# IDEAL AND NON-IDEAL GASES

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1 2/2016 ideal gas 1/8 IDEAL AND NON-IDEAL GASES PURPOSE: To measure how the pressure of a low-density gas varies with temperature, to determine the absolute zero of temperature by making a linear fit to your data assuming the gas is an ideal gas, and to investigate the difference in behavior between an ideal gas and a non-ideal (van der Waals) gas. APPARATUS: Ideal gas apparatus, pressure gauge, thermometer, hot plate, stand. INTRODUCTION: The Ideal Gas The ideal gas law states that PV = nrt where T is the absolute temperature measured in Kelvins. This equation is also called the equation of state of an ideal gas. It is a good description of most gases in the low-density regime where, on average, the gas molecules are far apart. The purpose of the experiment is to determine experimentally how the pressure P of a gas varies with temperature when no change is made in either the volume V or the number of moles of gas n.. Using this information, you will determine To, the absolute zero temperature measured in degrees Centigrade, by linear extrapolation of the ideal gas equation to where the pressure P of the gas is zero (no molecular bombardment of the walls of the gas container). In this experiment you will measure T C, the temperature in degrees Centigrade, which you will recall is related to T (in Kelvins) by T = T C - To (To is a negative number). The graphical technique you will use is to plot T C (vertically) vs. P (horizontally). The equation for the straight line fitted to the data points will have the form of a linear equation T C a bp T o V nr P (1) For P = 0, the "y" intercept To (in degrees centigrade) can be read directly from the equation. A Non-Ideal Gas We noted above that the ideal gas law holds for low-density gases. In theoretically deriving the ideal gas law it is necessary to make two assumptions -- the gas molecules are very small (they have no volume) and the molecules are non-interacting (there is no force between them). In 1873 the Dutch physicist van der Waals derived an equation of state without these assumptions. It is known as the van der Waals equation of state: P a v b RT, (2) v 2 where a and b are constants chosen to agree with experiment, and v is the molar specific volume -- the volume of the container divided by the number of moles of gas inside. b is also a molar specific volume and represents the total volume per mole of gas that is inaccessible to other molecules because it is already occupied by a molecule -- you can t have two molecules at the same place at the same time. If the molecules in the gas have a

3 2/2016 ideal gas 3/8 Recall that the meaning of To is that there is a 68% probability that the true value of To lies in the range T o To. If R 2 = 1.000, take R 2 to be equal to The second method is to gather the values of To -- {Toi} -- collected by N of your classmates and calculate the average <To> and the standard deviation: 1 To N i (T oi T o ) 2. (5) The meaning of To is that if one of you do the same measurement one more time, there is a 68% probability that your result would fall in the range T o To. In addition there is a 68% probability that the true value of To falls in the range T o T o N.

5 2/2016 ideal gas 5/8 changes to improve the readability and appearance of your graph. Make copies for you and your partner. The linear fit procedure of OriginPro provides values of the fitting parameters together with their standard errors. Record the fitted values of To and its error (σ To ) calculated by OriginPro. Give the instructor your value after you calculate so that it can be given to the other groups. Note that this line is a huge extrapolation into a region where the equation for the straight line does not apply. A real gas will condense into a liquid and then freeze to a solid as the average thermal energy per molecule decreases with falling temperature. 3. For the remainder of this experiment you will use OriginPro to calculate how the van der Waals equation of state predicts the pressure of oxygen (air) will vary as temperature is decreased. See item 5 in introduction to Origin Pro.pdf for help on how to manipulate worksheet. First set up a worksheet in OriginPro as shown below to calculate the pressure as a function of temperature for an ideal gas and a van der Waals gas. Extend your calculation down to 0 K. For your report you will compare the predictions for the two gases starting with a very low density gas (where you would expect the two to agree) to those for a very dense gas (where the ideal gas law should break down).

6 2/2016 ideal gas 6/8 IDEAL GAS LAW Name: Section: Partners: Date: DATA: DO NOT print the table which you are using to make the plot. 1. Plot Tc (y axis) vs. P (x axis) using OriginPro. Attach a copy of the plot to your report. 2. Enter the results of linear fit procedure using OriginPro for absolute zero on the Celsius scale: To(experiment): ± % 3. Evaluate the percent deviation from accepted value of o C, (T o(experimental) ) % 4. Record the absolute zero temperatures of the other groups below. 5. Evaluate the error σ <To> in T o using Eq. (5). σ <To> = Is the meaning of σ <To> consistent with the values of T o shown in 4 above? Explain. Is your determination of T 0 within the error σ <To>?

7 2/2016 ideal gas 7/8 THE VAN DER WAALS GAS Set up the worksheet shown above in the section on procedure to compare the ideal and van der Waals gases. You will study how the temperature dependence of the two gases changes as you increase their density in your calculation by adding more molecules to the metal sphere on the ideal gas apparatus. The variable that reflects this increase in density is the molar specific volume, v = V/n. Begin by inserting the value of v for standard temperature and pressure that you calculated in the preliminary question. For questions 2-5, extrapolate from T= 273 K and 363 K to zero pressure to find absolute zero. 1. For v = m 3 /kmole (STP), what is the maximum percent difference between pressures of the ideal and van der Waals gases has over the temperature range you used experimentally in the first part of this lab? 2. Increase the density in your calculation to ten times STP. What are the new pressure at T = 273 K and the extrapolated absolute zero temperature (at P = 0)? Solve the linear fit equation in the form, P = P 0 + st, for T abs-zero with P = 0. P (at T = 273 K) = T abs-zero = K 3. With this increased density, make a ten times larger. What are the new pressure and the extrapolated absolute zero temperature? P (at T = 273 K) = T abs-zero = K 4. Return a to its original value and make b ten times larger. What are the new pressure and the extrapolated absolute zero temperature? P (at T = 273 K) = T abs-zero = K 5. How much of the error that you found in your experimental value for absolute zero can be attributed to our assumption that air is an ideal gas? Use the calculated pressure values and extrapolate to zero pressure to find the calculated temperature and compare to zero.

8 2/2016 ideal gas 8/8 LAB REPORT Discuss, in a typed short paragraph, the systematic uncertainties in the experiment. Make quantitative estimates for reasonable variations. For example, assume the thermometer calibration is incorrect, so that 1.00 C actually corresponds to 1.01 C and calculate the error on absolute zero. For questions 2-4 in the van der Waals gas, discuss why the pressure and temperature changed as they did, again in a typed paragraph. For question 5 comment on whether the non-idealness of air was a significant error in your experiment.

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