q=0)

  13. Efficiencies of Real Heat Engines • No heat engine can ever have an efficiency greater than that of the Carnot (ideal) heat engine. • All real heat engines have losses (e.g. friction) and are therefore not reversible. • All real heat engines have efficiencies less than that of a Carnot engine operating between the same temperatures TH and TL. Alternate Statement of Second Law: • NO DEVICE CAN TRANSFORM A QUANTITY OF HEAT COMPLETELY TO WORK.

  14. Example: Electric Power Plant A conventional electric power plant uses super-heated steam at 550°C to turn a turbine, and condenses the steam using cooling water from a lake at 25°C. • What is the maximum possible efficiency of this power plant? • Assuming the maximum efficiency and a plant output of 300 MW, at what rate must energy be input to the boiler? • At what rate must waste heat be expelled to the lake?

  15. A reversible process … • proceeds slowly through equilibrium states. • could be reversed with no change in heat or work output. All real processes … • are irreversible and have additional heat losses (e.g. due to friction). • have efficiency ε < εideal

  16. A heat pump is a heat engine operating in reverse. Examples of heat pumps are refrigerators and air conditioners. Conservation of Energy: Energy In = Energy Out QL + W = QH Heat Pump

  17. Example: Refrigerator • A refrigerator pumps heat from the inside of the freezer (-5°C) to the room (25°C). What is the maximum coefficient of performance? • i.e. 8.9 Joules of heat would be pumped from the freezer for every Joule of work done by the compressor. (typical CP = 3-5)

  18. Entropy • Some processes that do not violate the first law of thermodynamics (conservation of energy) will never occur spontaneously. • Entropy (S) is a measure of the disorder or randomness in a system, and is a state variable (like P,V,T) that does not depend on the path taken.

  19. Entropy and Reversible Processes For any reversible process (e.g. Carnot cycle): The entropy of a system in a given state is independent of the path taken to get there, and is thus a state variable.

  20. Entropy and Equilibrium States • The entropy difference between two equilibrium states a and b does not depend on how the system got from a to b. Entropy is a state variable (like P,V and T)

  21. Second Law in Terms of Entropy ΔS = 0 reversible process ΔS > 0 irreversible process Example: Calorimetry Q = m c ΔT For small changes: dQ = m c dT

  22. Example: Problem 20-44 A 150-g insulated aluminum cup at 20°C is filled with 240 g of water at 100°C. • What is the final temperature of the mixture? • What is the total change in entropy as a result of the mixing process?

  23. Entropy and Disorder • Entropy is a measure of the disorder or randomness of a system. • All real systems tend to disorder - i.e. the entropy of the universe always increases in irreversible processes. • The entropy of a system may be reduced locally by doing work on that system. However, the total entropy of the universe still increases. • Although energy is a conserved quantity, it is always being degraded to its ‘lowest form’ (heat). • If the universe is finite, it will eventually reach the same temperature (heat death), at which point the energy will be unavailable for doing work.

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