Little do we know that indeed gases exhibit such intricate and sometimes, counter-intuitive, properties. Further, it must be known that it took many scientists across the years to even pinpoint and experimentally arrive at the conclusions we discuss in class.
It is amazing, however, that during those times when these scientists were analyzing the probable properties of gas, these same scientists were not totally aware of the existence of small particles which gas is consisted of. Nonetheless, when a more atomic level of understanding was achieved, these same properties were confirmed.
We start by stating the basic assumption that scientists before used to be able to pinpoint certain properties of gas. Scientists assumed that gases consisted of small perfectly non-elastic spheres. This means that when gas particles hit a barrier or another particle, they do not deform. Another assumption they made was the fact that all collisions between gas particles were perfectly elastic and that no energy is lost. Further, they assumed that Newton's Three Laws of Motion apply, that the distance between each particle is much larger than the size of one particle, that the gas particles are in a constant state of motion and do so in a random manner. And lastly, they assumed that there were no attractive and repulsive forces between each particle and its surroundings. These assumptions led to the idea of what is known to be an ideal gas.
So, under the assumption that some gas is ideal, then we can relate some of the quantities that can be measured from a gas. Such is said to be a state function. Examples of state functions for an ideal gas (and it can be proven to be so for an non-ideal one) is pressure $P$, volume $V$, temperature $R$, internal energy $E$, enthalpy $H$, Gibbs Energy $G$, entropy $S$, and etc.. State functions are called such because these values are dependent on the state of the object when these parameters were measured. Further, these state functions are such to be path independent, or that they do not depend on the 'path' taken to get some result.
After many years of analysis and experimentation, a group of scientists was able to come up with a group of statements with which we relate some of these state functions. This group of statements are known as the Gas Laws.
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| Figure 1: Schematic Diagram of Boyle's Experiment. (Source: General Chemistry by Ebbing, Gammon, et. al.) |
Robert Boyle was the first to discover the relationship between the pressure and volume enclosed by some gas. Boyle tested this by examining how the volume of gas changed as he added liquid mercury on a J-test tube.
Boyle initially added mercury onto a J-test tube. This created a partial vacuum of air inside enclosed by the mercury. Now, as Boyle added more and more mercury into the same test tube, he added pressure onto this gas. As a result, and as seen in Figure 1 on the right, the volume of air decreased more and more. This led him to state what is known now as Boyle's Law. Boyle's Law states that:
"When the gas is kept at a constant temperature, its volume is inversely proportional to the pressure exerted on it."This meant that $P\propto1/V$ and that the product of the gas' pressure and volume is equal to some constant. Simple as that.
After Boyle came Jacques Charles, who also discovered a relationship between two state functions of gas. This time, he focused on how the volume enclosed by a gas differs as its temperature changes under constant pressure. After experimentation, he realized that under constant pressure, the volume enclosed by a gas is directly related to its temperature. This now is known as Charles' Law. Charles' Law states that:
"The volume a certain amount of gas kept at constant pressure is directly proportional to its absolute temperature."The absolute temperature that is being mentioned in the law refers to the unit of temperature, Kelvin. This law just means that $V\propto T$ and that the quotient of the gas' volume and temperature is some constant.
One can illustrate this using the figure below:
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| Figure 2: Simple Illustration of Charles' Law. (Source: General Chemistry by Ebbing, Gammon, et. al.) |
In Figure 2, we have two colored balloons, one colored red and the other, blue. The picture to the right is when both of them are under the same temperature. Notice that they are similar in size, or volume. Now, when the blue balloon is dipped in liquid nitrogen, as in the left picture, we observe that its size decreased appreciably compared to the red balloon. This makes sense because the temperature liquid nitrogen is much less than the temperature of the room, hence by Charles' Law, the blue balloon must shrink.
Now, two other laws came into play. One of which is Gay-Lussac's Law and Avogadro's Law. To put it simply, Gay-Lussac's Law states that the pressure of a gas kept at constant volume changes proportionally to its temperature or $P\propto T$. This means that at constant volume, the higher the pressure on the gas, this higher it's temperature. Furthermore, Avogadro's Law states that the amount of gas enclosed $n$ in a volume changes directly or $n\propto V$.
The combination of these laws essentially resolves into what is known to be the Ideal Gas Law. Explicitly, the Ideal Gas Law states that:
\begin{equation*}
PV=nRT
\end{equation*}
where $P$ is in atmospheres, $V$ in liters, $n$ in moles, and $T$ in Kelvin. Furthermore, there is a constant $R$ which has a value of $8.3141598$ J$/$mol$\cdot$K.
We can relate the number of moles of a gas and the number of particles $N$ using the relationship that $n=N/N_\text{A}$ where $N_\text{A}$ is known to be Avogadro's Number, the number of particles in one mole. $N_\text{A}=6.0221408\times10^{23}$. Substituting this to the equation, we get:
\begin{equation}
\label{main} PV=N\,\dfrac{R}{N_\text{A}}T
\end{equation}
where we can define another constant $k$, the Boltzmann Constant, where $k=R/N_\text{A}=1.3806485\times10^{-23}$ J$/$K.
In this experiment, we will verify Boyle's Law and Charles' Law.
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| Figure 3: General Set-Up of the Experiment. The piston of the Mass Lifter Apparatus was held at maximum height while being connected to the Vernier LabQuest gas pressure sensor and air chamber using rubber tubing. The gas pressure sensor was then connected tot he Vernier LabQuest Interface. |
To verify Boyle's Law, water was brought to a boil, and the air chamber can was submerged in it. During this part of the experiment, the boiling water was kept at a constant temperature. The temperature of the boiling water was recorded, and different masses was placed on top of the mass lifter apparatus to see changes in the height of its piston. Masses used ranged from 50.0 g to 250.0 g in 50.0 g increments. Every after placing the mass on the piston, the resulting pressure reading on the Vernier LabQuest was recorded.
In the analysis of the data recorded, the inverse of the pressure of the gas was plotted as a function of the volume enclosed by it. The goal of this part of the experiment is to obtain a best estimate of the volume of the air chamber and the number of gas particles in it. The total volume of the gas enclosed is just the sum of the volumes of the mass lifter apparatus and the chamber, $V=V_\text{piston}+V_\text{chamber}$. Further, from Equation \eqref{main}, $1/P$ is related to $V$ by:
\begin{equation}
\dfrac{1}{P}=\dfrac{1}{NkT}\cdot V
\end{equation}
which is linear, and hence a linear fit of the form $y=Ax+B$ can be used. From this same equation we can solve for the volume of the air chamber and the number of gas particles. From the fit, the equation, and the relationship, we can calculate for $N=1/AkT$ and $V_\text{chamber}=B/A$.
To verify Charles' Law, the same air chamber was submerged in hot water and measuring the initial pressure of gas in the piston of the mass lifter apparatus. Now, for 3 minutes, in 20 second intervals, small chunks of ice was continually being added onto the water while the height of the piston of the mass lifter apparatus and the temperature of the water was being recorded.
In analyzing the data which was recorded, the temperature of the gas was plotted as a function of the volume enclosed by it. Same as before, the goal of this part of the experiment is to obtain a best estimate of the volume of the air chamber and the number of gas particles in it. From Equation \eqref{main}, $T$ and $V$ is related by:
\begin{equation}
T=\dfrac{P}{Nk}\cdot V
\end{equation}
which, again, is linear, and thus a linear fit of the form $y=Ax+B$ can be used. From the fit, $N=P/kA$, and $V_\text{chamber}=B/A$
After analysis of data, the following plots for $1/P$ vs. $V$, and $T$ vs. $V$ were obtained:
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| Figure 4: Obtained Data for Boyle's Law in the Experiment. The fit is given to be $1/P=0.0606339V+4.794459\times10^{-6}$. From the figure it is observed that an increase in pressure result in the decrease of volume, thus supporting Boyle's Law. |
From the plots, it is clear that both laws are exhibited by gases. In Figure 4, it is observed that under constant temperature, as the reciprocal of the pressure of the gas increases, that is, its pressure decreases, the volume of the gas increases. It is stated in Boyle's Law that when the gas is kept in constant temperature, then the pressure of the gas changes inversely with its volume. The temperature of the air chamber was kept at constant throughout the experiment since it was submerged in hot water, hence the trend shown in this figure coincides with the theoretical trend.
From these plots, the number of particles and volume of the chamber was obtained, and is presented as follows:
From here, it is seen more or less that the data is precise in terms of number of particles (one being approximately half of the other), and the magnitudes of the volume of the air chamber is not precise. The number of particles is expected to be precise as this is dependent mostly on constants, however, the magnitude of the air chamber is not. The volume of the air chamber is dependent on the slope and $y$-intercept of the graph itself. Hence the degree of error is much great compared to that for the number of particles. Most likely this is due to the inaccurate reading of the height of the piston of the mass lifter apparatus and temperature on the thermometer. It was hard to distinguish the bottom of the piston as it was black and the bottom was not flat, but rather conical, thus judging the actual bottom is difficult. Further, the temperature reading on the thermometer is always unstable, changing values far from each other to the point that averaging is useless as there are many uncertainties that are put into play. This could've affected the experimental data very much.
Generally, I am ok with this experiment. I am ok with this experiment because this is what we will be presenting by the end of the laboratory sessions, hence I must be able to grasp all relevant information now, to be prepared for the presentation.
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| Figure 5: Obtained Data for Charles' Law in the Experiment. The fit is given to be $T=4,570,603V-12.708333$. From the figure it is observed that an increase in temperature results in an increase in volume, thus supporting Charles' Law. |
In Figure 5, it is observed that in constant pressure, the temperature of the gas changes directly with its volume. Charles' Law states that in constant pressure, the temperature of the gas is directly proportional to the volume it encloses. Pressure was kept constant throughout this part of the experiment since the piston of the mass lifter apparatus was made sure to stay at its maximum height just before the experiment started. This shows that the experimental trend agrees with the theoretical trend.
From these plots, the number of particles and volume of the chamber was obtained, and is presented as follows:
| Number of Particles | $V_\text{chamber}$ (m$^3$) | |
|---|---|---|
| Boyle's Law | $3.2254684\times10^{21}$ | $7.9073779\times10^{-5}$ |
| Charle's Law | $1.6078207\times10^{21}$ | $2.7804497\times10^{-6}$ |
| Average | $2.4166445\times10^{21}$ | $4.0927115\times10^{-5}$ |
From here, it is seen more or less that the data is precise in terms of number of particles (one being approximately half of the other), and the magnitudes of the volume of the air chamber is not precise. The number of particles is expected to be precise as this is dependent mostly on constants, however, the magnitude of the air chamber is not. The volume of the air chamber is dependent on the slope and $y$-intercept of the graph itself. Hence the degree of error is much great compared to that for the number of particles. Most likely this is due to the inaccurate reading of the height of the piston of the mass lifter apparatus and temperature on the thermometer. It was hard to distinguish the bottom of the piston as it was black and the bottom was not flat, but rather conical, thus judging the actual bottom is difficult. Further, the temperature reading on the thermometer is always unstable, changing values far from each other to the point that averaging is useless as there are many uncertainties that are put into play. This could've affected the experimental data very much.
Generally, I am ok with this experiment. I am ok with this experiment because this is what we will be presenting by the end of the laboratory sessions, hence I must be able to grasp all relevant information now, to be prepared for the presentation.
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