This is called the ideal gas equation. Here, P is the pressure in Pa, V is the volume in m^{3}, n is the number of moles, T is the temperature in Kelvin, and k is the ideal gas constant, which has a known value of 1.3806488 x 10^{-23}.

Several simpler gas laws arise from the ideal gas law. Two of these are Boyle’s Law and Charles’ Law. Each of these describes the relationship between two of the variables described in the ideal gas equation. Robert Boyle discovered that under certain conditions, volume and pressure follow an inversely proportional relationship. Mathematically,

where C_{b} is equal to nkT. This is known as Boyle’s Law.

Charles’ Law describes that the volume of a gas is directly proportional to the temperature. Mathematically,

where C_{c} is equal to nk/P.

The goal of the experiment is to verify Boyle’s and Charles’ Law through the calculation of the volume of an airtight chamber by plotting the relationship between volume and pressure, and volume and temperature.

The materials used in the experiment were a heat engine, an air chamber can, rubber tubings, a Vernier LabQuest with a gas pressure sensor, a thermocouple, some ice, a pot, and a gas stove.

First, the pot was filled with water and placed on the stove to boil. For the verification of Boyle’s Law, the air chamber can was connected to the heat engine using rubber tubing, then submerged in the pot filled with boiling water. The piston of the heat engine was lifted to its maximum height, then the gas pressure sensor to the Vernier LabQuest. The initial pressure reading was then recorded. The temperature was also recorded using a thermocouple. Masses were then placed on the piston, in increasing increments of 50g. As each increment of mass was added, the change in height of the piston was recorded. Because the diameter of the piston is already known, the change in volume may be calculated. This was plotted against the change in pressure in the piston. The graph may be seen below.

From the high R^{2} value, it is evident that the relationship between pressure and volume is indeed quite linear. For this part of the experiment, the slope of the equation is equal to 1/nkT, and the volume of the air chamber itself is theoretically equal to the quotient of the y-intercept and the slope. By calculation, the number of moles in the chamber was 1.12 x 10^{19}, and the volume of the air chamber was 4.74 x 10^{-4} m^{3}.

These results were then compared to those obtained from a set of procedures involving Charles’ Law instead. The pot was removed from the stove, and instead of increasing the pressure on the piston, several chunks of ice were added to the pot, and the change in temperature was measured against volume. The graph of the results is shown below.

For this test, the number of moles is equal to the product of the slope and the recorded pressure divided by k, while the volume of the air chamber is simply equal to the y-intercept. The results of both experiments are tabulated below.

The deviation between the two values for the number of moles is quite large, but for the volume of the air chamber it is quite close. This makes sense, as the sheer largeness of numbers involved in measuring the number of moles makes it more prone to larger differences in data. However, on a whole, the experiment may deemed successful considering the error-prone set-up.

Sources of error include: improperly sealed connections of rubber tubing, inconsistent pressure and temperature readings, the fact that the system was an open one and heat could escape at any time, and limitations in the measurement of volume due to a relatively unclear scale on the piston.

]]>

Heat is quantified through the following relationship:

Where Q is heat measured in joules, m is mass, c is the specific heat capacity, and ΔT is change in temperature of the object. The objective of the experiment is to find the specific heat capacity of two metals, namely aluminum and copper, by exploiting this relationship.

Another phenomenon is essential to finding specific heat capacity: The Zeroth Law of Thermodynamics. The Zeroth Law of Thermodynamics states that if A is in contact with B, and B is in contact with C, is A and B are in thermal equilibrium, then A and C are in thermal equilibrium as well.

The experimental set-up consisted of an isobaric coffee-cup calorimeter filled with tap water. The metal, which was originally immersed in boiling water, was placed in the calorimeter. The change in temperature of the water is measured using a thermocouple. After this, using the known specific heat capacity of water, the heat of the water is calculated. By the Zeroth Law of Thermodynamics,

when there is a metal present. The heat of both the calorimeter and the water must be known to be able to calculated the heat of the metal. By extension, the heat capacities of each are needed to calculate their respective values for heat. When there is no metal, a calibration curve may be obtained to calculate the heat capacity of the calorimeter. This is done by mixing hot and lukewarm water. Change in temperature is tracked with reference to time, and a calibration curve is formed by linearizing the following relationship:

The y-intercept of the curve is equal to the exponential of the final temperature as time approaches infinity. Once the change in temperature of both the tap and hot water have been calculated, their heats may be obtained, and therefore the heat capacity of the calorimeter. Once this value is known, any change in temperature in the water in the calorimeter due to an immersed metal may be used to calculate the heat capacity of that metal.

Three calibration trials were done. 60 mL of both tap and hot water were used for each trial. Knowing the density of water, this means that the mass of each was 60g, for a total of 120 g. The specific heat capacity of water is also known to be 1 Jg ^{0}C^{-1}. Below is a representative calibration curve.

The heat capacities calculated for the three trials were the following:

Trial | Heat Capacity of the Calorimeter (J/g • ^{0}C) |

1 | 26.16 |

2 | 38.07 |

3 | 48.75 |

The average value obtained was 37.66. This result was far from precise as manifested by the large range in values, which had the experimenters quite confused. Still, this average value was determined to be the heat capacity of the calorimeter.

Knowing the mass of the metal, the volume of the water in the calorimeter, the density of water, and the specific heat of water, and measuring the changes in temperature in the water using the thermocouple, the specific heat capacity of the metals was then calculated. The following results were obtained:

Metal | Aluminum | Copper |

Calculated Specific Heat Capacity | 1.003 | 0.3895 |

Percent Error | 11.52% | 0.9002% |

From these results, it is safe to say that the specific heat capacities of both metals were accurately calculated, even with the imprecise measurement of the heat capacity of the calorimeter. Still, error may have arisen from a multitude of sources: the makeshift nature of the coffee-cup calorimeter and the fact that it was not a sealed system and that heat may have escaped, the fact that the thermocouple could only measure accurately to the ones digit, inaccuracies in the measurement of the water since it cannot be transferred perfectly from vessel to vessel, and many more.

In conclusion, the experiment has been deemed a success, and the method used valid.

]]>

Since we were children, we’ve known what thermometers did. These simple devices are used in a multitude of fields — from medicine, to chemistry, to the culinary arts. Many often overlook the science behind the measurement of temperature, because it is such a basic concept.

When an instrument, such as a thermometer, comes in contact with an object, heat from the object transfers to the thermometer, due to thermal equilibrium. This change in temperature is not instantaneous. A graphical representation of this relationship may be seen below.

This is described by the following function:

Where τ is the thermal time constant, t is the time elapsed, and T is the temperature. Substituting t for τ, this function takes the following form:

We may also substitute multiples of τ into the equation. The following functions result:

Because of the exponential nature of the function, as the time value gets higher, the change of the function with respect to time decreases. As time approaches infinity, T(t) = T_{f}. However, if too much time elapses, too much heat may escape, and error becomes more likely. Since the change in temperature is grows insignificant as time elapses, it is good practice to read the measurement when the time elapsed is between 3τ and 5τ. The goal of the experiment is to find the value of τ for an alcohol thermometer.

For this experiment, the temperature function of an alcohol thermometer was found by testing its behavior for both heating and cooling. For heating, a cup filled with tap water was placed on top of a small stove, then thermometer was held in place in the cup until the temperature ceased to rise. This value was recorded as T_{f}. The thermometer was then submerged in a cup of ice, and its temperature was recorded as T_{i}. Using these values, τ and its multiples were calculated using the above table. The thermometer was then submerged back into the hot cup, and when the thermometer read each temperature value in the table, the time was recorded.

These results were then linearized, where the negative of the inverse of the slope is the value for τ. Representative graphs for both heating and cooling may be seen below, as well a table showing the τ of each trial.

At first glance, both seem to be quite accurate, with high R^{2 }values. This indicated a high degree of precision. However, the resulting values for τ indicate otherwise.

In this table, all values for τ are indicated in seconds. This result is quite alarming, considering that the inherently linear nature of measurement of the alcohol thermometer should indicate a uniform τ value across all trials. The values for heating are quite consistent, with an average deviation of only 0.071. For cooling, on other other hand, the average deviation was 3.739. This is quite a large variation in values, and is indicative of inaccurate results with low precision. From this data, the value of τ cannot be conclusively determined.

Several sources of error may have caused this. The set-up itself was quite crude, and the thermometer was held in place by a person holding one end. Since the set-up was an open system, the temperature may have fluctuated due to the loss of heat to the surroundings. The source of heat, the stove, for the heating trials was consistently on. Since a constant source of heat was applied, the results were more precise. Compare this to the cooling set-up, which consisted of an ice cube that was melting, and it is easy to tell that the source of heat for the cooling set-up was not at all consistent. The thermometer may have also been placed incorrectly, since the ice cubes created many spaces that may have resulted in fluctuating temperatures within the cup itself.

In conclusion, the experiment was not successful because the τ values for heating and cooling were inconsistent. For future experiments, it is highly recommended that a cooling set-up with a more consistent temperature is used. For example. a cup of liquid water surrounded by ice may suffice.

]]>

This is a quote from “On the Nature of Light and Colors”, a lecture by Thomas Young, which originally proposed the dual nature of light through the double slit experiment. To build on this, an experiment based on these concepts was also performed by Physics 103.1 students. The main goal of the experiment was to analyze the effects of slit width and the number of slits on a light source.

A laser diode that emitted red light, which has a wavelength of approximately 650nm, was used as the light source. To ensure precision, specialized single slit and multiple slit disks were used to observe the effects of different types of slits on the behavior of the light. The entire set-up was perched on an optical bench, and a white paper screen was placed a distance away to provide a clear backdrop for the light. Lastly, a 12-inch ruler was used for measurement purposes.

For the first test, a single slit was used to calculate the wavelength of the diode. Two slits were used, the first with slit width (a) of 0.02mm and order (m) of 1. For the second, a=0.04mm and m=2. The length of the central maximum was measured by marking its ends on the white paper using a pencil, then measuring the distance between each mark.

The experimental wavelength was then calculated using the formula below:

The table below shows the results of the experiment:

It is obvious that the trial where a=0.04mm and m=1 is the far more accurate and precise trial. The high error from the other trial may be attributed to experimenter’s error in measuring the length of the central maximum. As the experimenter who measured this herself, this is very frustrating. The relief, however, is that the later parts of the experiment do not reflect this at all! If anything, These results confirm that the equation seen above does hold.

Another angle was also used to test this relationship. The experimental value for these tests is now the wavelength of the diode. The results may be seen below.

These results were extremely accurate and precise considering that our equipment consisted of a pencil and a ruler! This further confirms the equation above.

An interesting observation is that there clearly are smaller and dimmer fringes, multiple diffraction envelopes, on either side of the central maximum. This phenomenon seems to be counterintuitive; most would assume that the resulting pattern would be a single strip of light. It hints at the fact that light’s behavior as a wave is an inherent part of its nature.

Next, we moved on to the multiple slit disk to test the behavior of double slit interference. Aside from the disk, the same set-up and method of measurement was used.

Within the diffraction envelope, there are peaks in brightness, as well as dark areas. The bright spots indicate constructive interference, while the dark spots indicate destructive interference. This is demonstrates double slit interference, and is the exact phenomenon that Thomas Young used to demonstrate light as a wave.

For this initial trial, the slit width was calculated. The equation seen above still holds for double-slit interference. The results may be seen below.

The results are extremely accurate and precise, as with the previous results.

Another aspect of this experiment that was analyzed was the relationship between the slit width and the distance in between the slits and the resulting interference.

This experiment demonstrates several phenomenon. If the slit width is held constant, then the number of fringes within the central maximum is dependent on the distance between the slits, where increasing the distance increases the number of fringes. However, if slit width is manipulated, is appears that the number of fringes decreases as the slit width increases. The central maximum also appears to be dependent only on the slit width. Lastly, the fringe width seems to depend only on the distance between the slits, where an increase in slit distance means a decrease in fringe width.

In conclusion, the results of the experiment confirm the fact that nature behaves as a wave, as manifested through diffraction, and constructive and destructive interference patterns.

]]>

The first part of the procedure was to align the beam of light that would be used for the rest of the experiment. The result was a single beam of light perfectly aligned with the 0^{0}-0^{0 }axis of the optical disk.

The first part of the experiment tested the accuracy of the alignment of the optics. This was done by placing a plane, convex, and concave mirror on the optical disk. The placement ensured that the point where the light met the mirror was exactly at the center of the disk, and that the mirror was aligned to the 90^{0}-90^{0 }axis. It was a painstaking process for both me and my group mates. The experimental set-up was so sensitive, especially considering that our bare hands were not the most precise means to manipulate it.

The results may be seen below.

As seen from the table above, the optics were now completely aligned. The interesting part, however, is not that the plane mirror have the same angles of incidence and reflection, but that the convex and concave mirrors displayed this as well. Because infinitesimally small planes essentially comprise curved surfaces, the angles of incidence and reflection of curves consequently follow the same laws as those for plane surfaces. This is provided that the beam of light is relatively thin and the point of reflection is perfectly perpendicular to the said beam.

Next, the index of refraction of a cylindrical lens was calculated. The driving concept here is Snell’s Law. It is described in the equation below:

where theta 1 and theta 2 describe the angle of incidence and refraction, respectively, and n1 and n2 describe the indices of refraction of the incident and refractive mediums respectively. First, the experiment was performed with the beam striking the plane side of the lens.

The data was then graphed and linearized according to Snell’s Law.

The slope of this equation is equal to the ratio of the index of refraction between the two mediums. At STP, the index of refraction for air, the incident medium, is about 1. Therefore, the index of refraction of the cylindrical lens is 1.4767. The resulting equation also has an R^2 value of 1. This shows that the experiment had a high degree of precision.

Next, the same experiment was performed, except the lens was rotated 180 degrees, with the curved surface facing the beam.

The data is shown in the graph below. It was analyzed in the same manner as the results from the tests using the flat side of the lens.

The high R^2 value also denotes a high degree of precision. An interesting observation arises: the value of the index of refraction, which is now 0.5938, has changed after the change in the orientation of the lens as it is hit by the incident beam. This may be attributed to the change in angle due to the change in the apparent geometry of the lens.

The critical angle is defined to be the angle at which the medium no longer refracts light. In this instance, it was experimentally determined to be 42.5 degrees.

]]>

I’m a second year Applied Physics major, studying at the University of the Philippines Diliman National Institute of Physics. It’s difficult, but studying the branch of science that encompasses any and every measurable phenomenon can be overwhelming sometimes. I’ll be writing about what happens during my lab classes, where I and a couple of other friends will be learning the practical basics of optics and thermodynamics.

One branch of physics is mechanics, the study of the motion (or sometimes, lack thereof) of material objects. Kinetic energy is one of the hallmark concepts this interesting field. It is often described as the energy associated with motion. For example, as a freely falling object drops to the ground, potential energy is converted to kinetic energy, until suddenly the body stops, and all that kinetic energy is converted back to potential energy.

Kinetic energy, in a poetic sense, is associated with being dynamic. As motion hits a crescendo, and kinetic energy is at its maximum, the potential that a body once possessed is completely used up. I have lived my life with people telling each other that they have great *potential,* but to me, this doesn’t matter until that potential is used for something great.

I hope to live my life in a kinetic way, converting all the potential that I have into something bigger than myself.

Exciting times ahead!

]]>