tp03: The Ideal Gas Law Jack Lyes 18/01/2014 The main objective of this experiment was to calculate a value for absolute zero, the temperature at which a gas exerts zero pressure. This was achieved by measuring the temperature and pressure of a gas as it was heated, and extrapolating the linear fit of the data in order to give a temperature value associated with zero pressure. Absolute zero was calculated to be 262 C, with an uncertainty of ±13 C. This is consistent with the given value of absolute zero, 273 C. 1 Introduction The ideal gas law combines Charles Law and Boyle s Law, and relates the temperature and pressure of an ideal gas. Although an ideal gas is a theoretical concept, the law can be used to approximate the behaviour of gases under certain conditions. The law states: T = V P, (1) nr where T is temperature, P is pressure, V is volume, n is the number of moles of gas, and R is the universal gas constant. This law requires certain assumptions in order for a gas to be considered ideal : Gas molecules move in constant linear random motion. Gas molecules behave as rigid spheres. Pressure is due to collisions between the molecules and the container walls. All collisions are elastic. The temperature of the gas is proportional to the average kinetic energy of the molecules. There are no intermolecular forces between molecules except when they collide. The volume occupied by the molecules themselves is negligible relative to the volume of the container. The law shows that the absolute pressure of an ideal gas is directly proportional to the absolute temperature, for a fixed mass of gas. The aim of this experiment was to calculate the value, in degrees celsius, of absolute zero; that is, the temperature at which a gas exerts zero pressure. This was done by measuring the linear relationship of temperature and pressure as a gas cooled. 2 Experimental method There were two main parts to the experiment. The first involved measuring the temperature and pressure of a gas as it cooled. The second part involved varying the gas density, and measuring the pressure 1
of the gas at certain temperatures. The following is a list of the apparatus used: A temperature and a pressure sensor, mounted into a hollow copper sphere, containing a fixed mass of air. A clamp and clamp stand. A water bath. A thermometer A heating pad. A computer with DataStudio[1] software to record and plot temperature and pressure data. 3. In DataStudio, open a temperature vs. time and a pressure vs. time graph, to plot simultaneously. 4. Place the sphere in the water bath, ensuring that it is completely submerged, held by a clamp. 5. On the computer, click start to start recording data. 6. When the digits on-screen stop changing, note the temperature and pressure in a table. 7. The water bath must then be cooled, which can be done by adding ice cubes, or cold water. Wait until the temperature drops by around 10 C and then repeat the previous step. 8. Repeat the previous steps, decreasing the temperature until the water cannot be cooled further. Stop recording data. 9. With the recorded data, plot a temperature vs. pressure graph for the gas, applying a linear fit. 10. Repeat the above process. Part One In order to calculate absolute zero, the pressure and temperature of a gas must be measured as it cools. This linear relationship can be plotted, and the graph extrapolated to find the temperature at which pressure is zero. 1. Using the heating pad, heat a bath of water to the range 50 80 C, ensuring that it is turned off after the water is heated. 2. Ensure that the hose fitting and stereo plug are connected to the pressure and temperature sensors respectively, connect the sensors to the computer, and open DataStudio, including digits display. Part Two In part two of the experiment, the density of the gas in the sphere is varied. This is done by connecting the hose only after the sphere has been heated or cooled to a certain temperature, and then keeping the hose connected. Data is recorded, the hose is disconnected, the sphere s temperature is then altered and the process is repeated, so that measurements can be taken at a range of densities of gas. This works on the theory that for a lower temperature of gas in a volume, there are more moles of gas, and therefore a higher density. 1. Prepare three water baths, one hot, one at room temperature, and one cold. The temperature of the hot bath should be within 50 80 C, and the cold bath within 0 10 C. 2. Connect the temperature sensor, leaving the hose fitting disconnected. 2
3. Open DataStudio, again opening digit display, a temperature vs. time graph and a pressure vs. time graph 4. Submerge the sphere in the cold water bath, connecting the hose fitting to the pressure sensor. 5. Click start to start recording data, and note the values of temperature and pressure when the temperature stops changing. cases, the uncertainties were in the range ±0.01 0.40 units, thus making the uncertainties of ±0.005 units negligible. 3 Results 6. Without disconnecting the hose, submerge the sphere in the room temperature water bath, and repeat the previous two steps. 7. Again without disconnecting the hose, submerge the sphere in the hot water bath, repeating the previous step. 8. Stop recording data, then plot a graph of temperature vs. pressure, applying a linear fit. 9. Repeat the above procedure twice more, once starting with the room temperature water bath, and once starting with the hot water bath, plotting the data as temperature vs. pressure graphs. When carrying out this experiment, it is important to remember that water in the region 50 80 C can be particularly dangerous, as it may burn the skin. More importantly, the heating pad reaches temperatures in excess of 200 C, which can cause serious burns. An extra level of caution is required when handling this equipment. Apart from this, there are no other considerable safety hazards associated with the experiment. In the experiment, there is an inherent uncertainty of ±0.005 units associated with each measurement. This is due to the precision of the sensor equipment. Other uncertainties [see result tables] resulted from inaccuracies in reading from the temperature vs. time and pressure vs. time graphs, i.e. fluctuations arose when the temperature was expected to remain constant. This led to the averages of several peaks and troughs being taken, and the associated uncertainty noted, which is calculated as half of the average distance from a peak to a trough. In these Figure 1: Part One: Temperature vs. Pressure as gas cools Figure 2: Part One: Temperature vs. Pressure as gas cools (Repeat) Once the graphs have been plotted [see results tables for data], they should show an approximate lin- 3
ear relationship between pressure and temperature [Figure 1. and Figure 2]. The linear fit of the points can be extrapolated to give the value of absolute zero; absolute zero is represented by the temperature associated with the x-intercept of the graph. The x- intercept of the graph in Figure 1. is at 273.0 C, and at 251.2 C. By computing the uncertainties associated with each data value, the uncertainty in the gradient can be calculated, which can in turn be used to calculate the uncertainty in the x-intercept, and hence the uncertainty in the estimated value for absolute zero. The first graph gives a value of 273±2 C and the second a value of 251 ± 13 C From varying the density of the gas in the second part of the experiment, three temperature vs. pressure graphs were produced [Figures 3,4,5.] using a linear fit of the data, and then extrapolated. Each line has a different gradient, yet their x-intercepts are approximately equal. They were calculated using the same method as in part one to be 268 ± 8 C, 260 ± 31 C, and 260 ± 9 C respectively. 4 Discussion Taking all of the estimated values for absolute zero and calculating the mean gives a value of 262 ± 13 C. This result is just over 10 C out from the current given value of absolute zero, which is 273.0 C. However, the given value of absolute zero does lie within the bounds of the uncertainty. In part two, the density of the gas in the sphere was varied. This means that the number of moles of gas in the sphere was not constant throughout the experiment, and changed, depending on the initial temperature of the sphere. From the relation T = V nr P, and by comparing it to the equation P = mt of the plotted lines, one can deduce that the gradient, m, is equal to nr V, where V is the volume of the sphere, n is the number of moles of gas, and R is the universal gas constant. By equating the gradient of each line from part two to nr V, the number of moles of gas present in the sphere for each of the repeats, and their associated uncertainties, could be calculated. This gave (3.3±0.1)x10 5 mol for the sphere which was initially cold, (3.1±0.3)x10 5 mol for the sphere which was initally at room temperature, and (2.6 ± 0.1)x10 5 mol for the sphere which was initially hot. This is consistent with the theory that for a fixed volume of gas, a lower temperature will result in a higher density and a higher temperature will result in a lower density, as the number of moles of a gas present is directly linked to the gas mass. 5 Conclusions The aim of this experiment was to calculate a value for absolute zero, and to compare the calculated value with that of the standard given value. The experiment was repeated in order to reduce uncertainty, and a value was produced that is consistent with the given value of absolute zero. The secondary aim was to determine the relationship between the number of moles of a gas and the gas temperature. The calculated values were consistent with the theory, however the uncertainty in the measurements was too high to confirm the theory. More repeats would be necessary in order to reduce uncertainties and produce more accurate results. References [1] DataStudio c 1996-2014. PASCO Scientific 4
Figure 3: Starting with cold water bath Figure 4: Starting with room temperature water bath Figure 5: Starting with hot water bath 5