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Now, let's apply the property relation to see the pressure dependence of Cp.
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Cp is (dH over dT) at constant pressure. Let's start from enthalpy as a function of temperature and pressure.
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Then, the total differential of enthalpy is like this. (dH over dT) at constant P times dT
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+ (dH over dP) at constant T times dP. This is Cp by definition.
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Then from the exactness of a state function such as enthalpy, differentiation of Cp
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with respect to pressure at constant T is the same with differentiation of (dH over dP) at constant T
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with respect to temperature as shown in this equation.
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So the temperature dependence of this function, dH over dP, gives pressure dependence of Cp.
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Let's look at the temperature dependence of this function.
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Previously we have shown that (dH over dP) at constant T is - T times (dV over dP) at const P + V.
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Insert it into this equation, we just derived.
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Then, we need to differentiate (V - T times (dV over dT) at const P) with respect to temperature.
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Expand the equation. It becomes (dV over dT) - (dV over dT) - T times second derivative of V
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with respect to T. Cancel out the same things.
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Then, dCp over dP at constant T is - T times second derivative of V with respect to T at constant P.