What is the minimum temperature for an ideal gas following a circular path?

[kalbuskj31]

Homework Statement

2 moles of an ideal gas, P1 = 10 atm V1 = 5L, are taken reversibly in a clockwise direction around a circular path given by (V-10)^2 + (P-10)^2 = 25. All I have left to solve for is the minimum temperature.

Homework Equations

PV = nRT

The Attempt at a Solution

I found T1 = PV/nR = (10atm * 5L) / (2 moles * .08206) = 304.66K

From the equation I found out that the center of the circle is (V, P) = (10,10) with a Radius of 5.

From there I broke the circle up into 4 different sections starting at (5,10) and going clockwise. I found the max by saying that P = V and between (10, 15) and (15,10),

(V-10)^2 = (P-10)^2 = 12.5
P-10 = 3.536 Vmax = Pmax = 13.536 and Vmin = Pmin = 6.464

PV = nRT

Tmax = (13.536^2)/(2 moles * .08206) = 1116K (correct answer)
Tmin = (6.464^2)/(2moles * .08206) = 255K (Answer:Tmin = 225K)

Did I miscalculate where PV is a minimum between the section of (10,5) and (5,10) of the circle?

 

[presbyope]

Did I miscalculate where PV is a minimum between the section of (10,5) and (5,10) of the circle?

No you're fine. Mathematically you are minimizing $PV=(5\sin{t} +10)(5\cos{t}+10)$. Graphing is easiest. For any real data that's what you'd have to do anyway.