Spring Coupled with Two Gases

Consider the system shown in Fig. 2.5. Here, two gases (1) and (2) are separated by a freely movable piston. Further the gas (2) is coupled by a movable piston with a spring. The spring is fixed on the other end. The total volume Vtot of both gases is the sum of their individual volumes Vi and V2 

The energy of the gas is a function of entropy S, volume V, and mol number n. Since we have two gases we will provide the indices (1) and (2). Thus 

We assume now that the walls of the systems are impermeable to entropy and matter. Further we consider a state of equilibrium. This assumption implies the constraints dS i = 0, dS 2 = 0, d i = 0, d 2 = 0. For this reason we may express the total energy merely as a function of the volumes, i.e.. 

Using the constraint equation (2.56) together with the method of undetermined multipliers, 

without knowing the special form of the energy of the spring, we come to the conclusion that in equilibrium the pressures of the gases must be equal, i.e., pi = P2. 

---