Problem: 1 mol. mono-atomic ideal gas has gone through the cyclic process as shown in the figure, and entropy temperature diagram S-T has been drawn below. The gas has passed through a circle with equation: \left(\frac{S}{S_{0}}-2\right)^{2}+\left(\frac{T}{T_{0}}-2\right)^{2}=1, where S_0 = 2R, T_0 is a constant. When the gas temperature, volume and pressure is T_0, V_0, p_0, the size of the entropy is taken as S_0.(equivalent to specifying point 0)
If it is required that this is a positive cycle, that is, a heat engine with gas as the medium in a cycle does work externally, determine the direction of the cycle in the diagram.
Find the corresponding heat engine cycle efficiency.
Find the maximum volume V_m/V_0 in this cycle.
Please provide a solution for this problem. I don’t have answer key for this, it is difficult to solve it.
Since the heat transferred to the gas is dQ=TdS, the total heat will be the area under the graph of T(S). For positive work we must have \oint T(S) dS >0 , and direction of cycle in the diagram will be anti clockwise.
Efficiency therefore can be found as \displaystyle \eta = \frac{\oint T(S) dS}{\int TdS} = \frac{2\pi RT_0}{\pi RT_0 + 8RT_0} = \frac{2\pi}{\pi + 8} (\oint T(S) dS is the area of the circle)
for 1 mol. monoatomic ideal gas entropy will be S= S_0 + R \ln\left(\left(\frac{T}{T_0} \right)^{1.5}\left(\frac{V}{V_0}\right )\right), from which we can find that \displaystyle V= V_0 \left(\frac{T}{T_0} \right)^{-1.5} \exp\left(\frac{S(T)-S_0}{R}\right), by differentiating this and using \displaystyle \frac{dV}{dT}=0 you will find the temperature and maximum volume.
(You will probably get a 4th degree equation which you can solve numerically by using Newton's method - Wikipedia)
**I am not sure if the problem was stated correctly because they said that for temperature T_0 we must have entropy S_0, but this does not correspond to the graph
One line I missed, it was written that When the gas temperature, volume and pressure are T_0, V_0 and p_0, the size of the entropy is taken as S_0 (equivalent to specifying point 0). Does this make any difference ?
Where would you rate its difficulty? I think your first two parts answers are correct… One of my friends solved the 3rd part and he got answer as V_m = 22.85V_0
Glad you liked it…
Actually, this was one of the only problems where I saw use of entropy-temperature graph. Maybe it is used more extensively in training camps etc. but I found this pretty unique nonetheless.
Hi! Damir has been at the IJSO* 2023 camp in Columbia. It was amazingly and he won bronze there:hot_face:!!!
*IJSO - International Junior Science Olimpiad, there are three subjects: Biology, Chemistry and, sure, Physics. And you must write three this subjects there
!!!