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Here, we are going to a derive useful formulation to calculate entropy changes easily.
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Let's set entropy function as a function of temperature and volume
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and formulate total differential of entropy function for ideal gas.
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This total differential can be used in calculating entropy changes with temperature and volume easily.
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The formulation starts from the internal energy function U,
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since U is a function of temperature and volume. dS by definition is delta Q over T.
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dU is delta Q - PdV by the first law of thermodynamics if P - V work is reversible.
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For one mole of ideal gas, dU is Cv dT, since U is a function of temperature only.
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Then, the first law becomes like this. Delta Q is dU + PdV and it is CvdT + PdV.
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Equation of state for ideal gas is PV equal to RT, so pressure P is RT over V.
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Insert this into the definition of entropy. Then, dS is (Cv over T) dT + (R over V) dV.
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We can use this total differential in deriving dependence of entropy on other variables.