Calculating pressure from a known equation of energy

AI Thread Summary
The discussion focuses on calculating pressure using energy equations during heat transfer transitions. The participant attempts to derive relationships between energy changes and pressure but struggles with integral calculations. Key equations involve the change in energy across states and the specific heat capacity, leading to insights about entropy changes. A potential relationship for pressure is proposed, incorporating temperature and volume dependencies. Despite progress, the participant remains uncertain about proving their findings and seeks further insights.
ForgetfulPhysicist
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Homework Statement
Heat Engine with working substance characterized by energy E = g T^a V, with a>1 and g>0 being known coefficients. The values of P1, V1 are also known. Find P2 /P1 in terms of the known quantities.
Relevant Equations
dE = TdS - PdV
Screen Shot 2023-03-18 at 7.19.06 PM.png


My attempted solution is as follows:

Obviously the heat transfer happens during transitions 1->2 and 3->1.

It's also clear that
P1 = P3
V1 = V2

E2 - E1 = Integral[T dQ , from state 1 to state 2]

E3 - E2 = - Integral[P dV , from state 2 to state 3]

E1 - E3 = Integral[T dQ , from state 3 to state 1] + 7 P1 V1

But I can't find a way to perform any of these integrals or make any progress on this problem.

An attempt to calculate pressure is stuck at: p = - (dE/dV)_S = g a T^(a-1) (dT/dV)_S V + g T^a
 
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ForgetfulPhysicist said:
E2 - E1 = Integral[T dQ , from state 1 to state 2]

E1 - E3 = Integral[T dQ , from state 3 to state 1] + 7 P1 V1
Check the units in these equations.
 
Philip Koeck said:
Check the units in these equations.
Yes that was a typo. They should be written:
E2 - E1 = Integral[dQ , from state 1 to state 2]
E1 - E3 = Integral[dQ , from state 3 to state 1] + 7 P1 V1

One idea: I can calculate Cv = (dE/dT)_V = a g T^(a-1) V , and I also know Cv = T (dS/dT)_V which helps me know a little bit about the change of entropy from state 1 to 2.... but I'm still stuck.
 
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ForgetfulPhysicist said:
Yes that was a typo. They should be written:
E2 - E1 = Integral[dQ , from state 1 to state 2]
E1 - E3 = Integral[dQ , from state 3 to state 1] + 7 P1 V1

One idea: I can calculate Cv = (dE/dT)_V = a g T^(a-1) V , and I also know Cv = T (dS/dT)_V which helps me know a little bit about the change of entropy from state 1 to 2.... but I'm still stuck.
I also made a little progress on this by using $$\left(\frac{\partial E}{\partial V}\right)_T=-\left[P-T\left(\frac{\partial P}{\partial T}\right)_V\right]=gT^a$$which leads to $$P=\frac{gT^a}{a-1}+Tf(V)$$
 
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I derived the equation for entropy variation also, but I've as yet not been able to figure out a way that it can be used to provide an answer to this problem: $$dS=\frac{ga}{(a-1)}d(T^{a-1}V)+f(V)dV$$
 
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From dimensional considerations, it makes sense to me that the function f(V) should be proportional to R/V, where R is the gas constant. If f(V) were equal to R/v, we would have $$\frac{PV}{RT}=z=1+\frac{1}{a-1}\frac{E}{RT}$$I'm unable to prove this yet, but I may proceed as if it is the case and see where it takes me.
 
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