Ph70A Spring 004 Prof. Pui Lam SOLUTION Reading Assignment #3 :h.3: (3-3,5,6,7,8), h4: (4-,,4,5,6) Homework #3: First Law and Second of Thermodynamics Due: Monday 5/3/004.. A monatomic ideal gas is allowed to expand at constant pressure of 3 atm from a volume of L to 3L. It is then cooled at constant volume to P=atm. (a) Draw a PV diagram depicting this process (b) alculate the work done by the gas? What is the work done by the environment? (c) Find the heat added (or removed) during this process P(atm) A 3 3 V(L) (b) Work done by the gas = PΔV From A to : W done by gas = (3atm)(3L L) = (3x0 5 Pa)(x0 3 m 3 ) = 600J From to : W done by gas = 0 Work done by the environment = -PΔV = 600J (c) Use the First Law: Monatomic ideal gas: E int = 3 nrt = 3 PV From A to : ΔE int = 3 x05 Pa 450J = Q 600J Q =,050J [( )( 3x0 3 m 3 ) ( 3x0 5 Pa) ( x0 3 m 3 )] = 450J. A monatomic ideal gas initially at 0o and 00kPa has a volume of 4L. It undergoes a quasi-static isothermal expansion until its pressure is reduced to 00 kpa. (a) What is the final volume? (b) Find the work done by the environment. (c) Find the heat add to the gas during this process.
(a) P f V f = n f RT f ; P i V i = n i RT i Isothermal T f = T i Also n f = n i P f V f = P i V i V f = P i V i = 00 4L = 8L P f 00 (b)w = V f Vi PdV = nrt V f Vi V dv = nrt ln V f = P i V i ln V f = (00kPa)(4x0 3 m 3 )ln = 554.5J V i (c) V i ΔE int = 3 nrδt = 0 (isothermal ΔT = 0) Q = W = 554.5J 3. Two moles of ideal monatomic gas have an initial pressure of atm and an initial volume of L. The gas is taken through the following quasi-static cycle: It expanded isothermally until its volume is 4 L, then it is heated at constant volume until its pressure is atm again, then it is cooled at constant pressure back to the initial state. (a) Show this cycle on a PV diagram (b) Find the temperatures at the three corners of this cycle. (c) alculate the total work done by the gas and the net heat added to the gas? P(atm) A 3 4 V(L)
(b) PV = nrt T A = (x05 )(x0 3 ) (8.3) T = T A = 4.07K = 4.07K T = (x05 )(4x0 3 ) = 48.4K (8.3) (c)w done by gas = W A > + W > + W >A = +nrt A ln V + 0 + P A (V A V ) = 400J ln + (x0 5 )( 3x0 3 ) = 00J V A W done by environment = 00J ΔE int = 0 for a complete cycle Q = -W = -00J 4. Two moles of ideal monatomic gas have an initial pressure of atm and an initial volume of L. It expands adiabatically to 4L. alculate the work done by the gas and compare the result with the work done isothermally in Q.3 Adiabatic Q = 0 W done by environment = ΔE int = 3 [ P f V f P i V i] P i = x0 5 Pa,V i = x0 3 m 3, V f = 4x0 3 m 3 5. A arnot engine works between two heat reservoirs at Th=300 K and Tc = 77 K. (a) What is its efficiency? (b) If it absorbs 00 J from the hot reservoir during each cycle, home much work does it do? (c) How much heat does it give off in each cycle? What is the coefficient of performance of this engine when it works as a refrigerator between these two reservoirs? (a) arnot engine: Efficiency =- T c = 77 = 0.743 = 74.3% T h 300 (b)efficiency W output Q h W output = Efficiency Q h = (0.743)(00J) = 74.3J (c) Q c = Q h W output =00J 74.3J = 5.7J arnot refrigerator: OP Q c W = 5.7J 74.3J = 0.346 3
6. A system absorbs 300 J from a reservoir at 300 K and 00 J from a reservoir at 400 K. it then returns to its original state, doing 00 J of work and rejecting 400 J of heat to a reservoir at temperature T. (a) What is the entropy change of the system for the complete cycle? (b) If the cycle is reversible, what is the temperature T? (a) Entropy is a state variable. The engine goes through a complete cycle ΔS engine = 0. (b)δs total = ΔS reservoir@300k + ΔS reservoir@400k + ΔS reservoir@t c + ΔS engine For reversible cycle, ΔS total = 0 ΔS total = 300J 300K 00J 400K + Q c T c + 0 = 0 From conservation of energy: Q h - W output Q c = 0 300J + 00J 00J Q c = 0 Q c = 400J 7. True or False. If it is True, give an example. If it is False, give a counterexample. a) Work can never be converted completely into heat. False. ounterexample: ompress an ideal gas isothermally (ΔU=0), hence Q=W. (b) Heat can never be converted completely into work. False. ounterexample: Expand an ideal gas isothermally (ΔU=0), hence Q=W. This is not in violation of Kelvin Statement: Heat can never be converted completely into work, without changing the system or its environment. In the isothermal expansion, the system has changed. (c) All heat engines have the same efficiency. False. The efficiency of a heat engine depends what type of cycle it uses and the two operating temperatures (Th and Tc). A arnot cycle has the highest efficiency for a given set of Th and Tc. (d) It is impossible to transfer a given quantity of heat from a cold reservoir to a hot reservoir. False. You can do it if you provide work such as a refrigerator or heat pump. (e) The coefficient of performance of a refrigerator cannot be greater than. False. The OP=Qc / Work. it can have any values except infinity. (f) All arnot engines are reversible. True. In a arnot engine, heat transfers takes place at constant temperatures (Th or Tc). If the temperatures of the hot and cold reservoirs are infinitesimally close to Th and Tc, then the process is reversible. (g) The entropy of a system can never decrease. False. ounterexample: onsider the system is gas. You remove heat from it, i.e. cooling it, its entropy decreases (The heat has to go somewhere else, the entropy of the environment will increase but the entropy of the system can decrease). (h) The entropy of the universe can never decrease. True. The universe contains everything. If entropy decreases in one region (heat removed), the heat has to go somewhere hence entropy must increase in another region. If this is done reversibly then ΔS=0 otherwise ΔS>0. 4
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