Boyle’s Law and Charles’ Law

 

Gases, like humans, behave differently at different conditions and situations.

This experiment, for me, is like a Psychology experiment wherein the behavior of gases will be observed at different conditions, particularly, the behavior of dilute gases at low pressures and high temperatures.

Ideal Gas Laws

The ideal gas laws were used to describe the behavior of gases assumed to be ideal at room temperature and pressure close to 1 atm. These observations led to the discovery of several properties of  these gases as well as their relationships to other entities such as volume, temperature and pressure through experiments. The equation

                                                                        PV = NkT                                                          (1)

was then established as the equation of state for an ideal gas after the results of Boyle’s, Charles’ and Gay-Lussac’s experiments were combined together; where is the pressure, V is the volume, is the number of particles, is the Boltzmann constant with a value of 1.3806488 × 10-23 m• kg • s-2 K-1 and T is the temperature. [1]

The goal of this experiment is to verify Boyle’s Law and Charles’ Law, and to calculate the number of particles in a gas using (1).

Boyle’s Law 

Robert Boyle studied the compressibility of gases in 1660. In his experiment, he used a cylinder with a piston and a gas immersed in a bath, to maintain the temperature of the gas constant. Different masses were placed atop the piston to produce pressure and the gas volume was measured. From the volume vs. pressure plot of his results, he concluded that the volume of the gas V is inversely proportional to its pressure P[2]

                                                                           V = NkT/P                                                      (2)

where NkT is the proportionality constant.

For the first part of this experiment, the goal is to verify Boyle’s Law and to calculate the number of particles in the gas using equation (2).

For the setup, a 3/4-filled beaker was placed into a pot with continuously boiling water. The air chamber can was connected to the mass lifter apparatus, without using any force to prevent air leakage, and placed to the hot bath. The piston of the mass lifter apparatus was then lifted to its maximum height. To the Vernier LabQuest, the gas pressure sensor was connected. The diameter, 0.0325 m, and maximum height of the mass lifter apparatus, 99.5 mm, were recorded and the temperature T,  353.15 K, was monitored using a thermocouple probe and recorded in the experiment.

12919307_1327389997278050_2085432922_n.jpg

Figure 1. Setup for the verification of Boyle’s Law; air camber can connected to the mass lifter apparatus, thermocouple probe and air chamber can inside the beaker inside the pot, gas sensor connected to the Vernier LabQuest and to the mass lifter apparatus

A standard mass of 50 g was then placed at the platform of the mass lifter apparatus and the height of the piston was recorded. This was done using 100 g, 150 g, 200 g, and 250 g standard masses.

111.jpg

As shown in table 1, as the standard mass placed on the platform of the mass lifter apparatus, the height of the piston decreases and the pressure increases. The volume V of the cylinder was calculated using

                                                                      V = πr2h                                                             (3)

where r is the radius of the mass lifter apparatus, 0.01625 m, and h is the height of the piston. It could be seen from here that as the pressure decreases, the volume of the cylinder increases.

1

Figure 2. Plot of Volume versus 1/Pressure

Figure 2 shows the linear plot of the volume of the cylinder versus the inverse of the pressure. From this graph, the equation of the best fine line is

                                                                             y = 28.44x -1.98 × 10-4                                                                       (4)

where the slope 28.44 is the probability constant NkT, and the y-intercept -1.98 × 10-4 is the volume of the air chamber can yet, since the value is impossibly negative, it could be implied that the volume of the chamber is almost zero. From here, we could get the number of particles N by dividing the slope by the Boltzmann constant k and the constant temperature T, 353.15 K, as 5.84 × 1021/mole.

Also the Rvalue of the best fit line is 0.988115 implying that errors have propagated. The possible sources of this deviation were: (1) the temperature reading is fluctuating, hence, not constant, (2) immersion of the thermocouple probe and air chamber can to the beaker, and (3) the gas is not that “ideal”.

Nevertheless, table 1 and figure 2 both show the trend that volume is inversely proportional to pressure thus, verifying Boyle’s Law.

Charles’ Law

Jacques Cesar Charles in 1780 studied the compressibility of gases through an experiment where a cylinder with a piston and a gas is immersed in water. A pressure is created by placing a mass on top of the piston, which is then held constant. The gas pressure was measured as the temperature of the water bath increases. From the results of this experiment, the equation of Charles’ Law was obtained as

                                                                     V = T • Nk/P                                                         (5)

which states that the volume V of the gas is directly proportional to its temperature  and the proportionality constant is Nk/P[3]

For the second part of this experiment, the goal is to verify Charles’ Law and to calculate the number of particles in the gas using equation (5).

As for the setup of this experiment, the air chamber can was placed into the beaker with the boiled water as hot water bath. The piston was then lifted to its maximum height, 99.5 mm, and the pressure P, 102700 Pa, inside was monitored constantly. An ice chunk is placed into the beaker. The temperature T read by the thermocouple probe and the height of the piston were simultaneously measured throughout the experiment as each ice chunk melted. Three more ice chunks were put singly into the beaker after one has melted and the procedure was repeated.

12966504_1327390083944708_2102988401_n.jpg

Figure 3. Setup for the verification of Charles’ Law; air chamber can inside beaker and gas sensor connected to mass lifter apparatus, gas sensor also connected to Vernier LabQuest

t2

Table 2 shows the decreasing temperature of the gas as more ice chunks were added. It could also be seen that the height of the piston decreases, so does the volume of the cylinder computed using (3) with the radius equal to 0.01625 m.

2

Figure 4. Plot of Volume versus Temperature

Figure 4 shows a linear plot of the volume of the cylinder versus the temperature of the gas. The equation of the linear fir is given by

                                                                y = 1.25 × 10-7x + 4.016 × 10-5                                                                   (6)

where the slope 1.25 × 10-7 is the proportionality constant Nk/P. This is used to calculate the number of particles N and the obtained value is 9.28 × 1020/mole. From the y-intercept, 4.016 × 10-5, the volume of the chamber is calculated as 3.910 × 10-10 mbut, since this is a very small value, the volume of the chamber is almost zero.

Moreover, the Rvalue of the linear fit is 0.9902 implying that errors were present in the experiment. The possible sources of errors for this experiment are; (1) the pressure reading is actually fluctuating, hence not constant, (2) the immersion of the air chamber can into the beaker is unfixed, (3) the difficulty in the simultaneous measurement of the height of the piston and the temperature T, (4) the gas is not that “ideal”.

If a mass is placed on top of the platform of the mass lifter apparatus, it will create pressure thus, making the pressure higher thus, decreasing the slope value.

Nonetheless, both table 2 and figure 4 follow the same trend that implies volume V of the cylinder is directly proportional to pressure P inside thus, verifying Charles’ Law. 

Gay-Lussac’s Law

Joseph Louis Gay-Lussac in 1809 discovered that at constant volume V and number of particles N, the gas pressure is directly proportional to the temperature given by the equation

                                                                      P = T • Nk/V                                                       (7)

where Nk/V is the proportionality constant, which is then called Gay-Lussac’s Law. [1]

To verify this law, setup an experiment similar to that of the verification of Charles’ Law but, fix the height of the piston to its maximum to have a constant volume. As ice chunks will be added to the beaker, the temperature will absolutely decrease and the pressure is also expected to decrease.

Van der Waals Equation

If gases are not ideally behaving in room temperature and pressure almost close to 1 atm, they are called real gases. The Van der Waals equation is the corrected equation of the ideal gas equation.

                                                    [P + a(n/V)2](V/n – b) = RT                                                (8)

where P is the gas pressure, V is the volume, R is the gas constant, T is the temperature, n is the number of moles, b is the correction for finite molecular size and its value is the volume of one mole of the atoms or molecules, and a is the correction for the intermolecular forces.[4] However, people tend to choose the ideal over the complicated reality so, let’s just assume everything is ideal.

To sum everything up, the results of the experiments conducted verified both Boyle’s Law and Charles’ Law and the numbers of particles N for both setup were calculated. Also, the values of Rcoefficients were close to 1 so, it could be concluded that the gases used in the experiment are desirably ideal.


References:

[1] Experiment 4: Gas Laws. Physics Laboratory Manual. National Institute of Physics, UP Diliman, 2013.

[2] De Leon, Nelson. “Elementary Gas Laws: Boyle’s Law.” Boyle’s Law. Indiana University Northwest. Web. 07 Apr. 2016.

[3] De Leon, Nelson. “Elementary Gas Laws: Charles Law.” Charles Law. Indiana University Northwest. Web. 08 Apr. 2016.

[4] “Van Der Waals Equation of State.” Hyperphysics. N.p., n.d. Web. 8 Apr. 2016.

[5] Young, Hugh and Roger Freedman. University Physics with Modern Physics 13th edition. San Francisco: Pearson education Inc., 2012.

 

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