Physics Thermodynamics - Finding Work

AI Thread Summary
The discussion revolves around calculating the work done by a diatomic gas undergoing a process where pressure increases linearly with temperature. The participants express confusion over the relationship between pressure and volume, noting that if pressure varies linearly with temperature, it leads to inconsistencies in the expected behavior of the gas. They explore the implications of different linear equations for pressure and how they affect the volume calculations necessary for determining work. One participant derives a complex expression for work but acknowledges potential errors and discrepancies with the expected answer. The conversation highlights the challenges in applying thermodynamic principles to non-constant volume scenarios.
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Homework Statement


A 1.20 mol sample of an ideal diatomic gas at a pressure of 1.20 atm (P1) and temperature of 380 K (T1) undergoes a process in which its pressure increases linearly with temperature. The final temperature and pressure are 680 K (T2) and 1.83 atm (P2).

(b) Determine the work done by the gas.


Homework Equations



(T2 - T1) / (P2 - P1) = constant

V2 / V1 = P1T2 / P2T1

PV = nRT

W = integral of PV

The Attempt at a Solution



The answer is supposed to be [nR(P1T2 - P2T1) ln(P2/P1)] / (P2 - P1)
but I can't figure out how to get there despite the hours I have poured into this problem. All I know is that volume is not constant in order for there to be work done. Also, I have the vague idea of finding the linear equation or relationship between T and P using that equation to plug in T in PV = nRT. Then, technically, there would be a graph for PV and I could solve for V1 and V2, which can be the limits of integration for the integral of PV (to get work). But isn't that way too complicated, especially as an integral...so I was wondering whether someone else has any idea.

Please help explain, and thank you!
 
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I certainly understand your frustration!
If P varies linearly with T, you would expect P = kT. Put that in V = nRT/P and you get that V is a constant, which is wrong. Looks like the question is inconsistent!

The other meaning of "linear" would be P = kT + b so V = nRT/(kT+b) which doesn't seem likely. It does give quite a complex answer.
Since P and V both vary, dW = PdV + VdP
I worked the integral quickly and got
W = -nRb(P2 - P1) + nR/k*[P2-P1 -ln(P2/P1)]
Could well be errors in that! It doesn't look much like the given answer, does it?
 
Ah, thank you so much for the insight. (I'll have to try and work that out when I can find the time.)
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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