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

2 2/2016 ideal gas 2/8 radius r, then when two collide the centers can only come with a distance 2r of each other. Thus if we want to consider the molecules to have no volume, we must subtract from the volume of the container the volume of a sphere of exclusion of radius 2r around each molecule, b 1 2 N o 4 3 2r 3, (3) where No is Avogadro s number, the number of molecules in a mole. The 1/2 occurs because only the hemisphere facing the colliding molecule is effective in excluding it. The term a/v 2 that is added to the pressure of the gas in the container arises from a very short range attractive force between molecules known as the van der Waals force. When a molecule is not near the wall of the container, it is surrounded by molecules and, on average, this force cancels out. But near the wall the molecule experiences a net force pulling it away from the wall, which reduces the force (pressure) it exerts on the wall when it collides. This is the force that makes (some) gases liquefy at low temperature. The values for a and b are determined experimentally for each gas. For oxygen a = 1.38 x 10 5 P m 6 /(kmole) 2 and b = m 3 /(kmole). [P Pascal = N/m 2, where N is Newton.] For the purposes of your calculations remember that R is = 8314 J/(kmol).K. Avogadro s number is 6.02 x molecules per mole and the volume occupied by one kmole of a gas at standard temperature and pressure (0 o C and 760 torr = 1 atm = 1.01 x 10 5 P) is 22.4 m 3. [Note: the mole is defined to be the quantity of material whose mass in grams is equal to the molecular weight of the material. The proper SI unit is not the mole, but rather the kmole = 1000 moles. The mass of a kmole in kilograms equals the molecular weight of the material, the number of molecules in the kmole is 6.02 x and the standard volume is 22.4 m 3.] Eq. (2) gives a fairly good description of a real gas. Other equations of state such as the Beattie-Bridgman equation (which uses five adjustable parameters instead of two) give even better agreement with experiment. Error Analysis: Evaluating the least square fitting error. Your measurement of absolute zero isn t of much use unless you can estimate the error in your measurement. In this section we will discuss two ways to estimate the error. The first involves using just your data set, while the last involves combining the data for the various groups in your class. 1. This method is to use the correlation coefficient R 2 that OriginPro (see links to Introduction to OriginPro and Example project file in OriginPro ) calculates to find the error. We will leave an explanation of why this method (and the next) works to a more advanced course. If you fit a set of N data points{pi}to a straight line, Tc= To + mp, where m is the slope, then the error (standard deviation) in To is T o m N P i tan(cos 1 R). (4) N

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.

4 2/2016 ideal gas 4/8 PROCEDURE: SPECIAL PRECAUTIONS The hot plates are not waterproof. Do not spill water on them. Be careful of the hot water and hot bucket. Do not tamper with the valve on the bulb holding the gas. Do not allow the black plastic on the thermometer to touch water. Note: the pressure gauge reads in units of mm Hg (torr). The ideal gas equation uses SI units. To convert the pressure to SI units, use 1 torr = N/m 2 = P. 1. This portion of the experiment takes a lot of time, much of which is waiting. While waiting for the water to heat start setting up the spreadsheet described in step 3. Assemble the apparatus on the ring stand. (See the figure below.) Make sure that the bulb does not touch the bottom or sides of the can. Fill the bucket with ice water. Make sure the water is at least 4 cm over the top of the bulb. Don t add a lot of ice; it ll take longer for your water to heat if you add too much. The temperature of the bath is raised by turning on the hot plate to the highest temperature setting. Record in parallel columns the absolute pressure (mm Hg = torr) and temperature ( o C). To collect the data open Logger Pro, click on Data Collection (a small clock icon) and set duration to 90 minutes (more than expected time needed to complete data collection) and sampling rate to 6 samples/minute. After the data is collected, you can export it as an Excel file. Now import this file to OriginPro (see introduction to OriginPro.pdf. Also make sure to delete the column labels before importing the excel file). There may be a temperature gradient between the bottom and top surfaces, and the thermometer might not read exactly the bulb gas temperature. So, cover the bath top while taking readings. When you get to about 88 C, shut the hot plate off and allow it to cool. Don't let the water boil. Be careful not to touch or spill the hot bucket. 2. Using OriginPro, plot the temperature T (in C) on the y-axis as a function of the pressure P (torr = mm Hg) on the x-axis. Use OriginPro to fit your data to a linear equation. On your graph, you may need to change the range of the axes so that you can see where the y-intercept lies (x = P = 0). Add a title to your graph and make other style

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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