Equilibrium of Volume Energy

We still consider two gases as shown in Fig. 6.2 however, we will use now the subscripts and superscripts that are common in thermodynamics. We will refer to the gas in the left chamber with ( ) and the gas in the right chamber with ( ), and consequently, we will index all the parameters that will refer to these gases in this way. 

if only the exchange of volume energy, i.e., the energy of compression and the energy of expansion is in between the two gases, is allowed, we will rewrite 

Equation (6.7) states that the pressure in the left chamber and the pressure in the right chamber must be equal. We have used the condition that the energy must be stationary and the sum of the volumes must be constant to obtain the condition of equilibrium. We can use both requirements to check if there is in fact equilibrium. This means 

We can change the volume of the gas in the left chamber by d V and the volume of the gas in the right chamber by — dV . In this way, the condition V + V = Vtot = C is fulfilled. If we are doing so, then we should stay at constant energy with respect to the first total differential, i.e., dU = 0. 

We can check if the pressure in the left chamber is the same as the pressure in the right chamber. 

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