Physics (10th Edition)

Published by Wiley
ISBN 10: 1118486897
ISBN 13: 978-1-11848-689-4

Chapter 14 - The Ideal Gas Law and Kinetic Theory - Problems - Page 385: 29

Answer

0.09%

Work Step by Step

Let's apply the ideal gas law $PV=nRT$ to the gas to find the volume of the gas. $PV=nRT=>V_{gas}=\frac{nRT}{P}-(1)$ We can write the desired percentage as follows, Percentage = $\frac{V_{atoms}}{V_{gas}}\times100\%$ Percentage = $\frac{(\frac{4}{3}\pi r^{3})nN_{A}}{V_{gas}}\times100\%-(2)$ (1)=>(2), Percentage = $\frac{(\frac{4}{3}\pi r^{3})nN_{A}}{\frac{nRT}{P}}\times100\%=\frac{(\frac{4}{3}\pi r^{3})N_{A}P}{RT}\times100\%$ Let's plug known values into this equation. Percentage = $\frac{x\frac{4}{3}\pi(2\times10^{-10}m)^{3}(6.022\times10^{23}mol^{-1})(1.01\times10^{5}Pa)}{(8.31\space J/mol\space K)(273\space K)}\times100\%=0.09\%$
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