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Entropy as a function of temperature and volume

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

We can express the entropy as a function of temperature and volume.

It can be derived from the combination of the first and the second law for the closed system. For ideal gas the temperature dependence of entropy at constant volume is simply Cv over T. The volume dependence of entropy at constant temperature is R over volume for ideal gases.

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Thermodynamics in Energy Engineering

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