**Introduction: **

Basketball is an internationally known sport and is practiced by several different groups of people around several different areas in the world; in fact, it is one of the most popular sports in the world at the moment. To ensure a proper game of basketball, you need to have a good amount of elasticity from the basketball – as without it, the ball wouldn’t bounce to the proper height.

This phenomenon of elasticity occurs because of the laws of ‘kinetic theory’, where, because “the molecules [of the gas inside the ball / system] are in constant, random motion and frequently collide with each other and with the walls of any container,” – (NASA, N.A) the ball has enough pressure to push back on the ground and return to a fraction of the same height it was released from. This constant motion and frequent collision within the walls of the particles causes an outwards push on the object and is the pressure of an object (measured in kilo Pascals).

Now, if you maintain the same number of particles inside the ball but decrease the volume – you will see that the pressure inside of the object will increase. This is because the particles will achieve a greater rate of collision with the walls and each other because they have less empty space to travel – consequently giving the basketball more force to exert outwards / more pressure.

To represent this scenario in mathematics, a scientist named Robert Boyle derived the mathematical relationship between Pressure [kPa] and Volume [L] in 1662, where Pressure [kPa] x Volume [L] = a Constant [K] (PV = K). However, years later, in 1834, a scientist named Benoît Paul Émile Clapeyron would discover that this equation would only work if the temperature and number of moles (a specific number of particles in a substance – 6.023 x 10^{23} particles) in the system remained the same / AKA the Ideal Gas Law.

In this lab, my group and I are going to investigate the relationship between Volume and Pressure; we will use the Boyle’s Law Apparatus to test the theory of Robert Boyle. In this, we need to manipulate the variable of volume to see how the pressure of the air in the system increases/decreases at a constant number of moles of air in the system.

When conducting this experiment, we are expecting the data points to follow an inversely proportional pattern – due to the Pressure being multiplied by Volume being equal to a constant. We are also ignoring change in temperature inside the system.

**Variables:**

Independent – Volume (0.020, 0.025, 0.030, 0.035, 0.040, 0.045, 0.050, 0.055, 0.060) [L]

Dependent – Pressure [kPa],

Control – Volume of air in system, Temperature of air in system (theoretical)

**Hypothesis:**

As the volume of the air in the system increases at a steady increment, then the pressure of the system increases inversely proportionally.

**Materials:**

- Boyle’s Law Apparatus
- Air Pump
- Excel

**Safety:**

- When releasing the air to record the different sets of data for variable of volume, make sure to open the valve to a small degree because if you release the liquid too fast, you may miss the desired volume data point and/or cause formulation of bubbles and have to pump back up.
- When pumping inside the Boyle’s Law Apparatus, make sure to pump slowly, because doing it too fast will create bubbles inside the apparatus and may fluctuate the data points to a small extent.
- When trying to get to a desired volume data point, make sure that the bottom of the meniscus rests at the measurement line, because only then will your data be most precise.

**Procedure:**

- Connect the air pump to the Boyle’s Law Apparatus
- Open the valve of the Boyle’s Law Apparatus
- Use the air pump to get the volume of the air in the system to 20ml by placing the bottom of the meniscus at the measurement line
- Close the valve of the Boyle’s Law Apparatus
- Record the pressure of the system onto the notebook
- Repeat steps 2-5 for volumes of 25ml, 30ml, 35ml, 40ml, 45ml, 50ml, 55ml and 60ml for the air
- Repeat step 2-6 for a total of 3 trials
- Graph the data

**Observations and Data:**

Volume VS Pressure [PSI converted to kPa] Chart

Pressure [PSI] | Pressure [kPa] | ||||||

Volume [L] | Trial 1 | Trial 2 | Trial 3 | Average | Deviation | Averaged Pressure [kPa] | Deviation |

0.020 | 47.8 | 47.9 | 48.0 | 47.9 | 0.10 | 330 | 0.689 |

0.025 | 38.2 | 38.1 | 38.2 | 38.1 | 0.05 | 263 | 0.345 |

0.030 | 31.9 | 31.9 | 32.0 | 31.9 | 0.05 | 220 | 0.345 |

0.035 | 27.1 | 27.0 | 27.1 | 27.1 | 0.05 | 187 | 0.345 |

0.040 | 24.0 | 24.0 | 23.9 | 24.0 | 0.05 | 165 | 0.345 |

0.045 | 21.0 | 21.0 | 21.0 | 21.0 | 0.05 | 145 | 0.345 |

0.050 | 18.9 | 18.9 | 19.0 | 18.9 | 0.05 | 130 | 0.345 |

0.055 | 17.0 | 17.1 | 17.0 | 17.0 | 0.05 | 117 | 0.345 |

0.060 | 15.5 | 15.5 | 15.5 | 15.5 | 0.00 | 107 | 0.000 |

Volume VS 1/Pressure [Inverse Proportionality] Chart

Volume [L] | 1/Pressure [kPa] |

0.020 | 0.003030 |

0.025 | 0.003802 |

0.030 | 0.004545 |

0.035 | 0.005348 |

0.040 | 0.006061 |

0.045 | 0.006944 |

0.050 | 0.007692 |

0.055 | 0.008547 |

0.060 | 0.009345 |

Constant (Product of P x V =) Fluctuations VS Volume Measurement Chart

Volume [L] | Constant [P x V] |

0.020 | 6.6 |

0.025 | 6.575 |

0.030 | 6.6 |

0.035 | 6.545 |

0.040 | 6.6 |

0.045 | 6.525 |

0.050 | 6.5 |

0.055 | 6.435 |

0.060 | 6.42 |

Average | 6.533 |

Range | 0.18 |

**Graphs**

**Analysis:**

As you can see on graph 1, as the volume of the system increases, the pressure decreases at an irregular increment. The graph itself has a curved slope which almost represents the arc of a circle; proving that this graph is representing an illustration of inverse proportionality between the x variable and the y variable (volume and pressure respectively).

Therefore, logically, if we graph Volume VS 1/Pressure – we get a linear line, as shown on graph 2. For graph 1, we can also see that as the volume (x) of the air increases incrementally, the difference in increment of pressure (y) starts to decrease as shown in the table below – which portrays how as the air becomes increasingly compressed, the pressure of the air starts to increase at a greater magnitude.

Volume VS Pressure Increment Difference Chart

Volume Comparison [L] | Increment Difference in Pressure [kPa] |

0.020 to 0.025 | 67 |

0.025 to 0.030 | 43 |

0.030 to 0.035 | 33 |

0.035 to 0.040 | 22 |

0.040 to 0.045 | 20 |

0.045 to 0.050 | 15 |

0.050 to 0.055 | 13 |

0.055 to 0.060 | 10 |

Furthermore, on graph 1, we can also observe how there are no error bars present in the graph. This is because on ‘Volume VS Pressure [PSI converted to kPa] Chart’, even though are deviations set for the measurements, they are far too minuscule to be visible on the graph itself – which is why graph 1 does not contain any error bars.

However, when looking at the deviation shown on the chart mentioned above, you can observe that the least precise data point is the point where the air is being compressed to 0.020L – which un coincidentally creates the most pressure (deviation of +/- 0.689 kPa), and similarly the most accurate data point is of the air being compressed to 0.060L – creating the least pressure (deviation of +/- 0.000 kPa); while all other data points have a deviation of +/- 0.345 kPa.

Lastly, when looking at the product Constant from using Boyle’s Law, you can see that the range of the constant varies at only 0.18 (6.6 – 6.42) and the average of the constant is 6.533; which is only a deviation of 2.80% between it and the range – signifying that our data is extremely precise throughout the different data points.

Additionally, for graph 2, it is observable that the line is, in essence, almost perfectly linear; the data points’ linearity angles at around 40 degree and is in illustration of the inversely proportional relation between Pressure and Volume, where 1/Pressure multiplied by Constant = Volume (K/P = V) – giving a straight line. In this graph, each time the volume of the air increases by 0.005L from the starting at 0.020L, the increment of 1/Pressure also increases at a rather steady rate of 0.000790 with a range of 0.000170 – giving the line an almost perfect linear structure.

Volume VS 1/Pressure Increment Difference Chart

Volume Comparison [L] | Increment Difference in 1/Pressure [kPa] |

0.020 VS 0.025 | 0.000772 |

0.025 VS 0.030 | 0.000743 |

0.030 VS 0.035 | 0.000803 |

0.035 VS 0.040 | 0.000713 |

0.040 VS 0.045 | 0.000883 |

0.045 VS 0.050 | 0.000748 |

0.050 VS 0.055 | 0.000855 |

0.055 VS 0.060 | 0.000799 |

Additionally, an R squared value as well as the equation of the trendline is included for the Volume VS 1/Pressure Graph [graph 2]. Regarding the R squared value, you can observe how it displays 0.9996/1.0000 at the bottom right – which means that our data is only 0.04% off from the perfect trendline.

This means that our data is extremely precise because the consecutive data points line up as intended [look at ‘Volume VS 1/Pressure Increment Difference Chart’ above] with an increment difference range of 0.000170.

Moreover, according to the extrapolation shown in graph 2, as well as the equation of the trendline, it is noticeable that when the Volume of the Air in the System (x axis) is 0.000, the 1/Pressure is equal to a number less than 0; specifically (-0.0002) AKA y-intercept = (-0.0002), which is not possible as something multiplied by 0 is equal to 0, however, since this is what we found from our practical observations, this value is quite accurate because it shows that the trendline is only a small value off from perfection.

**Conclusion:**

In conclusion, my hypothesis of “as the volume of the air in the system increases at a steady increment, then the pressure of the system increases inversely proportionally,” was deemed correct. I say this because as you observed in graph 1, as the Volume of the Air in the System (x axis) got compressed from 0.060L down to 0.020L, the Pressure of the System (y axis) decreased from 107 kPa to 330 kPa respectively in the pattern of an inverse proportionality.

After graphing the data points for 1/Pressure VS Volume, the illustration for the relationship between both x and y variables was vivid because of the linearity of the consecutive data points as a whole.

The results of the experiment were the exact same as how I predicted them to be, because the figure of the data for both graphs (Pressure VS Volume and 1/Pressure VS Volume) came just as I found during my research during the introduction segment, with the Pressure V Volume Graph looking like the arc of a circle, and 1/Pressure V Volume Graph being a linear line.

The method used to find the data for this lab was very precise. Regarding the R squared and y-intercept for graph 2, the precision of the data points was extremely high because the R squared value was off by a miniscule 0.04% / 0.0004 – showing that the data fit the trendline extremely well, and the y-intercept was also located at (-0.0002), which in the practical sense of collecting data is extremely precise as well because of how close it was a perfect 0.0000.

Lastly, according to Boyle’s Law of PV = K, we calculated that the constant (as an average) that we got was 6.533 with a range of 0.18 – which once more signifies the degree of precision that our lab execution had. In our lab, the reason behind why our data was so accurate was because of the apparatus we use to get this data. The Boyle’s Law Apparatus was an extremely precise piece of equipment which displayed the Pressure in up to 3 significant figures, and Volume to 3 significant figures as well.

This made it much easier for my group and I to get the amounts down to the exacts in order to get precise data. However, one limitation on our data was that even though it was relatively easy to collect, often times we would need to restart specific parts of the procedure such as pumping the oil using an air pump, to compress the air back to 0.020L – which was the data point with the most deviation due it its high pressure.

This occurred because, the pressure on this data point was so high we would need to pump quite fast – which would create bubbles in the system and mess up the data. Even though this could be fixed most of the time by opening the valve of the apparatus, it would cause us to spend longer amounts of time to gather data and may have left the formation of singular bubbles in the system.

Another limitation of using this apparatus was that as soon as we would reach 0.015L or 0.065L of air volume, the pressure on the gauge would either appear below the marked measurements, or above the marked measurements section – which prevented us from gather 10 different data points, but rather letting us gather 9.

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