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While studying the book Heat and Thermodynamics by Zemansky and RH Dittman, in the topic 'equation for a hydrostatic system' (page no. 88) it was given

red squared dv/dt

in equation 4.12, when we take Pressure P constant, how the dv/dt (marked in red sqaure) got converted into partial derivative? (marked in red square) enter image description here

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  • $\begingroup$ V is a function of P, T,and you are treating one variable as a constant $\endgroup$
    – Kashmiri
    Aug 18, 2020 at 8:29
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    $\begingroup$ Please use Mathjax to typeset your equations $\endgroup$
    – Arnav Mahajan
    Aug 18, 2020 at 8:43

1 Answer 1

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Write $$dV=\left(\frac{\partial V}{\partial P}\right)_TdP+\left(\frac{\partial V}{\partial T}\right)_PdT,$$ then $$\frac{dV}{dT}=\left(\frac{\partial V}{\partial P}\right)_T\frac{dP}{dT}+\left(\frac{\partial V}{\partial T}\right)_P.$$ If the pressure is constant $dP=0$, and the first term drops out $$\frac{dV}{dT}=\left(\frac{\partial V}{\partial T}\right)_P.$$

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