Calculating pressure from a known equation of energy

In summary: I also made a little progress on this by using dimensional analysis . This helped me to identify the function f(V) which is proportional to R/V, and it also gave me an equation for the pressure, which is P=\frac{gT^a}{a-1}+Tf(V).
  • #1
ForgetfulPhysicist
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2
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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  • #2
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.
 
  • #3
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.
 
Last edited:
  • #4
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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  • #5
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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  • #6
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.
 

1. How do you calculate pressure from a known equation of energy?

To calculate pressure from a known equation of energy, you will need to use the formula P = E/V, where P is pressure, E is energy, and V is volume. You will also need to make sure that the units for energy and volume are consistent.

2. What is the relationship between pressure and energy?

Pressure and energy have an inverse relationship. This means that as pressure increases, energy decreases, and vice versa. This relationship is described by the equation P = E/V.

3. How does volume affect pressure in the equation of energy?

In the equation P = E/V, volume has a direct relationship with pressure. This means that as volume increases, pressure decreases, and vice versa. This is because as volume increases, the same amount of energy is spread out over a larger area, resulting in a decrease in pressure.

4. Can you use any unit of measurement for energy and volume in the pressure equation?

No, it is important to use consistent units for energy and volume in the pressure equation. For example, if energy is measured in joules, volume should be measured in cubic meters. Using different units can result in incorrect calculations.

5. How can the pressure equation be used in real-world applications?

The pressure equation can be used in a variety of real-world applications, such as calculating the pressure inside a container or the pressure exerted by a gas. It is also used in industries such as engineering, chemistry, and physics to analyze and design systems that involve pressure and energy.

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