Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 18 - The Micro/Macro Connection - Exercises and Problems - Page 524: 50

Answer

$209^\circ \rm C$

Work Step by Step

We know that the thermal energy of an ideal gas is given by $$E_{th}=nC_{\rm V}T$$ where $C_{\rm V}$ is given by $C_{\rm P}-C_{\rm V}=R$ Hence, $$E_{th}=n(C_{\rm P}-R)T$$ So the temperature of the gas is then given by $$T=\dfrac{E_{th}}{n(C_{\rm P}-R)}\tag 1$$ We also know that the number of atoms of molecules of an ideal gas is given by $N=nN_{\rm A}$; where $N_{\rm A}$ is Avogadro's number, and hence $n=N/N_{\rm A}$. Plug that into (1); $$T=\dfrac{N_{\rm A}E_{th}}{N(C_{\rm P}-R)} $$ Plugging the known; $$T=\dfrac{(6.022\times 10^{23})(1)}{(1\times 10^{20})(20.8-8.31)} $$ $$T=\color{red}{\bf 482}\;\rm K=\color{red}{\bf 209}^\circ C$$
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