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## Homework Statement

"A car has a six-cylinder Otto-cycle engine with compression ratio r = 10.6.

The diameter of each cylinder is 82.5 mm.

The distance that the piston moves during the compression stroke (see fig. 1) is 86.4 mm.

The initial pressure of the air-fuel mixture (at point a in fig. 2) is 8.50 x 10^4 Pa and the initial temperature is 300K (the same as the outside air).

Assume that 200 J of heat is added to each cylinder in each cycle by the burning petrol and that the gas has C

_{V}= 20.5 J.mol/K and γ = 1.40."

(a) By considering the efficiency of the engine, calculate

(i) the total work done in one cycle in each cylinder of the engine, and

(ii) the heat released when the gas is cooled to the temperature of the air outside.

(b) Calculate the volume of the air-fuel mixture at point a in the cycle.

(c) Calculate the pressure, volume, and the temperature of the gas at points b, c, and d in the cycle. In a pV-diagram, show numerical values of p, V and T for each of the four states.

(d) Compare the efficiency of this engine with the efficiency of a Carnot-cycle engine operating between the same maximum and minimum temperatures

## The Attempt at a Solution

(a)(i) has me completely baffled. I understand that the work is the area bounded by the two adiabats and the vertical isochors, but I don't see how this is related to the efficiency if η = 1 - r

^{1-γ}= 1 - (|Q

_{C}| / Q

_{H}).

That relation gives me a value for (a)(ii) of |Q

_{C}| = Q

_{H}*r

^{1-γ}- 200*10.6

^{-0.4}= 77.8 J, though; is this on the right lines?

(b) I tried to use the relationship for an adiabatic process TV

^{γ-1}= constant, so:

T

However, the 'V' terms just cancel here. I then considered the ideal gas equation V=nRT/P, but there is no value for the number of moles of working substance._{a}(rV)^{γ-1}= T_{b}V^{γ-1}(c) I think I will be able to do on my own once I am pointed in the right direction for (b); right now I feel like I am missing information I need to be able to do the question, but filling in the first few gaps should help me enough.

(d) Again, once I actually have the value the maximum temperature I can use the Carnot efficiency η = 1 - T

_{C}/T

_{H}to compare the value of η ≈ 0.611 for this Otto cycle. I suspect it should be lower given the nature of the Carnot engine?

My biggest problem with thermodynamics at the moment is a massive unfamiliarity with many of the key relations between variables; I have a nagging feeling I'm either overlooking the obvious or am unable to find the relationship that would make the questions tractable in my notes.

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