Question
Background:
The equation of state for an ideal gas is given by the well known Ideal Gas Law:
pV = nRT, where R = 8.314 J/K
The equation of state for an ideal gas is given by the well known Ideal Gas Law: pV = nRT, where R = 8.314 J/K·mol and n is the number of moles of gas in the system. An adiabatic process is one where no heat flows into or out of the system. This may be because the system is well insulated or, as in this case (approximately), the process occurs too fast for any appreciable heat to flow. When an ideal gas undergoes an adiabatic process from state i to state f, the pressure and volume are constrained by: pf Vf ? = pi Vi ? where ? = Cp /CV, Cp is the heat capacity at constant pressure, and CV is the heat capacity at constant volume. The work done by an ideal gas when undergoing an adiabatic expansion (or contraction) from state i to state f is given by: Wi?f = pV ?(Vf1-? -Vi1-? )/(1-?). An isobaric process is one where the pressure of the system does not change. The heat flow into the gas undergoing an isobaric expansion (or contraction) from state i to state f is given by: Q = nCp(Tf -Ti) The work done by an ideal gas when undergoing an isobaric expansion (or compression) from state i to state f is given by: Wi?f = p(Vf -Vi) Assume that you have N2 as your working fluid. For N2: Cp = 6.95 cal/mol·K and CV = 4.96 cal/mol·K. Assume further that the weight has a mass of 53.9 g. Use g = 9.809 m/s2. You measure the diameter of the syringe plunger and calculate the area to be 1.75 cm2. The maximum possible efficiency of a heat engine is given by the Carnot engine, which operates between two heat reservoirs, a cold one with temperature TC and a hot one with temperature TH. The efficiency of a Carnot engine is given by Theoretical Maximum Efficiency = eCarnot = (TH-TC)/TH We measure the efficiency of an engine by dividing the net work performed by the heat engine W by the heat energy input QH from the hot reservoir during the so called power stroke. In this case, this is the step b rightarrow c. The measured efficiency is determined by Measured Heat Engine Efficiency = emeasured = W/QH You start with an initial gas volume of 19.01 ml, an initial gas pressure of 0.97x 105 Pa, a cold bath temperature TC of 0.96 degree C, and a hot bath temperature TH of 80.99 degree C. [NOTE: 0 degree C = 273.15 K] What is the maximum efficiency for a heat engine operating between these two temperatures? [HINT: Consider a Carnot cycle.] % How many moles of gas are in the system? moles Fill in missing information in the tables below: What is the total work done by the system? mJ What is the actual measured efficiency of this heat engine? %
Explanation / Answer
(a) Maximum efficiency = 1-(Tc/Th) = 1-((0.96+273)/(80.99+273)) = 22.6 %
(b) Moles = PV/RT = 0.97(atm) * 0.01901(l) / 0.0821*(0.96+273) = 8.2 x 10^-4 moles
(c) Using this equation, PV=nRT, and substituting n=8.2 x 10^-4 and the appropriate missing quantities everytime, we get
P in state b = 1.000139 atm = 100013.9 Pa
V in state c = 0.02383 l = 23.83 ml
T in state d = 350.84 K = 77.84 degrees C
Using delta Q = delta U + delta W (First Law of Thermodynamics),
b-> c is an isobaric process, so deltaU = 0; Q = U+W = 0+W = 522.1 mJ = 0.124 cal
d-> a is also an isobaric process, so, deltaU=0; Q=U+W; Q=0+W => W = -0.433 cal = -1811.6 mJ
(d) Total work done = -40.5+522.1+51.9-1811.6 = -1278.172 mJ
(e) Actual efficiency = Work done/Heat supplied = Work done/(Qh) = 1278.172/522.1 = 2.45 %
