Proof question related to the Ideal Gas Law

[Kajan thana]

A cylinder contains an initial volume V1 = 1m^^3 of a perfect gas at initial pressure p1 = 1 bar, confined by a piston that is held in place by a spring. The gas is heated until its volume is doubled and the final pressure is 5 bar. Assuming that the mass of the piston is negligible and that the initial force on the spring is zero, show that the pressure and volume V are related according to: 𝑝 − 𝑝1 ∝ 𝑉 − 𝑉1

I tried to work this backward so 𝑝 − 𝑝1 = k (𝑉 − 𝑉1) where k is the constant. After that, I don't know how to go about solving this question. Are we assuming the temperature remains constant in this question?

 

[Chestermiller]

From freshman physics, we learned how do do an equilibrium force balance. So, if the area of the piston is A and the displacement of the piston (and spring) is x, what is your equilibrium force balance?

 

[Charles Link]

One observation is that the external pressure must also be ## p_1 ##. The rest is basically an ideal spring problem.
Note: They could also ask you to compute the final temperature w.r.t. the initial temperature, (assuming ideal gas law), and they could even ask you to compute the spring constant ## k ## in terms of the other parameters including the area).

 

[Kajan thana]

Forces need to balance so it will follow as this pA = kx + P₁A and we know that change of volume, V-V₁ is proportional to x where x is the length of the spring that is getting compressed. With a few rearrangements, I should get the answer.

 

[Charles Link]

Note that you know the constant of proportionality here: ## V-V_1=Ax ##.

 

[Kajan thana]

Note that you know the constant of proportionality here: ## V-V_1=Ax ##.

Thank you Charles and Chestermiller

 

[Charles Link]

Let's see your final result=what do you get for ## C ## where ## p-p_1=C(V-V_1) ##?

 

[Kajan thana]

From freshman physics, we learned how do do an equilibrium force balance. So, if the area of the piston is A and the displacement of the piston (and spring) is x, what is your equilibrium force balance?

Let's see your final result=what do you get for ## C ## where ## p-p_1=C(V-V_1) ##?

Does C = k/A² where k is the spring constant and A is area of the piston?

 

[Kajan thana]

Does C = k/A² where k is the spring constant and A is area of the piston?

Thank you again.. you are a star..