$\require{mhchem}$ Dumas Method === ## Introduction The Dumas method is a simple and direct way to determine the molecular weight of a volatile organic liquid. By itself, the Dumas method is not highly accurate and is used only to obtain approximate results. However, the Dumas method may be used in combination with an empirical formula, obtained by making an elemental analysis of the compound, to yield a molecular formula and an accurate formula mass for the vapor of the organic liquid. The method is based on the ideal gas law. $$ PV = nRT \hspace{10mm} \text{Eqn. 1} $$ The ideal gas law relates the pressure of a gas ($P$), the volume of the gas ($V$), the number of moles of the gas ($n$), and the temperature of a gas ($T$). The variable $R$ is the Gas Law Constant and has a value of $0.0821 \, \frac{\textsf{L} \cdot \textsf{atm}}{\textsf{K} \cdot \textsf{mol}}$ . It's important to note that no real gas is perfectly ideal, including the organic vapors we will be working with in this lab. However, when the pressure of a gas is low and the temperature of the gas high, the ideal gas law is a reasonable approximation for describing the state of a gas. The Dumas method is performed by placing a volatile (easily evaporated) organic liquid in a vented flask. It is important to determine the mass and internal volume before adding the organic liquid. The flask is then heated gently in a water bath until all of the liquid has been fully converted to vapor. The excess organic vapor will force out the other gasses present in the flask while filling the container. After a short period of time the organic vapor will reach equilibrium with the atmosphere and the rate of vapor escaping the flask will slow down significantly. The pressure of the vapor is now roughly equal to the atmospheric pressure. $$ P_\textsf{vapor} \, = P_\textsf{atm} \hspace{10mm} \text{Eqn. 2} $$ The volume the vapor occupies is equal to the internal volume of the flask. The internal volume of the flask is not the same as the value printed on the flask and varies slightly between individual flasks of the same type. $$ V_\textsf{vapor} \, = V_\textsf{flask} \hspace{10mm} \text{Eqn. 3} $$ The temperature of the organic vapor can be measured directly. Slow careful heating is required in order to get an accurate temperature as the vapor will continue to get hotter and hotter until nearly no vapor remains in the flask. Once the escaping gas comes to equilibrium and the temperature is measured, the flask is loosely sealed and allowed to cool until the remaining vapor condenses back to a liquid. The flask and remaining organic liquid is then weighed. The mass of the remaining liquid ($m$) is determined by subtracting the mass of the flask. Since we now know the pressure, volume, and temperature of the gas, we can calculate the number of moles of organic liquid left in the flask. $$ n = \frac{PV}{RT} \hspace{10mm} \text{Eqn. 4} $$ The formula mass ($M^w\,$) of the organic liquid can be estimated by dividing the mass of the remaining liquid by the number of moles of liquid. $$ M^w = \frac{m}{n} $$ ### Sample Problem 1 A volatile organic liquid is placed in a vented flask in a water bath. The flask is heated until all of the organic liquid is volatilized. Use the measurements recorded in the following table to determine the formula mass of the liquid. | $P_\textsf{atm}$ | $V_\textsf{flask}$ | $T$ | $m$ | | :--------: | :------: | :-----: | :----: | | 746.2 torr | 218 ml | 99.5 °C | 989 mg | Our first step is to convert our pressure, volume, and temperature values to be consistent with the units of the gas law constant. $$ 746.2 \, \textsf{torr} \left( \frac{1 \, \textsf{atm}}{760 \, \textsf{torr}} \right) = 0.9818 \, \textsf{atm} $$ $$ 218 \, \textsf{ml} \left( \frac{1 \, \textsf{L}}{1000 \, \textsf{ml}} \right) = 0.218 \, \textsf{L} $$ $$ 99.5 + 273.15 = 372.7 \, \textsf{K} $$ We can now use the ideal gas law to calculate the number of moles of liquid. $$ n = \frac{(0.9818 \, \textsf{atm})(0.218 \, \textsf{L})}{\left(0.0821 \frac{ \textsf{L} \cdot \textsf{atm}}{\textsf{mol} \cdot \textsf{K}} \hspace{1mm} \right) ( 372.7 \, \textsf{K} )} = 0.00699 \, \textsf{moles} $$ Finally, we can calculate the formula mass of the liquid. $$ M^w = \left( \frac{989 \, \textsf{mg}}{0.00699 \, \textsf{moles}} \right)\left( \frac{1 \, \textsf{g}}{1000 \, \textsf{mg}} \right) = 141 \frac{\textsf{g}}{\textsf{mol}} $$ ### Sample Problem 2 This example illustrates how to use an elemental analysis in combination with a Dumas determination, to improve the accuracy of both. The elemental analysis of the same organic compound used in the Sample Problem 1 shows it to be composed of: 17.8 mass% C, 2.18 mass% H, and 78.5 mass% Cl. Determine the empirical formula for the compound from the elemental analysis. Use the results of this analysis, in combination with the answer to Sample Problem 1, to determine the correct molar mass and molecular formula for the organic liquid. To determine the empirical formula of the compound from the elemental analysis, it is convenient to assume that we have 100 g of sample. We can then calculate the mass of each element in our sample. $$ 100 \, \textsf{g} \left( \frac{17.8 \, \textsf{g} \, \ce{C}}{100 \, \textsf{g}} \right) = 17.8 \, \textsf{g} \, \ce{C} $$ $$ 100 \, \textsf{g} \left( \frac{2.18 \, \textsf{g} \, \ce{H}}{100 \, \textsf{g}} \right) = 2.18 \, \textsf{g} \, \ce{H} $$ $$ 100 \, \textsf{g} \left( \frac{78.5 \, \textsf{g} \, \ce{Cl}}{100 \, \textsf{g}} \right) = 78.5 \, \textsf{g} \, \ce{Cl} $$ Next, we can calculate the number of moles of each element. $$ 17.8 \, \textsf{g} \, \ce{C} \left( \frac{1 \, \textsf{mole} \, \ce{C}}{12.01 \, \textsf{g} \, \ce{C}} \right) = 1.48 \, \textsf{mole} \, \ce{C} $$ $$ 2.18 \, \textsf{g} \, \ce{H} \left( \frac{1 \, \textsf{mole} \, \ce{H}}{1.008 \, \textsf{g} \, \ce{H}} \right) = 2.16 \, \textsf{mole} \, \ce{H} $$ $$ 78.5 \, \textsf{g} \, \ce{Cl} \left( \frac{1 \, \textsf{mole} \, \ce{Cl}}{35.45 \, \textsf{g} \, \ce{Cl}} \right) = 2.21 \, \textsf{mole} \, \ce{Cl} $$ We must now determine a multiple that we can multiply by each molar quantity that will yield a whole value. This can be done by inspection, it is usually easiest to start by dividing each value by the smallest among them to yield one element with a value of 1 mole. $$ \frac{1.48 \, \textsf{mole}}{1.48 \, \textsf{mole}} = 1.00 \, \textsf{mol} \, \ce{C} $$ $$ \frac{2.16 \, \textsf{mole}}{1.48 \, \textsf{mole}} = 1.46 \, \textsf{mol} \, \ce{H} $$ $$ \frac{2.21 \, \textsf{mole}}{1.48 \, \textsf{mole}} = 1.49 \, \textsf{mol} \, \ce{Cl} $$ Inspecting the result, it is pretty clear that multiplying each of these values by two will result in values very close to whole values. $$ 1 \, \textsf{mol} \, \ce{C} \times 2 = 2 \, \textsf{mol} \, \ce{C} $$ $$ 1.46 \, \textsf{mol} \, \ce{H} \times 2 = 2.92 \, \textsf{mol} \, \ce{H} $$ $$ 1.49 \, \textsf{mol} \, \ce{Cl} \times 2 = 2.98 \, \textsf{mol} \, \ce{Cl} $$ It is not at all surprising that we cannot achieve perfectly wholenumbers. This is due to error in our initial measurements. We will therefore round these numbers yielding a 2 mole C: 3 mole H: 3 mole Cl ratio. This corresponds to an empirical formula of $\ce{C2H3Cl3}$. Remember that this is an *empirical formula*. The *molecular formula* for this compound could be any multiple of this configuration. The information gathered by performing the Dumas method will allow us to decide the true molecular formula. Consider the formula masses for various possibilities in Table 1. | Formula | Formula Mass | | :--: | :---: | | $\ce{C2H3Cl3}$ | $133.5 \, \frac{\textsf{g}}{\textsf{mol}}$ | | $\ce{C4H6Cl6}$ | $267.0 \, \frac{\textsf{g}}{\textsf{mol}}$ | | $\ce{C6H9Cl9}$ | $400.5 \, \frac{\textsf{g}}{\textsf{mol}}$ | | $\ce{C8H12Cl12}$ | $534.0 \, \frac{\textsf{g}}{\textsf{mol}}$ | **Table 1:** Formula Mass of various possible configurations of empirical formula $\ce{C2H3Cl3}$ The Dumas method suggested our organic liquid has a formula mass of $141 \frac{\textsf{g}}{\textsf{mol}}$. This is closest to 133.5 $\ce{\frac{g}{mol}}$. Therefore, we can conclude that the true molecular formula for our unknown liquid is $\ce{C2H3Cl3}$ :::info 👉️ Don't be surprised that this does not add up to exactly 100 mass %. Experimental measurements can be "perfect" only by accident. Some experimental error is expected. ::: :::info 👉️ Note that we will assume that the compound is only comprised of carbon, hydrogen, and chlorine. ::: ## Materials - 125-ml Erlenmeyer flask - Copper Wire - Tin foil square - Hot plate - Ring stand - Clamp - Tongs - Ice bath ## Procedure :::danger 🔥➡️🧯Organic liquids are generally very flammable. Organic vapors even more so. It is important to be very careful when applying heat to these compounds. All sources of ignition (fire, electrical outlets. sparks, etc.) should be kept well away from your set up. Your flasks should only be heated in the water bath. ::: ### Part One: Determining Pressure, Temperature, and Mass 1. Measure 1. Obtain a square of aluminum foil, a length of soft copper wire, and a clean, dry 125 mL Erlenmeyer flask. The aluminum foil and wire will be used to close the flask opening. 3. Place the aluminum foil on a piece of cardboard or pad of paper and pierce a tiny hole through the center with a needle. Only the very tip of the pin should penetrate the foil. 4. Weigh the flask alone and record the mass on your data sheet. 5. Weigh the flask, aluminum foil, and copper wire all together. Do not fold the foil or wire. Record the weight on the Data Sheet. 6. Using a graduated cylinder, obtain about 5 mL of unknown sample. 7. Write down the elemental analysis of your unknown. 9. Add your sample into the Erlenmeyer flask. Place the aluminum foil over the flask mouth, centering the tiny hole. Fold down the sides to cap the flask opening. Wrap the copper wire over the foil around the neck of the flask and twist the ends to secure it. 9. Place the flask into the water bath. 10. Fill a test tube with a small amount of ice and water. Wipe any excess water from the outside of the tube. Hold the tube over the pinhole of your flask. The escaping vapors will condense on the outside of the tube and aid in visualizing the escaping vapors. 11. Once all of the liquid has boils away and the rate of escaping vapors has slowed, turn off your hotplate and allow one additional minute for the temperature of the water bath and flask to equilibriate. 12. Note the temperature of the water bath with your thermometer and remove the flask from the water bath. Allow the flask to cool briefly on a ceramic tile. 13. Place your flask into an ice bath. The remaining organic vapor will condense inside the flask. While the flask cools check the barometric pressure inside the room. 14. After ten minutes of cooling, all of your organic liquid should have condensed inside the flask. Remove the flask from the water bath and wipe any excess water from the outside of the flask. 15. Take your flask and liquid to the balance and determine its mass. 16. Dispose of your organic liquid in the appropriate waster container. **DO NOT POUR IT DOWN THE DRAIN.** ### Part Two: Determining Volume 1. Clean and dry your flask. Take the flask to the balance and record its mass. 2. Fill the flask carefully with water. 3. Place the flask on the balance and use a wash bottle to top off the flask right to the brim 4. Record the mass of the water and flask. **DO NOT OVERLOAD THE BALANCE BY EXCEEDING ITS UPPER LIMIT.** :::danger :boom: You must use the larger, open-topped balances to make this measurement. The filled flask will be too heavy for the typical lab balances. ::: 6. Measure the temperature of the water inside the flask and use the density table below to determine the volume of the flask ## Data Sheet {%pdf https://public.chemnotes.org/lab/sheets/dumasmethod.pdf %} ## Water Density | $\Large\frac{\textsf{g}}{\textsf{ml}}$ | 0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | | :-----------------: | :------: | :------: | :------: | :------: | :------: | :------: | :------: | :------: | :------: | :------: | | 15 | 0.999099 | 0.999084 | 0.999069 | 0.999054 | 0.999038 | 0.999023 | 0.999007 | 0.998991 | 0.998975 | 0.998959 | | 16 | 0.998943 | 0.998926 | 0.998910 | 0.998893 | 0.998877 | 0.998860 | 0.998843 | 0.998826 | 0.998809 | 0.998792 | | 17 | 0.998774 | 0.998757 | 0.998739 | 0.998722 | 0.998704 | 0.998686 | 0.998668 | 0.998650 | 0.998632 | 0.998613 | | 18 | 0.998595 | 0.998576 | 0.998558 | 0.998539 | 0.998520 | 0.998501 | 0.998482 | 0.998463 | 0.998444 | 0.998424 | | 19 | 0.998405 | 0.998385 | 0.998365 | 0.998345 | 0.998325 | 0.998305 | 0.998285 | 0.998265 | 0.998244 | 0.998224 | | 20 | 0.998203 | 0.998183 | 0.998162 | 0.998141 | 0.998120 | 0.998099 | 0.998078 | 0.998056 | 0.998035 | 0.998013 | | 21 | 0.997992 | 0.997970 | 0.997948 | 0.997926 | 0.997904 | 0.997882 | 0.997860 | 0.997837 | 0.997815 | 0.997792 | | 22 | 0.997770 | 0.997747 | 0.997724 | 0.997701 | 0.997678 | 0.997655 | 0.997632 | 0.997608 | 0.997585 | 0.997561 | | 23 | 0.997538 | 0.997514 | 0.997490 | 0.997466 | 0.997442 | 0.997418 | 0.997394 | 0.997369 | 0.997345 | 0.997320 | | 24 | 0.997296 | 0.997271 | 0.997246 | 0.997221 | 0.997196 | 0.997171 | 0.997146 | 0.997120 | 0.997095 | 0.997069 | | 25 | 0.997044 | 0.997018 | 0.996992 | 0.996967 | 0.996941 | 0.996914 | 0.996888 | 0.996862 | 0.996836 | 0.996809 | | 26 | 0.996783 | 0.996756 | 0.996729 | 0.996703 | 0.996676 | 0.996649 | 0.996621 | 0.996594 | 0.996567 | 0.996540 | | 27 | 0.996512 | 0.996485 | 0.996457 | 0.996429 | 0.996401 | 0.996373 | 0.996345 | 0.996317 | 0.996289 | 0.996261 | | 28 | 0.996232 | 0.996204 | 0.996175 | 0.996147 | 0.996118 | 0.996089 | 0.996060 | 0.996031 | 0.996002 | 0.995973 | | 29 | 0.995944 | 0.995914 | 0.995885 | 0.995855 | 0.995826 | 0.995796 | 0.995766 | 0.995736 | 0.995706 | 0.995676 | | 30 | 0.995646 | 0.995616 | 0.995586 | 0.995555 | 0.995525 | 0.995494 | 0.995464 | 0.995433 | 0.995402 | 0.995371 |