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Welcome to thermodynamics in energy engineering week 6.
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We are going to demonstrate the usefulness of the property relations by applying them to various situations.
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Here is the first example to demonstrate the usefulness of the property relations.
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We are going to calculate the entropy change of isothermal pressure change from 1 to 10 atm.
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Pressure changes at constant T, so we need to know dS over dP at constant T.
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dS over dP at const T can be obtained from dG. Look at the two independent variables. They are T and P.
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Which thermodynamic potential has T, P as variables? It's G.
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Applying the exactness property to G gives this property relation.
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-(dS over dP) at constant T is (dV over dT) at constant P. Let's first assume that our system is the ideal gas.
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For ideal gas, (dV over dT) at constant P is (R over V) from the equation of state.
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So the entropy change is now integration of - (R over P) dP from 1 to 10 atm. The result is -19.14 J/mole.k.
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Second, let's assume our system as a solid under isothermal compression.
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The property relation also holds in this case. Here, the equation of state for ideal gas is not valid.
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Instead, we can use the alpha v, the volume expansion coefficient defined like this.
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Alpha v is (1 over V) times (dV over dT) at constant p. So (dV over dT) at constant P is alpha v times V.
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The entropy change is then integration of V times (alpha v) dP from 1 to 10 atm.
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If V and alpha v do not depend on P or if the pressure range of interest is small, we can regard
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V and alpha v as constant. So the entropy change is - V times alpha V times pressure change.
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Let's insert the real value of copper to get actual entropy change.
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For copper, specific volume is 7.09 times 10 to -6 cubic meter per mole,
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and the linear expansion coefficient is 1.67 times 10 to -6 meter per meter K.
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The relation between volume expansion coefficient
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and the linear expansion coefficient is alpha v is three times alpha L.
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The entropy change with this value is - three times (7.09 times 10 to -6) times (1.67 times 10 to -6) times
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(10-1) atm times unit conversion factor (1.013 times 10 to 5 ) newton (meter squared) per atm.
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The entropy change is 3.24 times 10 to -4 J/mole.K.
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Which is much much smaller than the entropy change of ideal gas under the same compression.