#physics 103.1

For a while now, we’ve been talking about temperature and how it influences the world around us. This time we will talk about how temperature, among other factors, influences the gases around us. 

Let us observe gas in a fixed container and what happens when we change its temperature; a balloon on a hot summer’s day for instance. Like we described before, temperature is the measurement of the average kinetic energy of the molecules of a substance. The higher the temperature, the faster the molecules move. So as the balloon heats up while it’s under the sun, the air molecules inside it move faster, hitting the walls of the balloon more frequently. This increases the balloon’s pressure, as pressure is the amount of force exerted per given unit of area. This also causes changes in the balloon’s volume, the balloon being forced to expand to accommodate the increase in pressure. Pressure and volume are at a balance though, because as one increases, the other would decrease. However, we all know that if the pressure inside the balloon gets too much, the balloon being only able to expand to a certain extent, it would finally *POP*. 

These three factors, temperature, pressure, and volume, are interrelated to one another through what we know as the Ideal Gas Laws. An ideal gas is a theoretical gas that is composed of many randomly moving point particles that do not interact with one another except when they collide elastically (kinetic energy is conserved). Although it is a theoretical model, it is used to describe dilute gases at low pressures and high temperatures, as gases in these conditions have properties very similar with the ideal gas model. Many common gases can be considered to be ideal gases at room temperatures and pressures close to 1 atm. 

There are three main laws composing the Ideal Gas Laws, each relating a different pair of factors among temperature, pressure, and volume, and each having been discovered experimentally. Boyle’s Law relates volume to pressure and states that they are inversely proportional to one another. Charles’s Law shows how volume is directly proportional to temperature. The last of the laws, Gay-Lussac’s Law, states that pressure is directly proportional to temperature. These three laws can be summarized in the ideal gas equation:

where P is the pressure in Pascals, V is the volume in m^3, N is the number of gas particles, k is the Boltzmann’s constant (k = 1.3806488 × 10−23 m2 · kg · s−2K−1), and T is the temperature measured in Kelvin.

In this experiment, we will be experimentally verifying the former two laws, namely Boyle’s Law and Charles’s Law, using our own set-up.

METHODOLOGY

We tackle verifying the two laws separately but each uses the same instruments and measuring devices. 

Our container of gas in this experiment was the mass lifter apparatus attached to an air chamber via rubber tubing. The mass lifter apparatus had a diameter of 0.0325 m as written on its dimensions as well as gradations of 1 mm measured from 0 to 100 mm. The air chamber (in grey) would be in direct contact with our heat/ice bath, varying the temperature of gas through this contact. 

The mass lifter apparatus was also attached to a Vernier LabQuest with a gas pressure sensor, measuring the pressure within the apparatus in kPa to a smallest reading of 0.01 kPa.

The hot bath consisted of a beaker ¾ filled with water, in which the air chamber was placed, within a pot ¾ filled with water over a hot plate. The water was made to boil and kept boiling throughout the entire experiment.

In the succeeding part of the experiment, the hot bath was removed from the hot plate and chunks of ice were placed directly in the water of the beaker, to cool the gas within the air chamber. Temperature was measured through a digital thermometer with lowest reading of 0.1 degrees Celsius.

Boyle’s Law

To verify Boyle’s Law, we constructed a set-up such that temperature could be kept constant while we varied pressure and volume. The air chamber, connected to the mass lifter apparatus through rubber tubing was, placed in the hot bath. The piston of the mass lifter apparatus was lifted to maximum height. Temperature was measured and kept constant at 353.65 K. 

A 50 g standard mass was placed on the platform of the mass lifter apparatus. After 5 seconds, the height h of the piston and the pressure P measured were recorded. This step was repeated for masses of 100 g, 150 g, 200 g, and 250 g.

Using the heights recorded, the volume of the cylinder, Vcyl, for each height was calculated. The reciprocal of the pressure readings, P^-1, in Pa were also computed. A plot of Vcyl vs P^-1 was graphed with a linear fit. It should be noted that the total volume of air in this experiment was the volume of the cylinder added to the volume of the chamber. Thus, the y-intercept would be the negative of the volume of the chamber. The number of particles N and the volume of the air chamber were calculated using the Ideal Gas Equation using the slope and y-intercept of the graph respectively.

Charles’s Law

To verify Charles’s Law, we constructed a set-up such that pressure could be kept constant while we varied temperature and volume. We removed the hot bath from atop the hot plate from the previous set-up. The piston was lifted to its maximum height. Pressure here was monitored to be a constant 101.61 kPa.

The temperature of the hot bath was measuring using a digital thermometer and the initial height h of the piston was recorded. Ice chunks were slowly added on the hot bath and h and T measurements at equal time intervals of 10 seconds until the set-up had cooled to close to room temperature.

A plot of V vs T was made with a linear fit with the equation and R^2 values present. The number of particles N and the volume of the air chamber were calculated using the Ideal Gas Equation using the slope and y-intercept of the graph respectively. 

We also note that if we increase the pressure in this experiment, the corresponding effect to its V vs T graph would be a decrease in slope. This could be seen by rearranging variables in the Ideal Gas Equations.

Gay-Lussac’s Law (optional)

Although not accomplished in this experimentation, an additional set-up to verify Gay-Lussac’s Law could be made using the materials used in this experiment by keeping the volume constant while pressure and temperature would be made to vary.

The air chamber would once again be placed in the hot bath and allowed to heat for a while. The hot bath would then be removed from atop the hot plate. The piston would be lifted to maximum height throughout the entire experiment. The initial temperature T and pressure P reading would be taken. Ice chunks would then be slowly added to the water of the beaker, and T and P readings would be taken at equal intervals of 10 seconds, until the set-up reaches near room temperature.

RESULTS AND DISCUSSION

The following graphs show the results of our experimentation.

This graph shows results of our verification of Boyle’s Law. The R^2 value describes the precision of the experiment, showing a moderately high precision. Using the slope of the experiment and our known measurement for the temperature in the Ideal Gas Equation, we could compute an N of 6.02E+21 particles of gas. The volume of the chamber can also be gathered from the y-intercept of the graph, and it is seen to be 0.000205557 m^3.

This graph shows results of our verification of Charles’s Law. The R^2 value is smaller, but still maintains to be moderately high. Once again, using the slope of the experiment and our known measurement for the pressure during the experiment in the Ideal Gas Equation, we find an N of 4.08E+21 particles of gas. The volume of the chamber can also be gathered from the y-intercept of the graph, and it is seen to be 0.000105002 m^3.

It is seen that despite the precision of the experiments as indicated by the R^2 values, some discrepancies occurred as the N and Vcham values are not the same. This could be attributed to some limitations with our experiment and what we could keep constant throughout the experiment. However, the results do remain in the same order of magnitude, describing some accuracy in experimentation.

CONCLUSIONS AND RECOMMENDATIONS

We conclude that the experiment were effective in verifying Boyle’s and Charles’s Laws, showing how volume is inversely proportional to pressure and how volume is directly proportional to temperature respectively. This was evident with the high precision of the linear fit of the graphs made as seen in the high R^2 values, which would only happen if these laws were valid. The experiment was not free of limitations, however. Discrepancies between the measured values of N and Vcham in Boyle’s and Charles Law (of 6.02E+21 particles and 0.000205557 m^3 in Boyle’s Law verification and of 4.08E+21 particles and 0.000105002 m^3 in Charles’s Law verification) show the limitations of our experiment. 

These discrepancies could be attributed to systematic faults in our experiment. It was observed that gas was leaking out of the air chamber and mass lifter apparatus, as the piston’s height would decrease over time even if no force was acting on the system. Although we attempted to overcome this by taking readings at fixed time intervals, this could have still led to the decrease in accuracy of the experiment as observed with the different N and Vcham readings observed. Another limitation would be that the fixed variables in either experiment (temperature for Boyle’s Law and pressure in Charles’s Law) were measured at single instances. The reality was that these factors still varied throughout the experiment. Since these were used in computation, this could have also had an effect on the precision accuracy of the experiment.

To resolve these, I would suggest that better equipment be used and all joints be properly sealed so as to limit the amount of gas exchange. It would also benefit the experiment if the measurements for the supposedly fixed constants would be taken over a longer period of time, and the value used would instead be the average of values over that time frame. With these suggestions, similar experiments could allow more accurate and precise manipulation of factors utilizing the gas laws. 

As gas remains one of the most chaotic substances in terms of movement, it is amazing that we were able to grasp the effect of temperature, pressure, and volume on them. Physics finds new ways to make even the most complex of ideas within our grasps. This experiment specifically shows how experimental physics can be on par with theoretical physics in discovering more and more about the world around us.

REFERENCES

1.     Physics 103.1 Lab Manual, T3 Gas Laws, National Institute of Physics, Philippines, 2013.

2.     R.A Freedman, H.D. Young, University Physics with Modern Physics 13th Edition, Chapter 18, Pearson Education, San Francisco, 2012.

On a cold rainy day, it’s nice to stay in bed, cuddled up with your blankets and pillows, and take naps with the warmth around you. Later on in the day, too lazy to cook your own food, you check your refrigerator for the leftover pizza you had the night before. It’s cold, and from experience, you would rather skip on chowing down on cold pizza when you have the opportunity to make it taste better. You pop the pizza into the microwave, and after around thirty seconds of anticipation, out comes a “fresh” slice, with its cheesy aroma reminding you why you love pizza so much. A bit too hasty, you took a bite, burning your tongue in the process. You quickly grab a glass of ice water to soothe your tongue, and take your food back to bed for the rest of your lazy day.

Hot and cold are words we use to describe the temperature of an object. Temperature comes in a balance. Neither hot nor cold are inherently bad. We use objects of both degrees to assist us in everyday life, from cold refrigerators keeping our food preserved to hot microwaves to heat up our food when it’s time to eat. When we talk about temperature though, we’re actually describing another characteristic of an object – its internal kinetic energy. In fact, temperature is known as the measurement of the average internal kinetic energy of molecules within an object. A high temperature or something that is hot has molecules moving quickly due to their higher average in kinetic energy. On the other hand, an object of low temperature or something that is cold has its molecules moving relatively slower due to their lesser average kinetic energy.

The heat that we feel coming from a warm slice of pizza is actually a result of the difference between the temperature of the pizza and its surrounding air. We can then define heat as the energy transferred between objects as a result of a difference in temperature. When two objects transfer heat, they reach a point where their temperatures are equal and therefore, no further energy can be transferred. This is called thermal equilibrium. 

Hot and cold though are rather qualitative terms. Something that is hot to one person might just be moderate to another. A Filipino whose natural environment is in the tropics would find winter much colder than someone who is already accustomed to the chilly weather. This is why we find ways to standardize our ways we measure temperature.

We actually have three major degree scales we use to measure temperature. These would be the Celsius (C), Fahrenheit (F), and Kelvin (K) scales. Celsius uses water as its basis, with 0 degrees as the freezing point of water and 100 degrees as the boiling point of water. Fahrenheit also follows this, having 32 and 212 degrees marking the two phase changes respectively. The Kelvin scale has 0 K being absolute zero or the temperature at which there is no kinetic energy of molecules within the object and all molecules are at rest. Celsius and Fahrenheit are used in everyday life, depending on a person’s location in the world; however, we in the science community use Kelvin as the SI unit for temperature as it closely relates to our definition of what temperature actually is.

Measuring temperature requires the use of a thermometer. When a thermometer is placed in contact with a substance, heat transfer occurs until the thermometer and the substance reach thermal equilibrium. The thermometer actually reads its own temperature in this case. It should be noted that placing the thermometer in contact with the observed substance changes the substance’s temperature. It is then essential that the transfer of heat to or from the thermometer remain minimal.

Heat does not transfer linearly, however. The greater the difference between the temperatures of the two objects, the faster heat is transferred. Once the difference decreases, the rate at which heat is transferred also decreases. This is visualized in the following graph:

Figure 1. Graph of temperature versus time in a heating process. 

We let T(t) denote the temperature at a given time t. The shape of this graph or how fast a thermometer reaches thermal equilibrium depends on the thermal time constant (tau). Using our knowledge of exponential functions, we can solve for T(t) through the equation

When time is equal to different multiples of tau, we get the following

We can then observe that accurate measurements for the final temperature of an object are measured at t values between 3 tau and 5 tau. Shorter than that and you would have an imprecise reading while longer than that would risk heat transfer between the system and its surroundings.

Typical thermometers used would include alcohol and mercury thermometers, where a thin glass cylinder is filled with the substance and scaled to provide its readings. How exactly can we get readings from these thermometers? Well when these measuring devices increase in temperature, the substances within them expand as a result of the molecules, with increase in energy, moving further away from one another. For thermometers with measurements in Celsius, the usual way of marking would involve allowing the thermometer to reach temperatures of 0 and 100 degrees, using ice and boiling water, marking these points, and creating smaller, proportional divisions. Since the length of the substance within the thermometers expands proportional to the temperature, this method of creating divisions would remain accurate.

In this experiment, we used the concept of thermal equilibrium to obtain the thermal time constants for an alcohol thermometer and a mercury thermometer.

 Methodology

Two setups were used for experimentation.

Figure 2. Hot setup of boiling water in a 3/4 filled pot.

The first was our hot setup. A pot was filled with water up to 3/4 of its capacity and left to boil. This water would be boiling throughout the entire activity so as to allow the water to remain at temperatures near the boiling point.

Figure 3. Cold setup of ice and ice water

The second was our cold setup. A beaker was filled with ice water and left to stand. The ice would allow the water to remain at temperatures close to the freezing point of water.

To determine the thermal time constants for the given thermometers, two procedures were conducted: the heating and cooling procedures. For each of the procedures, three trials were conducted for each of the alcohol and mercury thermometers.

For the heating procedure, a thermometer was first placed in the hot water, and the water’s temperature was recorded as Tf. The thermometer was then dipped in the cold water. The temperature in the cold setup was recorded as Ti. T(tau) was then computed using the above equation. With initial temperature Ti from the cold setup, the thermometer was placed into the hot setup and the time taken for it to reach T(tau) was recorded as the thermal time constant for the thermometer. It should be noted that time measurements were taken using a digital stopwatch of lowest increment of 0.01 s. It was also ensured that the thermometer did not touch the sides of any of the containers while any measurement was being taken.

For the cooling procedure, a similar process was conducted; however, Tf was initially taken as the temperature of the cold setup while Ti was taken as the temperature of the hot setup. T(tau) was computed for once again. With initial temperature Tf from the hot setup, the thermometer was placed into the cold setup and the time taken for it to reach T(tau) was recorded as the thermal time constant for the thermometer.

Results and Discussion

After conducting the trials for experimentation, the data was tabulated as seen below.

Tables 2 and 3 show the data for the alcohol thermometer. It can be observed that the time constant for cooling is significantly larger than the time constant for heating. We look towards our data for the mercury thermometer for any similar trends. At a range of 3 tau to 5 tau, it would then take 17.46 s to 29.1 s to get an accurate measurement for heating and 21.15 to 35.35 s to get an accurate measurement for cooling, both using an alcohol thermometer.

Tables 4 and 5 show the data for the mercury thermometer. It is noted that the thermal time constant is much smaller than that of the alcohol thermometer. At a range of 3 tau to 5 tau, it would take 5.16 s to 8.10 s to obtain an accurate measurement for heating and 6.57 s and 10.95 s to get an accurate measurement for cooling, both for a mercury thermometer. This coincides with known concepts, as mercury thermometers are used to measure much higher temperatures and thus, expand to greater increments much faster than alcohol thermometers.

It is also of note that once again, the thermal time constants for cooling are significantly greater than the thermal time constants for heating for both alcohol and mercury thermometers. Intuitively, it would be proposed that they would have equal time constants as they are of the same material. However, it can be concluded that the time constants for cooling would indeed be greater than the time constants for heating, and thus, it would take a longer time for an object to decrease its temperature than to increase its temperature by the same difference. This could be attributed to the cooling pattern resembling a logarithmic function with a floor value as opposed to the function given above.

Conclusion and Recommendations

It was discovered that the thermal time constants for the used alcohol thermometer were 5.82 s and 7.05 s for heating and cooling respectively while the time constants for the used mercury thermometer were 1.72 s and 2.19 s for heating and cooling respectively. The thermal time constants were greater for the alcohol thermometer, meaning it would take a longer time for the thermometers to get an accurate reading for the alcohol thermometer than the mercury thermometer. It would also take a longer time to get an accurate reading for a cooling procedure than a heating one.

The deviation between results was significant enough that the results gathered may not have been as precise as we would want. These would be results of reaction time, which for small times would have a greater impact, and heat transfer to the surrounding. To reduce the amount of errors and ensure that a greater precision is achieved, additional trials for each procedure and each thermometer should be conducted.

It is important that we understand the thermal time constants of our thermometers. This is so we can ensure accurate and precise temperature measurements in our experimentation. It is also important that we understand the limitations of our measuring instruments so we would choose the most appropriate ones given our procedures. With this knowledge, we can head into more complex thermodynamic experimentations with confidence in our basics.

References

1.     Physics 103.1 Lab Manual, T1 Temperature Measurement, National Institute of Physics, Philippines, 2013.