Experiment 9: Gas Laws

In the year 1834, a French scientist by the name Benoît Paul Émile Clapeyron discovered the Ideal Gas Equation, which finally put together the various behaviour of an ideal gas (a hypothetical gas whose molecules occupy negligible space, does not react with other substances, and obeys the gas laws perfectly) into one elegant equation

Where P is pressure Pascals, V is volume cubic meters, N is the number of molecules of gas, k is the Boltzmann constant at 1.3806488 × 10−23, and T is temperature in Kelvin. However, this brainchild was not born completely out of a dash of pure genius. Instead, it was the product of decades of independent scientific inquiry on the topic of gas and its behaviour. Namely three separate previously discovered gas laws can be attributed to the synthesis of the Ideal Gas Equation. The first is Boyle’s law, which was discovered in 1662 by Irish scientist Robert Boyle, it states that under constant temperature, as the pressure of a gas increases, its volume decreases. 

Where C is the proportionality constant NkT. The next law is what’s known as Charles’ Law which states that at constant pressure, as the temperature of a gas increases, so does its volume.

This time, the proportionality constant is Nk/P  . Finally, the third law known as the Gay-Lussac Law states that at constant volume, the pressure of a gas increases as its temperature increases. 

Here, the proportionality constant is set at  Nk/V. Combining these laws, one would eventually lead to the Ideal Gas Equation, which makes one wonder. What took them so long?

This experiment aimed to verify these laws as well as experimentally obtain the amount of gas molecules there are in the setup.

The experiment began with the analysis of Boyle’s Law. This was conducted by varying the volume of a sealed gas cylinder while monitoring the pressure change inside it. Meanwhile, the cylinder was connected to an air canister which was submerged in boiling water. This served as the heat source of the air inside the cylinder. Therefore, the total volume of the system was the cylinder and the canister combined. The results were then plotted on a volume vs. inverse pressure graph. Obtaining the equation of the line, the number of molecules within the gas chamber was computed for by equating the slope of the line to the proportionality constant NkT. Using the y-intercept of the equation, the volume of the canister was also obtained. The results of this are on Table W3.

Table W1: Diameter of the Piston

Diameter of Piston (m)

0.0325

Table W2: Boyle's Law

Height (m)	P (Pa)	V (m3)	1/P
0.098	114300	8.12987E-05	8.74891E-06
0.097	114720	8.04691E-05	8.71688E-06
0.095	115200	7.881E-05	8.68056E-06
0.092	115670	7.63212E-05	8.64528E-06
0.089	116200	7.38325E-05	8.60585E-06
Temperature  (K)	368.15

Table W3: Boyle's Law Linear Equation

Slope	0.0182
y-intercept	.000007
R2 Value	0.9754
N	3.58x1018
Volume of Chamber

Charles’ Law was next analysed using the same setup, but this time with constant pressure and varying temperature. Cubes of ice were dropped one at a time into the boiling water. Each time, the change in temperature and volume was recorded. The results were then plotted on a volume vs. temperature graph and the resulting linear equation was obtained. Using the same method as in Boyle’s Law, the number of molecules was once again obtained, as well as the experimentally derived volume of the canister. The verifying factor to these equations was the obtained experimental volumes of the canister for each gas law analysed. As it turns out,


Table W4: Charles' Law

Height (m)	T (K)	V (m3)
0.098	364.15	8.12987E-05
0.083	355.15	6.8855E-05
0.077	350.15	6.38776E-05
0.069	343.15	5.72409E-05
0.062	340.35	5.14339E-05
0.06	338.85	4.97747E-05
0.054	333.25	4.47973E-05
0.05	329.65	4.14789E-05
P (Pa)	113900

Table W5: Charles' Law Linear Equation

Slope	.000001
y-intercept	0.0003
R2 Value	0.9934
N	4.95x1022
Volume of Chamber

However, another factor can be attributed to the verification of the Gas Laws. Both experimental setups yielded best fits greater than 0.97 with Boyle’s Law at 0.9754 and Charles’ Law at 0.9934. This is testament to the accuracy of the predictions these laws make, thus strengthening their legitimacy.
On further thought, if a mass were placed on top of the piston during the Charles’ Law experiment, the pressure of the system would have increased. This would mean a decrease in the slope of the linear equation, but no change in its y-intercept.
It is evident in this experiment, that Gay-Lussac’s Law was overlooked. Should one find the need to verify this law as well, the same materials used would suffice. In order to do so, one would simply have to keep the volume constant by somehow keeping the piston from moving. Upon doing so, one can vary the temperature of the gas, while simultaneously recording the change in pressure of the system. Plotting the measurements on a pressure vs. temperature graph, then treating the graph with the same procedures as before, would yield the necessary information for this experimental setup. Overall, the computations for the number of molecules all yielded realistic results, the obtained canister volumes for each setup were consistent, and the R² values for each graph were satisfactory, thus verifying the Ideal Gas Laws. 

Experiment 9: Gas Laws

In the year 1834, a French scientist by the name Benoît Paul Émile Clapeyron discovered the Ideal Gas Equation, which finally put together the various behaviour of an ideal gas (a hypothetical gas whose molecules occupy negligible space, does not react with other substances, and obeys the gas laws perfectly) into one elegant equation

Where P is pressure Pascals, V is volume cubic meters, N is the number of molecules of gas, k is the Boltzmann constant at 1.3806488 × 10−23, and T is temperature in Kelvin. However, this brainchild was not born completely out of a dash of pure genius. Instead, it was the product of decades of independent scientific inquiry on the topic of gas and its behaviour. Namely three separate previously discovered gas laws can be attributed to the synthesis of the Ideal Gas Equation. The first is Boyle’s law, which was discovered in 1662 by Irish scientist Robert Boyle, it states that under constant temperature, as the pressure of a gas increases, its volume decreases.

Where C is the proportionality constant NkT. The next law is what’s known as Charles’ Law which states that at constant pressure, as the temperature of a gas increases, so does its volume.

This time, the proportionality constant is Nk/P  . Finally, the third law known as the Gay-Lussac Law states that at constant volume, the pressure of a gas increases as its temperature increases.

Here, the proportionality constant is set at  Nk/V. Combining these laws, one would eventually lead to the Ideal Gas Equation, which makes one wonder. What took them so long?

This experiment aimed to verify these laws as well as experimentally obtain the amount of gas molecules there are in the setup.

The experiment began with the analysis of Boyle’s Law. This was conducted by varying the volume of a sealed gas cylinder while monitoring the pressure change inside it. Meanwhile, the cylinder was connected to an air canister which was submerged in boiling water. This served as the heat source of the air inside the cylinder. Therefore, the total volume of the system was the cylinder and the canister combined. The results were then plotted on a volume vs. inverse pressure graph. Obtaining the equation of the line, the number of molecules within the gas chamber was computed for by equating the slope of the line to the proportionality constant NkT. Using the y-intercept of the equation, the volume of the canister was also obtained. The results of this are on Table W3.

Slope	0.0182
y-intercept	.000007
R2 Value	0.9754
N	3.58x1018
Volume of Chamber

Charles’ Law was next analysed using the same setup, but this time with constant pressure and varying temperature. Cubes of ice were dropped one at a time into the boiling water. Each time, the change in temperature and volume was recorded. The results were then plotted on a volume vs. temperature graph and the resulting linear equation was obtained. Using the same method as in Boyle’s Law, the number of molecules was once again obtained, as well as the experimentally derived volume of the canister. The verifying factor to these equations was the obtained experimental volumes of the canister for each gas law analysed. As it turns out,

However, another factor can be attributed to the verification of the Gas Laws. Both experimental setups yielded best fits greater than 0.97 with Boyle’s Law at 0.9754 and Charles’ Law at 0.9934. This is testament to the accuracy of the predictions these laws make, thus strengthening their legitimacy.
On further thought, if a mass were placed on top of the piston during the Charles’ Law experiment, the pressure of the system would have increased. This would mean a decrease in the slope of the linear equation, but no change in its y-intercept.
It is evident in this experiment, that Gay-Lussac’s Law was overlooked. Should one find the need to verify this law as well, the same materials used would suffice. In order to do so, one would simply have to keep the volume constant by somehow keeping the piston from moving. Upon doing so, one can vary the temperature of the gas, while simultaneously recording the change in pressure of the system. Plotting the measurements on a pressure vs. temperature graph, then treating the graph with the same procedures as before, would yield the necessary information for this experimental setup. Overall, the computations for the number of molecules all yielded realistic results, the obtained canister volumes for each setup were consistent, and the R² values for each graph were satisfactory, thus verifying the Ideal Gas Laws. 

Experiment 8: Calorimetry

Different substances require different amounts of energy (in the form of heat) to raise their temperatures to a certain level. For example, adding 1000 J of heat to oil wouldn't raise the oil's temperature as much as it would raise the temperature of an iron bar of the same mass with the same amount of heat. This property for a substance to thermally respond to an inflow of energy is called heat capacity. All the factors contributing to how well a substance responds to the gain or loss of heat energy can be expressed in a simple and elegant equation.

Where Q is the amount of heat involved when concurring a temperature change of T, m is the mass of the substance involved, and C is the substance's specific heat. In this experiment, we will be using calorimetry - a method to experimentally derive the specific heats of the substances in question. Namely, an aluminum bar and a copper bar.

Calorimetry involves the use of two other additional equations. One, being

where Q is the heat added to the calorimeter, T is the temperature change that its system undergoes, and C is its heat capacity, which is commonly referred to as the calorimeter constant. The last one is basically a restatement of the law of conservation of energy, wherein

Figure 1: Calibration Curve of Trail 3

The experiment began by placing 100 mL of tap water in a styrofoam cup, which served as the calorimeter. The initial temperature of the water was taken and recorded at 28 °C . Next, hot water at ~70 °C was quickly poured into the cup, and the change in temperature of the mixture was recorded over a span of 5 minutes in 30 second increments. The data was then plotted in a calibration curve. The linearization of this curve allowed for the computation of the ideal temperature of the mixture at the very moment the two different sources of tap water mixed. This temperature T(time = 0) would serve as the final temperature for the entire setup.

With the necessary data collected, the Calorimeter constant was obtained. This was done for two other trials. The three calorimeter constants were then averaged, obtaining an experimental value of 91.9528694 J/g°C.

The second part of the experiment aimed to obtain the heat capacities of two different metal bars; copper and aluminum. In order to do this, both bars were heated to 100°C then placed in 200 mL of water in the calorimeter. The necessary data was gathered, then using the same equations, the specific heats of the two metals were obtained.
The values were found to be:

Copper - 0.43957641 J/g°C
Aluminum -  0.51007307 J/g°C

Compared to their known values, the result for copper exhibited a 13.89% error. Meanwhile, Aluminum's results showed a 43.33% deviation. Many aspects of the way the experiment was carried out were prone to error. One such factor could be the crude nature of the calorimetry set up (styrofoam cups, cold air leaking into set up), which would ultimately affect the accuracy of the results. Another could be the fact that a lot of the experiment involved things that needed immediate or even instantaneous measurement. However, since actions such as mixing two water sources took time, the required data was not able to be measured as quickly as needed. Nonetheless, with already such a crude setup, the deviations were not as far as expected, making percent errors as high as 43.33% relatively satisfactory.

Sources:

[1] "Experiment 8: Calorimetry." Physics 103.1 Experiment Manual. N.p.: n.p., n.d. N. pag. Web. 22 Apr.      2016.

Experiment 7: Temperature Measurement

Temperature and its concept is one of the most intuitive aspects to our lives as human beings. However, much of its behaviour turns out to be not as intuitive as one might think. For one, very few laymen can correctly differentiate temperature from heat, in which temperature is not actually the "hotness" or "coldness" of things, but is in fact an indicator of the direction of heat transfer between two objects. Another non-intuitive aspect of temperature is that it cannot be directly measured. When two objects of differing temperature come into contact, heat will flow from one object to the other. When the exchange of heat no longer occurs, the two objects are in what we call thermal equilibrium. Both objects would be at the same temperature. According the the zeroth law of thermodynmics, if A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then it should follow that A is in thermal equilibrium with C. So when using a thermometer to measure an object's temperature, what is actually being measured is the thermometer's temperature. This does, however, come with consequences. The transfer of heat from the object to the thermometer would mean an decrease in the object's actual temperature. Another would be that getting a reliable temperature measurement is not instantaneous. Depending on the thermometer used, it will take time to make a good measurement since there will be a flow of heat from the object to the device. This may vary from a fraction of a second, to half a minute depending on the nature of the device used. Taking these factors into account, each material has what is called a thermal time constant. This is defined as the time it takes for an object to reach 63.2% of its final temperature. In this experiment, we explored, obtained and analyzed the various time constants of three different types of thermometers. Namely the alcohol thermometer, mercury thermometer, and the thermocouple. The method of choice was pretty simple. For a given thermometer, an initial temperature reading was measured in a glass of ice cold water. Afterwards, a final temperature reading was taken in a pot of boiling water. The desired temperature at the thermometer's yet unknown thermal time constant was solved using the equation



With the desired temperature at hand, the thermometer was brought to its initial temperature then quickly dunked into the pot of boiling water until it reached the temperature solved using the equation. The time it took for that to be achieved was recorded. This recorded time was the thermometer's thermal time constant! Three trials were done for the heating process, while another three trials were done for the reverse - the cooling process. In total, six trials were done for each type of thermometer. All results were tabulated below.

Table 1: Heating of Alcohol Thermometer

Trial	Final temp	Initial temp	T(τ)	τ(s)
1	92	2	58.88	5.38
2	92	2	58.88	5.68
3	92	2	58.88	5.7

Table 2: Cooling of Alchohol Thermometer

Trial	Final Temp	Initial Temp	T(τ)	τ(s)
1	3	92	36.752	10.63
2	4	92	35.384	7.92
3	3	92	35.752	7.05

For the alcohol thermometer, thermal time constant for heating was generally consistent throughout the three trials. A more visible difference, however, was observed in the time measurement of the cooling process. Trial one differed from other two trials by roughly three seconds. Nonetheless, the general trend observed between both processes was that the cooling process took a longer time than the heating process. To account for the error observed, it should be noted that this particular experiment invited a big risk of human error, as the trials generally depended on a lot of error-prone methods of measurement, such as visually observing when the temperatures hit their marks, having to stop the timer once the measurer said so, etc.

Table 3: Heating of Mercury Thermometer

Trial	Final Temp	Initial Temp	T(τ)	τ(s)
1	94	4	60.88	8.87
2	94	1	59.776	10.08
3	94	1	59.776	10.9

Table 4: Cooling of Mercury Thermometer

	Final Temp	Initial temp	T(τ)	τ(s)
1	2	94	35.856	15.86
2	2	94	35.856	22.22
3	2	94	35.856	20.96

The mercury thermometer on the other hand did not display the same precision in results as the alcohol thermometer did; especially in the cooling process, where differences were as high as 6 seconds. However, it was again observed that the cooling process took longer than the heating process. Compared to the alcohol thermometer, the mercury thermometer also had larger thermal time constants. This can be attributed to the mercury thermometer’s bigger dimensions, which indicate more mass.

Table 5: Heating of Thermocouple

Trial	Final Temp	Initial Temp	T (tau)	t (s)
1	99.3	0.6	62.98	0.13
2	98.9	0	62.5	2.3
3	99.3	0.4	62.9	1.35

Table 6: Cooling of Thermocouple

	Final temp	initial Temp	T (tau)	t (s)
1	2	96	36.6	1.48
2	2	96.3	36.7	1.12
3	2	96.3	36.7	3.2

Compared to the last two types of thermometers, the thermocouple’s results varied drastically in duration. All trials for both heating and cooling lasted typically below the count of four seconds. Such results are consistent with existing theory on the transfer of heat. Two main factors can be attributed to such quick thermal time constants. One is the relative size of the sensory nodes of the device. Compared to the sizes of the alcohol and mercury thermometers, the thermocouple’s sensors (two thin copper wires) are only a fraction in overall mass and size. Even intuitively, one can see how it would take much less time to change its temperature. The second factor lies in the fact that the thermocouple is composed of material with particularly high thermal conductivity, which is copper. Copper’s thermal conductivity is set at more than 380 W/mK while the glass used in conventional thermometers is at roughly 1 W/mK. With these two factors at play, it is no doubt that the thermocouple would exchange heat at a faster rate, and thus reach thermal equilibrium faster. With such quick to occur time durations, it should be noted that the precision of the measurements were subjected to more risk of error. This accounts for the relatively greater variation between the results of each trial.

The applications of thermometers and the theory behind how they work can be and is in fact applied throughout our everyday lives. Take for example, the measurement of temperature of someone with a fever. Normally, engineers consider a duration of three thermal time constants to be a reliable time period of measurement. 

Sources:

[1] N.p., n.d. Web. <http://www.chemistryexplained.com/St-Te/Temperature.html>.
[2] "Physics 103.1 Experiment Manuals: Temperature Measurement." National Institute of Physics, n.d. Web.
[3] "Physics 103.1 Experiment Manuals: Heat Conduction." National Institute of Physics, n.d. Web.
[4] "Thermal Conductivity of Materials and Gases." The Engineering Toolbox. N.p., n.d. Web.