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 - Conceptual Questions - Page 522: 6

Answer

a) $2$ b) $2^2$

Work Step by Step

a) We know that the RMS speed is given by $$v_{\rm rms}=\sqrt{(v_{avg})^2}$$ Let's assume we have $N$ number of molecules, so the average speed of them is given by $$v_{avg}=\dfrac{v_1+v_2+...+v_N}{N}$$ Increasing the speed of each molecule by a factor of 2 means increasing the total average speed by 2. $$v_{avg}'=\dfrac{2v_1+2v_2+...+2v_N}{N}=2\dfrac{v_1+v_2+...+v_N}{N}=2v_{avg}$$ Hence, the RMS speed then is $$v_{\rm rms}=\sqrt{(v_{avg}')^2}=\sqrt{(2v_{avg})^2}$$ $$v_{\rm rms}=2\sqrt{(v_{avg})^2}$$ So, the RMS speed increased by a factor of 2. --- b) We know that the pressure due to the force of the molecules colliding with the walls is given by $$P=\dfrac{Nm }{3V}v_{\rm rms}^2$$ So when the RMS speed increases by a factor of 2, the pressure increases by a factor of $2^2=4$ since $P\propto v_{\rm rms}^2$
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