Fluid+Mechanics+(Ideal+Gas+Law)

(Po×Vo = nRT)/(Pf×Vf = nRT)→Vf = (Po×Vo)/Pf Taggart J. K. Price


 * Purpose **

The purpose of my project was to demonstrate a proficiency in Fluid Mechanics. More specifically, this project helps me present an in depth understanding of the ideal gas law. Through this project, I utilized the ideal gas law in order to predict variable changes in an experimental setting. By manipulating pressure in my equations, I can predict the changes in volume of a balloon under the same conditions.


 * Procedure **

The materials I used include:


 * [[image:photo (6).JPG width="240" height="315"]] || Glass Jar || This container is composed of a thick layer of glass, allowing use as a closed system in which to barometric experiments. The volume of the container is .004 m 3 . ||
 * [[image:photo (3).JPG width="240" height="314"]] || Pressure Sensor || Measuring in kilopascals, this sensor was small enough to fit inside the glass container. It has volume of approximately 9x10^-4 m^3. ||
 * [[image:photo (5).JPG width="240" height="315"]] || Temperature Sensor || Measuring in degrees Celsius, this sensor helped observe changes in temperature corresponding with pressure changes. It has a volume of approximately 4x10 -5 m 3 . ||
 * [[image:photo (4).JPG width="240" height="139"]] || Pressure Machine || This machine allowed manual manipulation of the pressure variable; during this experiment, it was used to remove pressure. ||
 * [[image:http://t3.gstatic.com/images?q=tbn:ANd9GcSvbXjxV0q1KbPwlf5ynjA9YNAeQ-I7DUm4XmAiaJ7Hsf6yb9Fr width="275" height="182"]] || Balloon || This balloon was used to observe volumetric changes in an isothermal process. ||

Used Equations:

** PV = nRT ** ** ρ = m/V ** An observation of the volumetric properties of a balloon in an isothermal process was not the original intent of this experiment. At first, I set out to confirm the relationship between changes in pressure and temperature in an isochoric process, subsequently confirming the gas constant. My first experiment was hindered at first by a tear in the tube to the pressure machine. No matter how much time I spent on the calculations, they were always wrong! When I realized that the tubing was torn, I cut out the ruined part and continued the experiment. However, the temperature was still remaining constant, and so my original goal for the project were dashed. These were complications in my control of outside variables; however, this led to my stumbling upon the balloon project. My original experiment used this specific version of the ideal gas law:

** ∆P×V = nR∆T → (∆P×V)/n∆T = R ** Upon starting this project, I realized that temperature was remaining constant, and due to this and the fact that the pressure machine was sucking out air from the system, I revised the equation to be:

** (∆P×V)/(∆n×T) = R ** With this realization, my original intentions of solving for the gas constant R were nullified. I continued research on the number of moles taken out of the system based on the changes in pressure in order to prove the constant temperature. Then, I decided that all this information could be used to create a new equation. This equation could observe the changes in a balloon system; while the pressure would be the same between the inside of the jar and the balloon, and the number of moles would remain constant within the balloon, I could conclude with an equation which could predict changes in volume proportional to changes in pressure:

** (P ** o ** ×V ** o ** = nRT)/(P ** f ** ×V ** f ** = nRT)→V ** f ** = (P ** o ** ×V ** o ** )/P ** f  After proving that temperature was constant, the experiment became simple. By inserting any final pressure, one could observe the dimensional change of the balloon from a state of equilibrium.


 * Analysis **

After failing to recognize the slow decrease in moles of air over time in the glass jar system, I found it appropriate to prove that the temperature remained constant. If I could prove the the temperature was constant, this would allow me to observe the relationships between volume and pressure in the balloon. A constant temperature would ensure that volume and pressure would be the only changing variables, as I could then determine the amount of air moles in the balloon. Though the temperature fluctuates minimally, it shows no correlation to the changes in pressure in the system. Experimentally, one could then conclude that temperature is a constant within the glass jar:


 * Pressure (Pa) || Temperature (°C) ||  || Pressure (Pa) || Temperature (°C) ||   || Pressure (Pa) || Temperature (°C) ||   || Pressure (Pa) || Temperature (°C) ||   || Pressure (Pa) || Temperature (°C) ||
 * 100,800 || 24.0 ||  || 52,790 || 23.7 ||   || 10,500 || 22.6 ||   || 8,620 || 23.0 ||   || 8,000 || 21.8 ||
 * 100,800 || 26.0 ||  || 50,000 || 22.0 ||   || 10,000 || 21.7 ||   || 8,600 || 23.2 ||   || 6,600 || 21.7 ||
 * 100,800 || 26.2 ||  || 46,900 || 23.6 ||   || 9,900 || 21.6 ||   || 8,590 || 22.7 ||   |||| Notice how at even 6% of ||
 * 100,00 || 23.0 ||  || 40,000 || 22.0 ||   || 9,300 || 22.7 ||   || 8,500 || 22.8 ||   |||| atmospheric pressure, ||
 * 90,000 || 23.0 ||  || 32,000 || 22.0 ||   || 9,270 || 23.8 ||   || 8,440 || 23.7 ||   |||| temperature still does not ||
 * 83,360 || 22.6 ||  || 30,000 || 22.0 ||   || 9,000 || 21.7 ||   || 8,400 || 23.4 ||   |||| change more than 4° C. ||
 * 80,000 || 23.3 ||  || 20,900 || 22.8 ||   || 8,800 || 22.2 ||   || 8,250 || 23.4 ||   ||||   ||
 * 70,000 || 24.1 ||  || 20,000 || 21.7 ||   || 8,750 || 23.0 ||   || 8,220 || 23.6 ||   ||||   ||
 * 60,000 || 22.7 ||  || 11,200 || 22.6 ||   || 8,700 || 23.0 ||   || 8,170 || 23.6 ||   ||||   ||

In mathematical calculations, the temperature can be found by determining the volume of air inside the balloon. This is done by emerging the slightly inflated balloon into a graduated cylinder and measuring the water displacement. With the balloon I used, the volume was approximately .008 m 3. However, one cannot accommodate the volume of the layer of rubber balloon around the air. For this reason, calculations can fluctuate drastically; the volume of the rubber has the potential to double or even triple the calculated temperature. Despite this, we can confirm that the overall temperature of the air inside the balloon matches the average temperature outside the balloon, if one chooses to ignore the changes made by outside variables:

** T ** calculated ** = [100,800 Pa × (8 × 10 -3 m 3 )]/[.32 mol × 8.314 J/mol*K] = 303 K = 30°C **

The volume of the balloon rubber plays a huge role in why this temperature is so different from the experimental average temperature; for this reason, we can conclude that actual temperature within the balloon is constant and approximately equal to that inside the jar. The fact that certain variables cannot be controlled allows for room to consider calculated anomalies as errors which can be disregarded.

From the determination that this is an isothermal process, it becomes simplistic to conclude the volume to pressure relationship in this system, as predicted in the procedures:

** (Po×Vo = nRT)/(Pf×Vf = nRT)→Vf = (Po×Vo)/Pf **


 * Results **

The actual implementation of the experiment proved an overall success. Upon decreasing the amount of pressure in the glass jar, one could observe an immediate expansion in the volume of the balloon. The starting pressure inside the jar was approximately 101,800 Pascals. At this time, the balloon was at a deflated state, displaying a minimum volume. I was able to decrease the pressure inside the jar to almost 5,000 Pascals. This is only a 20th of an atmosphere ; subsequently, the balloon inflated drastically in confirmation of the inverse relationship between pressure and volume.

More specifically, I was able to calculated the exact values within the rational function. The equation determining this system's isothermal properties is useful to determine proportional ratios between volume and pressure. However, it does not assist us in determining actual values, are the true functional relationship between volume and pressure. After calculating based on the variable values in equilibrium, I was able to conclude on the true volume versus pressure function:

**V = 58/P**

If one were to measure the volumetric properties of the balloon, the pressure inside the balloon could be calculated. Also, because of barometric properties, it can be concluded that this value also represents the pressure outside of the balloon, within the jar. Respectively, if one were to use a pressure sensor inside the jar, the measured pressure could be used to calculate the volume of the balloon. In this sense, isothermal processes are extremely efficient in their ability to ensure proper fluid mechanic relationships.

In conclusion, I can confidently say that I thoroughly understand the ideal gas law. My proficiency to calculate theoretical values for isothermal, isochoric, and isobaric processes proved extremely useful in my work as an AP Physics B student. However, had I ever chosen to question that these numbers and rules were applicable in real life? Of course! To follow scientific laws based on faith alone is against all the fundamentals of science; to seek answers and seek documented evidence. This experiment served as an affirmation of the ideal gas law as an implemented relationship in everyday life.