Ideal Gas Law. Name, Date, Partner Lab Section. Date, Partner and Lab Section

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1 Ideal Gas Law Jane Doe Physics 16, Tuesday Section Title Title Partner: Michelle Smith Sept. 12, 2006 Introduction: Name, Date, Partner and Name, Lab Section Date, Partner and Lab Section The purpose of this experiment is to study the conditions under which air behaves as an ideal gas. Measurements of air pressure and temperature are made at fixed volume and fixed moles. The analysis of the measurements will show that air obeys the ideal gas law. From the analysis an experimental value for the universal gas constant R is obtained. Theory: The Ideal Gas Law Subheading Subheading Ideal gases are gases in which the particles of the gas only interact through elastic collisions; there are no intermolecular forces between the particles. Real gases can approximate ideal gases at conditions of low pressure and high temperature where the intermolecular forces are negligible. Under those conditions the equation of state for an ideal gas holds: Equation reference number PV =nrt Equation reference number (1) where the pressure (P) and volume (V) are directly proportional to the temperature (T). The constant R is the universal gas constant and n is the number of moles in the volume. If V and n Variables defined in text Variables are defined held fixed in textand a series of measurements of pressure and temperature are made, then Eq. (1) predicts a linear relationship between the pressure and temperature. It is therefore expected that a plot of P vs. T will be a straight line with slope proportional to R. Using a regression analysis of the plot, the slope can be found, and an experimental value of R determined.

2 To test the ideal gas law, we used an air-tight cylinder with a movable piston. Due to its own mass, the piston exerts pressure on the gas in the cylinder below it. Once the system is in equilibrium, the pressure exerted by the cylinder is equal to the pressure of the gas. The total pressure is: P TOT =P atm gm A (2) Here g is the acceleration of gravity, M is the mass of the piston, A is the cross sectional area of the piston and P atm is the atmospheric pressure. The volume of the gas is proportional to the height h of the piston above the floor of the cylinder: V =A h (3) The number of moles n is determined by the volume of gas at standard temperature and pressure (STP) of K and 1 atm. At STP one mole gas occupies of volume of 22.4 liters. We can use this to define a unit of molar density: STP = 1 mole 22.4 liters (4) Therefore the number of moles is: n= V (5) By substituting Eq. (2), (3) and (5) into Eq. (1), we find the equation of state in terms of the measured quantities M and T: M = AR g T P A atm g =at b Note: no long derivation. Just Note: reference no long equations derivation. and Just give reference result. equations detailed calculation and give result. in notebook. detailed calculation in notebook. (6) M can be changed by placing additional mass on the piston. In this experiment, M is adjusted to keep the height constant at each temperature T. A measurement of M vs. T at fixed height is

3 equivalent to a measurement of P vs. T at fixed V. The slope (a) and intercept (b) can be found from a regression analysis of M vs. T, allowing R to be determined. Experimental Technique: The Double-Walled Cylinder Figure with relevant parts Figure labeled with relevant parts labeled Figure number Figure and caption number and caption Figure 1: Double-walled cylinder and piston The experiment setup consist of a double-walled cylinder shown in Figure 1. An air-tight piston of (5.000 ± 0.001) kg and cross sectional area A = (78.54 ± 0.31) cm 2 holds the gas inside. The space between the inner and outer walls is filled with water to maintain the temperature of the gas in the cylinder. The cylinder is placed on top of an electric heating coil which heats the water. After the setting of the coil is adjusted, the water and the gas in the cylinder are allowed to reach thermal equilibrium before making further measurements. This typically takes about 15 minutes. The water bath temperature is then measured (Table 1) using a thermometer, with an estimated uncertainty of ±2 K. The apparatus was initially setup at STP. The water bath between the two cylindrical walls was filled with ice water. The valve on the top of the piston was opened to allow air to flow freely between inside and outside the cylinder. A support block was placed between support

4 stand and the top of the cylinder. This ensures that the only pressure on the gas inside the cylinder was the atmospheric pressure. Two thermometers were used to measure temperature: one of the ice water bath and the other the temperature of the air inside the cylinder. The second thermometer was narrow enough to be inserted in the open valve without blocking air flow through the valve. After sufficient time had pass, the air in the cylinder was at STP. The molar density of the air is then given by Eq. (4). The thermometer in the valve was removed and the valve was closed off. The support block was removed and the piston was allowed to slowly descend into the cylinder. The total pressure on the air was given by Eq. (2). Assuming the temperature of the air remains constant, the molar density of the air was calculated using: gas = STP P TOT P atm (7) We measured the initial height h of the piston to be ( )cm from the base of the cylinder (see Fig. 1). The heating coil was turned on and a series of measurements was made by varying the temperature of the water bath using the heating coil. As the water bath warmed, the air inside expanded and the piston rose. The air inside the cylinder was allowed to reach thermal equilibrium with the water bath. Mass was then added to the support stand until the piston compressed the air to the initial height of 18.9cm. We determined the uncertainty in the mass by applying masses of 1 gram increments to the support stand to see if there was an observable change in the height. We estimated uncertainty in the mass to be ± 5 grams. Data, Analysis and Results: Measurements were made at a total of eight different temperatures. Table 1 is a summary of the data taken.

5 Table 1: Measurements of Temperature and Mass Table number and Table Titlenumber and Title Temperature (±2 K) Mass (±0.005 kg) Note: table has less than 10 entries. Note: longer table tables, has less like than motion 10 entries. sensor data, in longer spreadsheet. tables, like motion sensor data, in spreadsheet Column heading with variables, Column units heading andwith uncertainty variables, units and uncertainty Figure 2 is a plot of mass vs. temperature. The plot demonstrates the linear relationship between the mass and the temperature as predicted by the ideal gas law. Title vertical (y) vs. horizonal (x) Title vertical (y) vs. horizonal (x) Axes scale divided into intervals Axes scale of 1,2,5 divided or 10 into intervals of 1,2,5 or 10 Axes labeled with variable Axes labeled and units with variable and units Figure number and Figure caption number and caption Figure 2: Mass vs. Temperature - solid line is best fit to data A regression analysis was performed to determine the best fit slope and intercept.

6 Table 2 contains the results of this analysis: Note: just present results of Note: regression just present analysis. results of regression analysis. summary output from Excel in summary notebook. output from Excel in notebook. Table 2: Regression Analysis Results Slope a Intercept b (0.304 ± 0.012) kg/k (-83.7± 4.0) kg Using the best fit value of the slope a and Eq. (6), the universal gas constant R can be determined. The result for R is: Results always presented with Results uncertainty. always presented detailed J uncertainty with uncertainty. analysis in notebook detailed R meas = 8.50±0.40 uncertainty analysis in notebook mole K The uncertainty in R meas does not include systematic effects, such as possible deviations from standard temperature and pressure while the apparatus was being prepared. The atmospheric pressure can also be calculated, using the best fit value of the intercept b: This agrees well with the anticipated value of about 1 atm. P atm = 1.03±0.05 atm (9) (8) Conclusion: Under the present experimental conditions, air behaves like an ideal gas. At fixed volume and fixed moles, measurements of pressure vs. temperature exhibited a linear relationship, consistent with the equation of state for an ideal gas. From the analysis of mass vs. temperature an experimental value of R meas = (8.50 ± 0.40) J/(mole K) was obtained. This experimental value is consistent, within uncertainty, with the accepted value of R = 8.31 J/(mole K). The largest source of uncertainty in R meas comes from the slope of the regression analysis. The uncertainty in the slope is 3.9%, compared to the uncertainty in A of 0.4%. This uncertainty arises from the spread of the data in Fig 2, rather than from a systematic deviation from linearity.