An unknown gas $($i$.$e$.,$ cannot be treated as an ideal gas$)$ is contained in a well$-$sealed piston$-$cylinder device. The piston can be considered effectively massless. There is a spring on the backside of the piston that changes the force exerted on the piston as the gas volume changes. The spring is initially uncompressed and under mechanical equilibrium. The initial pressure of the gas is $P_0.$

a$)$ Draw two free body diagrams $($FBDs$)$ of the piston: one at the initial state with the spring in its equilibrium position, and one at some later state when the gas has expanded to some larger volume.

b$)$ Recall that the force of a typical spring is $F=kx,$ where $x$ is measured from the uncompressed equilibrium position of the spring. Knowing this, derive an equation for the pressure of the gas $(P_{\text{g}\text{a}\text{s}})$ as a function of the specific volume of the gas $(v_{\text{g}\text{a}\text{s}}).$ Your equation should include the initial pressure of the gas $(P_0),$ the initial specific volume of the gas $(v_0),$ the cross$-$sectional area of the piston $(A),$ the mass of the gas $(m_g),$ the spring constant $(k).$ That is$,$ derive a function of the form

$P_{\text{g}\text{a}\text{s}}=f(v_{\text{g}\text{a}\text{s}},P_0,v_0,A,m_g,k)$

Hint: Consider a force balance on the piston after it has moved upward some distance. Also, how do you relate a displacement of the piston to a change in volume of the gas for this geometry?

c$)$ Based on the form of your function, how does the pressure of the gas vary with specific volume? Quadratically? Logarithmically? Exponentially? Some other general functional relationship?

d$)$ Draw a P$-$v diagram for the gas as the system goes from the initial state $P_0,v_0$ to some final state $P,v.$ The shape of the curve $($e$.$g$.,$ linear, polynomial, exponential$)$ must be well defined to receive full credit.