Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.3 Partial Derivatives - 14.3 Exercises - Page 966: 89


$T\dfrac{\partial P}{\partial T} \dfrac{\partial V}{\partial T}=mR$

Work Step by Step

Since, $P= \dfrac{mRT}{V} ,V=\dfrac{mRT}{P}; T= \dfrac{PV}{mR}$ $\dfrac{\partial P}{\partial T}=\dfrac{\partial}{\partial T}[\dfrac{mRT}{V}]=\dfrac{mR}{V}$ ...(1) $\dfrac{\partial V}{\partial T}=\dfrac{\partial}{\partial T}[\dfrac{mRT}{P}]=\dfrac{mR}{P}$ ....(2) From the equatiosn (1) and (2) , we get $T \times \dfrac{\partial P}{\partial T} \times \dfrac{\partial V}{\partial T}=(T) \times \dfrac{mR}{V} \times \dfrac{mR}{P}$ Hence, $T \times \dfrac{\partial P}{\partial T} \times \dfrac{\partial V}{\partial T}=(\dfrac{PV}{mR}) (\dfrac{mR}{V}) (\dfrac{mR}{P})=mR$
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