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Equation of state for adiabatic, reversible changes of ideal gases

Equation of states for adiabatic, reversible changes of ideal gases
Then, let’s derive the equation of state for adiabatic, reversible expansion or contraction of ideal gas. The equation of state in this case describe the conditions of ideal gas during the adiabatic reversible expansion or contraction. Let’s regard this adiabatic, reversible expansion or contraction as the pressure change from P1 to P2 However, different from the adiabatic, reversible changes, there can be different pathways to change pressure from P1 to P2. First, think about the isothermal expansion and contraction. The pressure changes isothermally from P2 to P1 upon expansion or contraction For isothermal process of the ideal gas, dU is zero. By first law of thermodynamics, dU is delta Q + delta W.
So, delta Q equal to -delta W, and it is the P-V work PdV. Inserting the equation of states yields that Q is nRT natural log P1 over P2. The second process is the reversible, adiabatic change. In this case, P, T, V all changes different from the isothermal process where T is fixed. Since it’s adiabatic, delta Q is zero. Then by the first law, dU is -PdV and it is CvdT. P is (RT over V) from the equation of state for ideal gas. Separation of variables gives this equation. Integration on both side with respect to the corresponding variables results in this equation Cv times log T1 over T2 is R times log V1 over V2. Express this equation as index equation.
Then, it becomes T2 over T1 to Cv is V2 over V1 to R. Modify the equation a bit more. We know that Cp-Cv is R for ideal gas. And let’s define Cp over Cv as gamma. Then, R over Cv is gamma -1. Then we can rearrange the above equation as P1 times (V1 to gamma) is P2 times (V2 to gamma). In other words, P times V to gamma is constant. This is the equation of states for reversible adiabatic changes for ideal gas. To be noted here is that the PV equal to nRT, the general equation of state for ideal gas holds for all ideal gas, even for this reversible adiabatic changes.
However, this equation, P times (V to gamma) is constant is exclusively applicable to ideal gases under adiabatic, reversible changes Let’s draw the curves of the equation of states on P-V graph for both isothermal and adiabatic cases. Isothermal reversible change from P1 to P2 is like this. It follows the PV equal to nRT, so P is nRT over V. For adiabatic, reversible changes PV to gamma is constant, so P is constant C/V to gamma. The gamma, the Cp/Cv is always larger than 1 since Cp is larger than Cv. So the slope of this curve is steeper than the isothermal case with the index of 1. The area under this curve is the work done by the system.
Here, the orange area is the work done by the isothermal process. The purple area is the work done by the adiabatic process. Therefore, the adiabatic process does less work on the surroundings.

The equation of state for the ideal gases under adiabatic and reversible volume changes can be formulated in a very special form.

The physical variables of ideal gases under adiabatic and reversible volume changes can be related by different mathematical forms from the general equation of state, although the ideal gas law, the general equation of state for ideal gases, is always valid.

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Thermodynamics in Energy Engineering

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