Setup for Studying the Isothermal Process of an Ideal Gas
Abstract Generated abstract
The paper describes a laboratory apparatus for studying the isothermal behavior of an ideal gas and for measuring the work performed during an isothermal process. The setup uses a movable graduated burette connected by a flexible tube to an open mercury vessel, forming a manometer that allows simultaneous determination of gas volume and pressure as the burette is slowly displaced. The authors outline the experimental procedure and calculate work both by summing small pressure, volume intervals and by using the ideal-gas isothermal expression. A sample experiment with air shows approximate constancy of pressure times volume and gives work values differing by about 2.2 percent, illustrating the apparatus’s instructional accuracy.
Full Text
APPARATUS FOR STUDYING THE ISOTHERMAL PROCESS OF AN IDEAL GAS
G. I. FUKS, Yu. A. KOROLENKO
A significant feature of the apparatus is that it makes it possible to measure the work of an isothermal process. A schematic of the apparatus is shown in Fig. 1.
It consists of two glass vessels: burette 1 and open tube 2, connected to one another by flexible hose 3. Mercury is poured into the vessels.
Burette 1 has stopcock 4 at the top. The graduations of the burette make it possible to measure the volume of gas in it. With the aid of winch 5, burette 1 can be moved. Clamping screw 6 makes it possible to fasten the burette in any position. Tube 2 is stationary. With stopcock 4 closed, the system consisting of vessels 1 and 2 and tube 3 is a mercury U-shaped manometer measuring the pressure of the gas above the mercury level in burette 1.
At the beginning of the experiment, with stopcock 4 open, we lower burette 1 to such a position that the mercury displaces the air from it. Then we connect the burette to the vessel containing the gas under study and, by raising the burette with winch 5, draw into it the required quantity of gas. After this, we close stopcock 4.
Having lowered the burette to the lower position, we read off the initial gas volume \(V_1\) and the initial gas pressure
\[ p_1 = h_1 + B, \tag{1} \]
where \(h_1\) is the difference between the mercury levels in the burette and in the open tube, and \(B\) is the barometric pressure.
Fig. 1.
Raising burette 1 to some height, we cause the gas to expand and obtain a new state characterized by pressure \(p_2\) and volume \(V_2\). In this case, because the process is isothermal,
\[ p_1 V_1 = p_2 V_2. \tag{2} \]
If the expansion of the gas is small, then the work of expansion
\[ \Delta w=\int_{V_1}^{V_2} p\,dV \simeq p_{\mathrm{av}}(V_2-V_1), \tag{3} \]
where \(p_{\mathrm{av}}\)—the average pressure in the process—can be calculated with sufficient accuracy from the relation
\[ p_{\mathrm{av}}=\frac{p_1+p_2}{2}. \]
From several successive small displacements of burette 1 we compose the complete process. The work of the gas in it can be calculated as the sum of the works of the separate elementary processes, i.e.
\[ w=\int_{V_{\mathrm{init.}}}^{V_{\mathrm{fin.}}} p\,dV \simeq \sum p_{\mathrm{av}}\Delta V\ \text{kg·m}. \tag{4} \]
The process of change of state of the gas arising as a result of the displacement of vessel 1 is isothermal, since, for small slow displacements, the temperature of the gas has time to become equal to the temperature of the surroundings.
Since the process is isothermal, its work is
\[ w=\int_{V_{\mathrm{init.}}}^{V_{\mathrm{fin.}}} p\,dV = p_1 V_1 \ln \frac{V_{\mathrm{fin.}}}{V_{\mathrm{init.}}}\ \text{kg·m}. \tag{5} \]
Table 1
| \(10^6\cdot V\) | \(h=p_0-B\) | \(p\) | \(pV\) | Intervals of the process | \(\Delta V\cdot 10^6\) | \(p_{\mathrm{av}}\) | \(\Delta w\) |
|---|---|---|---|---|---|---|---|
| \(m^3\) | mm Hg | kg/m\(^2\) | \(m^3\) | kg/m\(^2\) | kg·m | ||
| 24,8 | −324 | 5720 | 0,142 | 1—2 | −2,1 | 5960 | −0,0125 |
| 22,7 | −288 | 6200 | 0,141 | 2—3 | −2,5 | 6635 | −0,0166 |
| 20,2 | −224 | 7070 | 0,143 | 3—4 | −1,6 | 7340 | −0,0118 |
| 18,6 | −185 | 7610 | 0,141 | 4—5 | −2,3 | 8090 | −0,0205 |
| 16,3 | −104 | 8570 | 0,140 | 5—6 | −2,2 | 9340 | −0,0210 |
| 14,1 | 0 | 10100 | 0,142 | 6—7 | −2,8 | 11310 | −0,0316 |
| 11,3 | +177 | 12520 | 0,142 | 7—8 | −0,6 | 12810 | −0,0077 |
| 10,7 | +220 | 13100 | 0,140 | 8—9 | −1,1 | 13850 | −0,0152 |
| 9,6 | +328 | 14590 | 0,140 | \(\sum \Delta w\) | −0,1369 kg·m | ||
| \((pV)_{\mathrm{av}}\) | 0,141 |
\[ w=p_1V_1\ln\frac{V_k}{V_1}=-0{,}134\ \text{kg·m}. \]
\[ \text{error } \Delta=\frac{w_p-w_t}{w_t}\cdot 100=2{,}2\%. \]
The results of one experiment are given in Table 1 and in Fig. 2. The working substance is air; atmospheric pressure \(B = 744\) mm Hg.
Fig. 2.
Comparison of the results according to (4) and (5) gives an idea of the accuracy of the experiment. The apparatus was made in the technical thermodynamics laboratory of the Department of Theoretical and General Heat Engineering of the Tomsk Polytechnic Institute.