Journal of National Taipei Teachers College, Vol. ⅩⅣ(Sep. 2001) 529~542
NATIONAL TAIPEI TEACHERS COLLEGE
529
A Connection of Ideal Gas Laws by Experiment
Tzyh-lee Chang, Pi-ling Chang and Herbert C. Cheung ∗
ABSTRACT
Generally, we need to perform different experiments in order to examine Boyle’s
law and the law of Charles and Gay-Lussac. In this work we proposed a method which
could allow us to not only investigate the preceding two laws, but also extend a
connection to all the other laws such as Avogadro’s law, Dalton’s law, combined law,
and the general form of ideal gas law. An analysis of the data obtained in this
experiment yields acceptable results. For example, the conversion constant between
the Celsius and Kelvin scales was found to be 1.15% less than the true value (=273.15),
and the universal gas constant was found to differ from the true value of 0.08206(Latm/K-mol) by 1.16%. If the idea of this experiment can be applied to design a more
feasible experiment for students to follow, students can not only learn all the ideal gas
laws, but also know how to apply them in practical situations.
Key words: Boyle’s law, the law of Charles and Gay-Lussac, Avogadro’s law,
Dalton’s law, combined law, the general form of ideal gas law.
∗
Tzyh-lee Chang: Associate Professor, Department of Natural Science Education, NTTC
Pi-ling Chang: Assistant Professor, Department of Nutrition Sciences, the University of
Alabama at Birmingham
Herbert C. Cheung: Professor, Department of Biochemistry and Molecular Genetics, the
University of Alabama at Birmingham
530 Journal of National Taipei Teachers College, Vol. ⅩⅣ
Journal of National Taipei Teachers College, Vol. ⅩⅣ(Sep. 2001) 529~542
NATIONAL TAIPEI TEACHERS COLLEGE
531
A Connection of Ideal Gas Laws by Experiment
Tzyh-lee Chang, Pi-ling Chang and Herbert C. Cheung ∗
I. INTRODUCTION
If one asks where the level of the water is inside a cylindrical bottle immersed
upside down in a tank of water, many students will simply reply that the level of the
water inside the bottle is the same as the water level of the tank. In fact, the water level
inside the bottle is below that of the water level of the tank. This simple demonstration
can motivate students to seek why there are differences in the water level. As a matter
of fact, it can be explained by the equilibrium of pressure which is reached between the
inside pressure of dry air plus vapor pressure of water in the upside down cylindrical
bottle and the external pressure of atmospheric pressure plus water pressure in the tank.
Furthermore, with a few measurements on the different depths of the water levels both
inside and outside the cylindrical bottle, one can quantitatively link the experimental
data to all the ideal gas laws. The experimental method presented is simple in theory,
safe and straightforward in practice.
II. THEORY OF IDEAL GAS LAWS
A sample of gas can be characterized by four variables: the volume V, the
temperature T, the pressure P, and the number of moles n. For an ideal gas, simple
∗
Tzyh-Lee Chang: Associate Professor, Department of Natural Science Education, NTTC
Pi-ling Chang: Assistant Professor, Department of Nutrition Sciences, the University of
Alabama at Birmingham
Herbert C. Cheung: Professor, Department of Biochemistry and Molecular Genetics, the
University of Alabama at Birmingham
532 Journal of National Taipei Teachers College, Vol. ⅩⅣ
relationships, known as the ideal gas laws, have been established concerning these
variables:
Boyle’s law
P1V1=P2V2
T and n constant
(1)
Avogadro’s law
V1/n1=V2/n2
P and T constant
(2)
Dalton’s law
P1/n1=P2/n2
V and T constant
(3)
The law of Charles
V1/T1=V2/T2
P and n constant
(4)
P1/T1=P2/T2
V and n constant
(5)
and Gay-Lussac
Combined law
P1V1/T1=P2V2/T2
n constant
These individual relationships imply a more general relationship: PV=nRT,
(6)
(7)
where R is known as the universal gas constant. In this experiment we can study all
these laws from the experimental data. In addition, we can also examine the quality of
data by comparing the derived conversion constant between the Celsius and the Kelvin
scale and the universal gas constant with well-known values of 273.15 and 0.08206Latm/K-mol, respectively.
III.EXPERIMENTAL SECTION
Mercury has commonly been used to study the Boyle’s law in general science labs
(Hermens, 1983; Hein, et al., 1992). However, these experiments must be performed
cautiously because spilled mercury can be very hazardous. To keep away from danger,
the use of hypodermic syringes was proposed instead (Davenport, 1962). Boyle’s law
was examined by taking the volume readings of a gas trapped in the syringe by piling
books onto the piston. In addition, Charles’ law could also be examined by taking the
volume readings of a gas in a syringe which was immersed in a liquid bath maintained
at various temperatures (Davenport, 1962). In our experiment we used water rather
than mercury. Thus our experiment has the same advantage of safety as the use of
hypodermic syringes. Furthermore, to the best of our knowledge our experiment has
been the first one reported so far that can lead all the ideal gas laws to be learned by
one experiment. In fact, the idea of our experiment was obtained from a previous
study about the relationship between candle flame and oxygen (Chang, 1999).
The equipment and materials used in our experiment include two thermometers (a
mercury thermometer and a digital thermometer, Sensortek model BAT-12), one
A Connection of Ideal Gas Laws by Experiment 533
barometer (Fisherbrand compact digital barometer), one measuring tape (or a ruler),
distilled water, two stands and two utility clamps, four graduated glass cylinders (2
2L, 1 1L, and 1 500mL), and a few volumetric flasks and pipettes. The experiments
were performed at two different temperatures. The first measurement was done in the
lab at 20.9°C and the second in the cold room at 4.8°C. These two temperatures
happened to be the lab temperature and the cold room temperature during the time of
data collection. It is purely for the purpose of simplicity to take the measurements at
20.9°C in the lab and 4.8°C in the cold room, respectively. Consistent results should
be obtained if data were taken at different lab and cold room temperatures.
Firstly, we filled the two 2L cylinders with distilled water till close to the brim,
then inserting a 1L cylinder and 500mL cylinder with their open ends down into the
2L cylinders, respectively.
It is important to hold the cylinder vertically when
inserting it into the water. Otherwise bubbles might incur suggesting air leakage and
thus contribute to the inaccuracy of data analysis. The data were taken by measuring
the distances between the open end of the upside down cylinder and the water levels
both outside and inside this cylinder. Four sets of data were obtained for each set-up
by inserting the cylinders into different depths under the water. We actually obtained
different depths by changing the amount of water using a pipette-aid filler/dispensers
rather than by moving the upside down cylinder to the desired positions.
In order to avoid the possible problem of equilibrium between water
vaporization and condensation, we waited for about one hour between each data taken.
After taking the data in the lab, we moved the whole set-up without removing the 1L
and 500mL cylinders from the 2L cylinders in which they were inserted to the cold
room and left them there for overnight. The procedures used in the cold room were
exactly the same as those used in the lab.
To monitor the temperature, the probe of the digital thermometer was inserted in
the middle of the upside down cylinder by using Scotch tape to fix its cable to the
wall of cylinder. The temperatures recorded directly from inside of the cylinder are
quite consistent with those recorded by using a mercury thermometer inserted in the
water surrounding the upside down cylinders. The temperatures varied within ±0.2°C
during the time of data collection in the lab and in the cold room.
534 Journal of National Taipei Teachers College, Vol. ⅩⅣ
IV. RESULTS
The data obtained from 1L cylinder and 500mL cylinder at lab temperature
(20.9°C) are shown in Tables 1 and 2, respectively. Similar data using the same two
cylinders at 4.8°C are also shown in Tables 3 and 4. The first two columns in these
four tables are experimental data, on which the calculated and measured data listed in
the last three columns are based. The values in the first columns are the distances (d)
between the open end of the upside down cylinder (1L or 500mL) and the water
surface outside the cylinder. The distances (w) between the open end and water
surface inside were also measured corresponding to each of the four different depths
shown in the first columns, and they are listed in the second columns in all four tables.
No bubbles were observed when inserting the cylinders into water thus indicating that
there is no gas leak. During the experiments, the atmospheric pressures remained as
995mb in the lab and 991mb in the cold room (both readings from the digital
barometer), respectively. We calibrated the digital barometer with a Fisher mercury
barometer.
A correction value of 5.331×10-3atm should be added to the above
readings.
Table 1 Data for 1L cylinder at 20.9°C and 995mb.
d / cm
w / cm
(d-w)÷1035.99 / atm
V/L
PV / atm-L
40.00
1.54
0.0371
1.1981
1.1982
35.00
1.35
0.0325
1.2035
1.1980
30.00
1.20
0.0278
1.2077
1.1965
25.00
1.04
0.0231
1.2125
1.1956
Table 2 Data for 500mL cylinder at 20.9°C and 995mb.
d / cm
w / cm
(d-w)÷1035.99 / atm
V/L
PV / atm-L
32.00
1.14
0.0298
0.61548
0.6110
28.00
1.02
0.0260
0.61770
0.6109
24.00
0.89
0.0223
0.62010
0.6109
20.00
0.77
0.0186
0.62230
0.6108
A Connection of Ideal Gas Laws by Experiment 535
Table 3 Data for 1L cylinder at 4.8°C and 991mb.
d / cm
w / cm
(d-w)÷1033.96 / atm
V/L
PV / atm-L
40.00
4.36
0.0345
1.1185
1.1290
30.00
3.95
0.0252
1.1305
1.1306
20.00
3.55
0.0159
1.1420
1.1315
10.00
3.14
0.00663
1.1536
1.1323
Table 4 Data for 500 mL cylinder at 4.8°C and 991mb.
d / cm
w / cm
(d-w)÷1033.96 / atm
V/L
PV / atm-L
32.00
3.48
0.0276
0.57502
0.5764
24.00
3.18
0.0201
0.57983
0.5769
16.00
2.88
0.0127
0.58493
0.5777
8.00
2.65
0.00517
0.58893
0.5772
The pressure exerted by the air trapped inside the upside down cylinder differs
from atmospheric pressure by an amount equal to the water pressure due to the
difference of water levels (=d-w).
This height should be added to atmospheric
pressure (both in atm) to give the total pressure on the gas sample. The height in atm
is shown on the third column in each of the four tables. The value of 1035.99 shown
in the headings of the third columns in Tables 1 and 2 and the value of 1033.96 in
Tables 3 and 4 allow for conversion of the heights in cm of water to atmospheres.
They are obtained from the following eq: 101,325Nt/m2 = ρ(kg/m3)×g(m/s2)× H(m),
where g = 9.80m/s2, and ρ is the density of water.
3
°
3
The values of ρ are
°
0.9980137g/cm at 20.9 C and 0.9999654g/cm at 4.8 C, respectively (Weast, 1986).
The fourth columns on each of the four tables represent the volumes of trapped
air inside the cylinders corresponding to different settings.
To more accurately
measure the volume of the trapped air we fill the cylinder with water using volumetric
flasks and pipettes to the heights corresponding to the readings listed in the second
columns of all four tables (i.e., the distances from the open end).
The total pressure is the sum of atmospheric pressure and the pressure due to the
difference of water levels (data in the third columns of all tables).
The partial
pressure P due to dry air is corrected by subtracting the vapor pressure of water from
536 Journal of National Taipei Teachers College, Vol. ⅩⅣ
the total pressure. The vapor pressures of water are 18.536mmHg at 20.9°C and
6.453mmHg at 4.8°C, respectively (Weast, 1986). The products of P and V are listed
in the last columns. Here we are only concerned about the number of moles of dry air
because it is assumed to be the same for all the four different settings, whereas the
number of moles of water vapor is different and thus the total number of moles of
gases is different. Note that the number of moles of dry air related to the data in
Tables 1 and 3 is the same, and so is that related to the data in Tables 2 and 4.
V. DISCUSSION
1. Boyle’s Law
The average values of the products PV are shown as follows:
1L cylinder at 20.9°C
°
500mL cylinder at 20.9 C
°
1L cylinder at 4.8 C
°
500mL cylinder at 4.8 C
<P1V1> = 1.1971±0.0012
(8)
<P2V2> = 0.6109±0.0001
(9)
<P3V3> = 1.1308±0.0014
(10)
<P4V4> = 0.5771±0.0005
(11)
The constancy of these PV products (Tables 1 to 4) proves the validity of eq (1)
and thus the validity of Boyle’s law.
However, since the pressure and volume
changes are not large, good results can be obtained under conditions of stable
temperature and atmospheric pressure only. The subscripts specify the cylinder and
the experimental conditions under study. It is convenient to use them in the following
discussion.
2. Avogadro’s Law
From the values of PV shown from eqs (8) to (11), we can see that PV = C′(T,n),
where C′(T,n), is a constant that depends on temperature and the number of moles of
dry air. Since the data in Tables 1 and 2 were obtained under the same temperature,
then from Boyle’s law, we know that under 1atm and 20.9°C, the volume of trapped
dry air is equal to the value of the product PV. Thus the ratio of the value in eq (8) to
that in eq (9) should reflect some relationship between the moles of dry air in 1L
cylinder and 500mL cylinder. The ratio is 1.9595. The same line of reasoning can be
used to find a ratio of 1.9596 for the same two cylinders under 1atm and 4.8°C. These
A Connection of Ideal Gas Laws by Experiment 537
two ratios are very close to the ratio of the volume of 1L to 500mL cylinders
(1247cm3/637cm3 = 1.96). From these close ratios, we can deduce that the volume of
dry air is proportional to its number of moles under conditions of constant
temperature and pressure. Actually, this is consistent with Avogadro’s law which
states that under conditions of constant temperature and pressure, equal volumes of
gases contain equal number of particles. Therefore, the dependence of C′(T,n) on the
number of moles of dry air can be written as C′(T,n) = nC(T), where C(T) is a
constant that depends only on temperature. So far the equation of state can be further
described as PV = nC(T), which satisfies both the laws of Boyle and Avogadro. The
relationship of P1V1/n1 = P2V2/n2 under constant temperature will be used next.
3. Dalton’s Law of Partial Pressure
We notice that Dalton’s law of partial pressure has already been assumed, since
we calculate the pressure of dry air by deducting the vapor pressure of water from the
total pressure. Dalton’s law of partial pressure states that under conditions of constant
volume and temperature, the number of moles of different molecules are responsible
for the partial pressure exerted by them. We can actually examine this by the same
method as above (however, now we assume the volume of trapped dry air to be 1L
rather than the pressure at 1atm as we did previously). Under conditions of the same
volume for trapped dry air (=1L), we have
<P1V1>/<P2V2> = n1/n2 = P1(assuming V1=1L)/P2(assuming V2=1L) = 1.1971/0.6109
= 1.9595
(12)
Similarly,
<P3V3>/<P4V4> = n3/n4 = P3(assuming V3=1L)/P4(assuming V4=1L) = 1.1308/0.5771
= 1.9596
(13)
Note that n1 = n3 and n2 = n4. If the gas molecules are assumed to behave ideally, then
the consistent results shown in eqs (12) and (13) may be evidence of the validity of
Dalton’s law.
538 Journal of National Taipei Teachers College, Vol. ⅩⅣ
4. The Law of Charles and Gay-Lussac
This law states that at constant pressure the volume of a given mass of gas is
directly proportional to its temperature on the Kelvin scale. By assuming that the
pressure is at 1atm,
<P1V1>/<P3V3> = V1/V3 = 1.1971/1.1308 = 1.0586 = (t1+α)/(t3+α),
(14)
where α is the conversion constant between the Celsius and Kelvin scales, and t1 and
t3 are the temperatures in degrees Celsius. The values of t1 (=t2) and t3(=t4) are 20.9°C
and 4.8°C, respectively. The value of α found is 270. Similarly,
<P2V2>/<P4V4> = V2/V4 = 0.6109/0.5771 = 1.0587 = (t2+α)/(t4+α).
(15)
Here the calculated value of α is also 270. These values differ from the known value
of 273.15 by 1.15%. The significance of α is that at -α(°C) the volume of dry air is
zero (i.e., no dry air is in the gas phase).
The law of Charles and Gay-Lussac also states that at constant volume the
pressure of a given mass of gas is directly proportional to its temperature on the
Kelvin scale. Assuming that the volume is fixed at 1L, we can obtain similar results
as above by a similar line of reasoning.
<P1V1>/<P3V3> = P1/P3 = 1.1971/1.1308 = 1.0586 = (t1+α)/(t3+α),
(16)
<P2V2>/<P4V4> = P2/P4 = 0.6109/0.5771 = 1.0587 = (t2+α)/(t4+α).
(17)
Obviously, the value of α obtained is the same as above and it means that at -α(°C)
the pressure of dry air goes to zero, as it should be because there is no dry air in the
gas phase.
5. Combined Law
From the law of Charles and Gay-Lussac, we can see that for a given mass of dry
air PV/T = constant. Thus we can use this equation to find volumes under STP (1atm,
0°C) conditions.
The V1(STP) is found to be 1.112L by solving the following
equation: <P1V1>/(273.15+20.9) = (1atm)V1(STP)/273.15.
Since under STP
conditions one mole of ideal gas occupies 22.41L, n1 = V1(STP)/22.41 = 4.961×10-2moles.
By solving similar equations, we find V2(STP) = 0.5675L, V3(STP) = 1.111L,
V4(STP) = 0.5671L, and the corresponding numbers of moles are n2 = 2.532×10-2moles,
n3 = 4.958×10-2moles, and n4 = 2.530×10-2 moles, respectively. The values of V1(STP)
A Connection of Ideal Gas Laws by Experiment 539
and V3(STP) or (n1 and n3) are close to each other, as they should be, and so are the
values of V2(STP) and V4(STP) or (n2 and n4).
6. A More General Form of Ideal Gas Law
For all the laws discussed above, we observe that they are all special cases of a
more general form of ideal gas law, which can be written as PV = nRT, where R is
the universal gas constant. The value of R can be found from the number of moles by
solving R = (<P1V1>– <P3V3>)/ [(n1/2+ n3/2)(t1–t3)] = (1.1971–1.1308) /[(4.9594×10-2)(20.9–
4.8)] = 0.08295 (L-atm/K-mol) for the case of 1L cylinder. A similar computation yields R =
0.08307 (L-atm/K-mol) for the case of 500mL cylinder. The average value differs from the
true value of 0.08206 (L-atm/K-mol) by 1.16%.
VI. CONCLUSION
Ideal gas laws are discussed in every basic physics and chemistry textbook. It is
fundamental to know them before a further understanding of gaseous behavior is
obtained. The experiment described here provides a straightforward way to reveal the
validity of Boyle’s law and the law of Charles and Gay-Lussac. Subsequently, by
applying these two laws, we are able to connect all the other ideal gas laws including
Avogadro’s law, Dalton’s law, combined law, and the general form of ideal gas law.
Although the experiment presented here can connect all the laws, we notice that the
change in the volumes of trapped air corresponding to different settings is not very
significant. Moreover, this experiment may not be feasible for most of the students
because there might not be a cold room available for them. It is our next goal to
design a more feasible experiment for improvements.
ACKNOWLEDGMENT
Tzyh-Lee Chang gratefully acknowledges the support from National Taipei
Teachers College during his sabbatical leave. He also appreciates Professor Cheung’s
cordial invitation to his laboratory at the University of Alabama in Birmingham.
540 Journal of National Taipei Teachers College, Vol. ⅩⅣ
REFERENCES
Chang, T. L. (1999). An extension of the relationship between candle flame and oxygen - a
quantitative study related to the contents of the learning activity 2 of unit 3 in elementaryschool natural science textbooks volume 9. J. NTTC, 12, 301-316. (in Chinese)
Davenport, D. A. (1962). Hypodermic syringes in quantitative elementary chemistry
experiments: I, the gas laws. J. Chem. Educ. 39(5), 252-255.
Hein., M., Best, L. R., Miner, R. L., & Ritchey, J. M. (1992). College chemistry in the
laboratory. 5th Ed., Pacific Grove: Brooks/Cole Publishing Co., p. 123-140.
Hermens, R. A. (1983). Boyle’s law experiment. J. Chem. Educ. 60, 764.
Weast, R. C., Ed. (1986). CRC handbook of chemistry and physics (67th Ed.). Boca Raton:
CRC Press, Inc., pp. D-190, F-4.
國立台北師範學院學報，第十四期（九十年九月）529～542
國立台北師範學院
541
一個連結所有理想氣體定律的實驗
張自立、Pi-Ling Chang 、Herbert C. Cheung ∗
摘
要
一般需要進行不同的實驗來驗證波義耳定律和查理–蓋、呂薩克定律。本文
中我們提出一個可驗證這兩定律又可延伸關連到所有其它定律如亞佛加厥定
律、道耳頓定律、組合定律和廣義理想氣體定律的新實驗。從實驗所得數據
中，我們得到不錯的結果，例如：分析得到的攝氏和凱氏溫標轉換值只與已知
值 (=273.15) 相 差 1.15% ； 另 外 ， 所 得 到 的 普 遍 氣 體 常 數 值 ， 也 只 與 已 知 值
0.08206(L-atm/K-mol)相差 1.16%。若本實驗的構想可被應用以設計一個對學生更
可行的實驗，則學生不僅能學到所有理想氣體定律，且能知曉如何實際應用這
些定律。
關鍵詞：波義耳定律、查理–蓋、呂薩克定律、亞佛加厥定律、道耳頓定律、組
合定律、廣義理想氣體定律。
∗
張自立：自然科學教育學系副教授
Pi-ling Chang:美國阿拉巴馬大學(伯明罕)營養科學系助理教授
Herbert C. Cheung:美國阿拉巴馬大學(伯明罕)生化和分子遺傳學系教授
542 國立台北師範學院學報，第十四期