deletions | additions       diff --git a/Ideal Gas Law Simulation Report.tex b/Ideal Gas Law Simulation Report.tex  index bd77857..f0c6387 100644  --- a/Ideal Gas Law Simulation Report.tex  +++ b/Ideal Gas Law Simulation Report.tex 
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\usepackage[utf8]{inputenc}  \fi  \fi  \usepackage{graphicx}  % We will generate all images so they have a width \maxwidth. This means  % that they will get their normal width if they fit onto the page, but  % are scaled down if they would overflow the margins.  \makeatletter  \def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth  \else\Gin@nat@width\fi}  \makeatother  \let\Oldincludegraphics\includegraphics  \renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=\maxwidth]{#1}} \usepackage{url}  \ifxetex  \usepackage[setpagesize=false, % page size defined by xetex  unicode=false, % unicode breaks when used with xetex 
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In this lab, I set out to create a 3D simulation of ideal gas particles  in a cubic container in order to experimentally determine the pressure  of the gas based on given circumstances, such as circumstances. From there, I planned to  explore  the the volume of the  container relationship between pressure  and the temperature of the system. volume.  To produce an accurate simulation, a replication of a real world  circumstance using programming, of a gas particle it is first necessary 
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written as $Pressure=F/A$. Force can also be described as change in  momentum over change in time: $F = \frac{\Delta p}{\Delta t}$. The  change momentum of a single particle equals its mass multiplied by its  change in velocity: ${\Delta p} = m\Delta v$ v$.  Since there is more than one particle in a system, the entire change in momentum is the combined  change in velocities of each particle that hits the specified area.  Thus, the following formula can be used to determine total force:  ~  $F = \frac{2m * \displaystyle\sum\limits_{0}^n v}{\Delta t}$ Where $n$ is the number of collisions and $v$ is the velocity of the  particle hitting the wall. Since the change in velocity is double the 
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Once the force has been computed using the momentum of the particles,  the pressure can then be determined with the initial formula $P = F/A$.  \begin{figure}[htbp]  \centering  \includegraphics{figures/Screen Shot 2013-01-19 at 6.37.29 PM/Screen Shot 2013-01-19 at 6.37.29 PM.png}  \caption{image}  \end{figure}  \section{Hypothesis}  Increasing the number of particles in the simulation will yield a  pressure closer to the actual value (determined using the Ideal Gas  Law). Furthermore, increasing the volume of a container will decrease  the pressure in the system.  \section{Method}  \subsection{Computing Pressure}  I started began constructing my simulation with '3D Balls Bouncing', a project  from Open Processing, as a base. Starting  off with a system that could  already handle 3D collisions of small objects inside a container was  necessary {[}see Failed Methods{]}. From there, I created small spheres  with the properties of an ideal gas. Their speed was roughly determined  on a Maxwell-Boltzman distribution and assuming that the most probable  speed ($V_{p}$) would occur the most often (calculated using the formula  $V_{p} = \sqrt{\frac{2kT}{m}}$. Next, I focused on one wall in the  container and tracked each time a particle collided with the area. I  could then compute the change in momentum over the area since I had the  masses and velocities of the particles. The simulation did not keep  track of total time since initiation, which is equivalent to time, but I  knew that aspects of the code were executed every frame. The project ran  at a fixed number of frames per second (60) so I designed this formula  to figured out the change in time:  $total\ time (seconds) = \frac{total\ frames}{frames\ per\ second}$.  I inserted that data into this formula  $F = \frac{2m * \displaystyle\sum\limits_{0}^n v}{\Delta t}$ (see  Introduction) to find the total force on the wall. To obtain the  pressure, I just divided the answer  by creating the area of the wall.  \subsection{Testing Accuracy}  Now that I was able to compute the pressure, I could test my hypothesis  by increasing the number of particles in the system and comparing the  pressure readings. I started off with 5 particles and then tried 10, 15,  and 20 particles. I calculated the total pressure of each system once  every 5 seconds for 20 seconds.  \subsection{Experimentally Assessing the Relationship Between Pressure  and Volume}  The next step of  my own experiment involved manipulating the volume of the  container while keeping the number of particles constant. In order to do  so, I altered one line of code in the box Class:  \texttt{int boxsize = 300;}  This variable alters the dimensions of the box (currently 300x300x300  pixels). I changed the boxsize to 400, 500, and 600, and calculated the  pressure every 5 seconds for 20 seconds.  Since the simulation could not handle a large number of particles, I  \section{Attempted \subsection{Failed  Methods} I originally designed my own 3D collision system but it was less  efficient so a computer could not render as many particles in the 
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Concisely state the results and what I learned  \begin{figure}[htbp]  \centering  \includegraphics{figures/figure_1/friedrich-wanderer-above-the-sea-of-fog-1200x1024.jpeg}  \caption{image}  \end{figure} \section{Improvements}  \begin{enumerate}  \item  Since the simulation could not handle a large number of particles, I  had to resort to using an unrealistically low number of particles  (e.g. 5 particles vs $5 x 10^{23}$).  \begin{enumerate}  \item  This also meant that my Maxwell--Boltzmann distribution would not be  as precise because if a particle obtained a truely outlier speed, it  would greatly affect the results  \end{enumerate}  \item  Add more improvements \ldots{}  \end{enumerate}  \section{Bibliography}  \begin{itemize}  \item  "Maxwell Speed Distribution Directly from Boltzmann Distribution."  Development of Maxwell Distribution. N.p., n.d. Web. 07 Mar.  2013.\textless{}\url{http://hyperphysics.phy-astr.gsu.edu/%E2%80%8Chbase/kinetic/maxspe.html}\textgreater{}.  \item  "Language Reference (API) Processing 2." Language Reference (API)  Processing 2. N.p., n.d. Web. 07 Mar. 2013.  \textless{}\url{http://processing.org/reference}\textgreater{}.  \item  "3D Balls Bouncing- OpenProcessing." 3D Balls Bouncing-  OpenProcessing. N.p., n.d. Web. 07 Mar. 2013.  \textless{}\url{http://www.openprocessing.org/sketch/20136}\textgreater{}.  \item  "Kinetic Theory." Wikipedia. Wikimedia Foundation, 03 Apr. 2013. Web.  07 Mar. 2013.  \item  "The Distribution of Molecular Speeds." ChemEd. University of  Wisconsin, n.d. Web. 10 Mar. 2013.  \textless{}\url{http://chemed.chem.wisc.edu/chempaths/GenChem-Textbook/Kinetic-Theory-of-Gases-The-Distribution-of-Molecular-Speeds-941.html}\textgreater{}.  \item  "Including Graphics in a LaTeX Document." Including Graphics in a  LaTeX Document. University Of Colorado Boulder, n.d. Web. 10 Mar.  2013.  \textless{}\url{http://amath.colorado.edu/documentation/LaTeX/reference/figures.html}\textgreater{}.  \end{itemize}  \end{document}