Intermediate Thermodynamics Questions & Answers  




If all molecules would always have a velocity vector pointing in the positive $x$ direction, then the average velocity in the positive $x$ direction would be $\overline{q}$ (with $\overline{q}$ the molecular speed). But, molecules move in all directions randomly, not just in the positive $x$ direction. When integrating and taking a time average over all molecules in a gas, then it can be shown that the average velocity in the positive $x$ direction is $\frac{1}{4} \overline{q}$. I can give you 0.5 point for this question. For more points, you need to express better what you don't understand.




To answer your question, no, you cannot substitute $\overline{u^2}$ by $\overline{q_x}^2$. This is because by definition: $$ \overline{u^2} \equiv \frac{1}{\Delta t}\int_0^{\Delta t} u^2 dt $$ $$ \overline{q_x} \equiv \frac{1}{\Delta t}\int_0^{\Delta t} \max(0,u) dt $$ That is, $\overline{u^2}$ is the average in time of the square of the $x$ component of the velocity while $\overline{q_x}$ is the average in time of the component of the velocity in the positive $x$ direction. Taking the square of $\overline{q_x}$ will give a totally different answer as $\overline{u^2}$: $$ \frac{1}{\Delta t}\int_0^{\Delta t} u^2 dt \ne \left( \frac{1}{\Delta t}\int_0^{\Delta t} \max(0,u) dt\right)^2 $$ You can find more information about how to integrate the latter integrals in some book on the “Kinetic Theory of Gases” — but this is beyond the scope of this course. I'll give you 1 point for this question — for more points, you need to formulate it correctly the first time with the right notation. For the second part of your question, please delete it and ask a new question below (only one question per post). 



In class, we found that $\xi=N \bar{q}_x$. We used dimensions to give us a hint only. Ultimately, the number of particules hitting the wall per unit time per unit area is equal to the number of particules per unit volume ($N$) times the average velocity of the particules perpendicular to the wall. The velocity perpendicular to the wall corresponds to the average velocity of the molecules along one direction ($\bar{q}_x$, or $\bar{q}_y$, but not $\bar{q}$..). I chose the positive $x$ direction for illustrative purposes — I could have chosen the negative $x$ direction, or the positive $y$ direction, and we would have obtained the same answer. I'll give you 0.5 point for this question.




You're on the right track. Consider one molecule. It can move in any direction with equal probability. Thus, to find the average velocity along one direction, you need to integrate the molecule's velocity in spherical coordinates (on the surface of a sphere). I'll give you 0.5 point for this question. I would have given more if you would have asked it the first time without using an attached image and if the post would be free of spelling mistakes.



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