Intermediate Thermodynamics Questions & Answers  


Now I understand your question better. The Boltzman constant $k_{\rm B}$ is equal to $\bar{R}/N_{\rm A}$ with $\bar{R}$ the universal gas constant, not to $R/N_{\rm A}$ ($R$ is the gas constant which varies from gas to gas). I can give you a bonus boost of 0.5 point for this question.




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.




There is a mistake in your question: I didn't say that $\overline{q}^2=\overline{u}^2+\overline{v}^2+\overline{w}^2$ but I said that $\overline{q^2}=\overline{u^2}+\overline{v^2}+\overline{w^2}$. Please edit your question and make sure that you use exactly the same notation as in class.. Also, to put some blank spacing in equations, use the tilde character. I will answer your question after you make these modifications.


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). 



Please write the mathematics a bit better. Use “\left[” and “\right]” instead of “[” and “]” and use “\times” instead of “*”.



$\pi$ 