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