(Lab 1) Measurement and Uncertainty: Density, volume, and propagation of error

Dealing with uncertainty

When we make measurements of the physical world, can we “know” the true quantity? The accuracy

of a measurement (how close it is to the true value) and the precision of a measurement (how many

significant digits in the quantity) are limited by many factors. For example: how refined is the

equipment or apparatus? How skilled is the observer? Are there inherent physical limitations?

Scientists report not just the quantity and units associated with a measurement, but the uncertainty in

the measurement as well (in this context, the terms “error” and “deviation” can be considered

synonyms of “uncertainty”). For example, the U.S. Mint reports the diameter of a standard quarter

dollar piece as 24.26 mm. What about the uncertainty in the measurement? If the uncertainty is not

provided, we must take a conservative approach in treating the final significant digit and realize this

measurement might be as high as 24.27 mm or as low as 24.25 mm. However, if the measurement had

been reported as 24.260±0.005 mm, the measurement would be said to have greater precision.

When we make a measurement, how do we treat its uncertainty? The two types of uncertainty are

referred to as systematic errors and random errors. Systematic errors result from a mis-calibrated

device or a measuring technique that will reliably give a measurement that is too high or too low when

compared with the true value. We typically deal with systematic errors through careful experiment

design. Random errors demand more of our attention and thought. There are three common origins of

random errors we should bear in mind in Chem 125/6, using our best judgment to decide which is

appropriate:

1) Instrument least count. The least count is the smallest division marked on an instrument. Most

meter sticks have a least count of 1/1000th of a meter, or 1.0 mm. An analog wall clock would have a

least count of 1/60th of a minute, or 1.0 s, so a digital stop watch with a least count of 0.02 s would be

more precise by comparison.

2) Estimated error. We have to use conservative judgment when making measurements. For example,

if we have a balance that does not respond to a mass less than 5.0 g while attempting a calibration, we

would have to account for this by estimating the error in any measurement using that balance as

±2.5 g (the interval is divided by 2 to reflect a total range, or 5.0 g in this example).

3) Average deviation. We can often deal with random errors in a statistical manner. If we make

repeated measurements several times, we can then find an mean and standard deviation for the dataset.

Both the mean of measurement x (written <x>) and standard deviation (written s) are easily calculated

using software functions. When making repeat measurements, we expect 68% of the values to fall in

the range of (<x> - s) to (<x> + s), and 95% of values to fall in the range of (<x> - 2s) to (<x> + 2s).

Example: Five of us measure the density of the same bar of gold, finding the values 19.25 g/mL, 19.33

g/mL, 19.32 g/mL, 19.23 g/mL, and 19.39 g/mL. Using software we find <x> = 19.30 g/mL and s =

0.06. Based on these data, we might expect 68% of further trials to give a result in the range of [19.24,

19.36] g/mL. We might also predict that 95% of further trials would give a result in the range of

[19.18, 19.42] g/mL.

Propagation of error

When we use two measurements in a calculation, we are comfortable performing arithmetic on the

quantities: simply add, subtract, multiply, or divide the numbers. When dealing with their

uncertainties, we must be more careful. If we add two measurements of length such as:

24.23±0.05 mm + 3.66±0.05 mm = ?

...we can deal with the quantities easily: 27.89. But what of the uncertainties? For addition and

subtraction, we would treat them like this:

√0.05

2+0.05

2 = 0.07

...making our new calculated value 27.89±0.07 mm

Note that in the method shown, the uncertainty is rounded by the rules of significant digits, and the

quantity is rounded to the same decimal place that the uncertainty was.

The general format for addition and subtraction is:

√(Δ x)2+(Δ y)2 = Δ z

We call Dx the uncertainty of measurement x, Dy is the uncertainty of measurement y, and Dz is the

uncertainty associated with the new, calculated quantity.

Things change slightly when we multiply or divide measurements. If we want to divide a mass of

3.0±0.5 kg by a volume of 1.5±0.2 L, we could once again be quick with the quantity: 2.0 kg/L.

The new uncertainty associated with our answer would be:

Δ z = 2.0√(0.5

3.0

)2

+(0.2

1.5

)2

= 0.4

…making our calculated density 2.0±0.4 kg/L. Once again, the uncertainty was rounded by the

rules of significant digits, and the quantity was rounded to the same decimal place to match.

The general form for error propagation when multiplying or dividing is:

Δ z = z √( Δ x

x)2

+( Δ y

y)

2

Laboratory

Hypothesis: We can identify the material a regular cube shaped object is composed of based on its density.

Experiment:

Determine the volume of ten different polyhedra supplied in the lab. Use both a standard ruler and a

Vernier caliper and recored the uncertainties in your calculations. Each measurement of a length,

width, height, or mass has an associated uncertainty; when those four quantities are multiplied (or

divided) by each other, we must be careful with how we treat the associated uncertainty (a.k.a. “error”)

as well. Your work will be evaluated in part based on your written propagation of error for your

conclusions.

Data: Carefully plan how you want to organize your length, width, height, and mass data for each of your ten

polyhedra. Determine the volume of each mystery polyhedron with both a typical ruler and Vernier

caliper. Recall that density is mass divided by volume, and remember to record the uncertainty for

each measurement you make.

Conclusions:

For your post-lab write-up, what do you think each of your ten cubes is composed of? Compare your

densities (with uncertainty intervals!) to those of published values for common materials. Cite your

sources. How to the measurements made with a typical ruler compare to those made with the Vernier

caliper? Which density measurements have the widest range of uncertainties, your individual

calculations obtained from propagated errors, or the class-wide data complete with standard deviations?

(Lab 2) Determining a Molecular Formula: waters of hydration

Waters of hydration

Chemists use the phrase salt to mean any ionic compound (no net charge) that results from a neutralization reaction of an acid and a base. Most salts are composed of a metal cation and a nonmetalanion, and are occasionally encountered as hydrates, meaning they have a number of water molecules associated with each formula unit of the salt.

One famous example of a hydrate is magnesium sulfate heptahydrate (common name: epsom salts) whose formula is:

MgSO4•7H2O

The bullet character followed by a coefficient of seven tells us that there are seven moles of water associated with every one mole of magnesium sulfate. In this example we might say that there are seven waters of hydration, which can often be removed from a sample by heating the sample: magnesium sulfate heptahydrate becomes anhydrous magnesium sulfate above 200 °C.

Laboratory

Hypothesis: We can determine how many waters of hydration are lost from a given hydrate sample by measuring the mass before and after heating.

Experiment: You will be assigned a hydrate sample whose formula (including waters of hydration) is known. Your task is to confidently identify the product obtained upon heating.

Obtain a hydrate sample along with an oven-dried crucible. Your GSI will handle the hot ceramic, though you will need to record the crucible's mass. Place a measured amount of your solid unknown hydrate in the crucible. The actual amount you aim to measure is not crucial, anywhere between 1 g and 2 g will do, as long as you recored the particular mass of hydrate sample that makes it into your crucible. Return the crucible & hydrate sample to the oven (via your GSI) and heat at ~200 °C for 30 minutes.

Your GSI will retrieve your heated crucible + sample from the oven and immediately place it on the balance you used to obtain the tared mass. Record this new, dehydrated mass of your crucible + sample.

Data: The crucial measurements we will make are the mass of your hydrated compound prior to heating, and the mass of your sample after heating. You will also know the identity of the molecular formula prior to heating.

One important (but often-overlooked) measurement you should have made is the mass of your oven-dried crucible; we sometimes refer to this as the “tare” or “tared mass” and it is useful to know in order to subtract the value from your final mass measurement to determine how much sample remains after heating.

Conclusions: For your post-lab write-up the big questions are: compared to your sample's original formula, how many waters of hydration did it lose, and what is the new formula? Some other things your reader might want to know: How did you perform your calculations-- can you explain them step by step? How confident are you in your calculations (or range of possibilities) for your hydrate? Based on in-class conversations with classmates investigating the same starting material, how did your sample behave compared to theirs? How do your data and conclusions compare with their work?

(Lab 3) Partial Pressures: collecting a gas over water

Collecting a gas over waterWe will perform a simple acid-base reaction:

CaCO3(s) + 2 HCl(aq) → CaCl2(aq) + CO2(g) + H2O(l)

We will collect the CO2 generated and measure its volume. The hydrochloric acid will be present in excess, while the calcium carbonate will be our limiting reactant. The calcium carbonate sample will not be 100% pure, however; it will be cut with calcium chloride. Your task will be to determine the percent composition of your calcium carbonate sample.

LaboratoryHypothesis: We can determine the percent composition of an unknown carbonate sample by combining a portion with strong acid and collecting the resulting carbon dioxide gas over water.

Experiment:

Fill a large tub of water with tap water and add 1 g of citric acid and 1 g of sodium bicarbonate. This CO2-saturated sample will be used throughout the experiment.

Assemble a gas-tight reaction vessel using two test tubes of different size. The smaller tube should contain ca. 200 mg of your solid unknown, while the larger tube should contain 10 mL 3 M HCl. Carefully slide the small tube into the large tube without splashing any of the acid. It is important that these two samples do not mix just yet. Measure the mass of this constructed reaction vessel; we will need to compare its total mass prior to a reaction taking place to its mass after.

Fill a shallow pan about two-thirds full with the CO2-saturated water, and ensure that CO2 gas has stopped evolving. Fill your 25 mL graduated cylinder by laying it horizontally in this water, then position it vertically, inverted, without removing the mouth of the cylinder. Secure the cylinder in place with a clamp, thread a piece of tygon tubing up into the submerged mouth of the cylinder, and it is ready to serve as your CO2 gas collector. Record the level of the water.

Secure a one-hole rubber stopper with a small piece of glass tubing into the mouth large of the large test tube of your reaction vessel. Connect the piece of tygon tubing to this glass tube. You now have a gas-tight reaction vessel whereby the gaseous product can be collected and measured. Gently agitate the reaction vessel so that some of the HCl solution mixes with your solid sample. As the production ofCO2 slows, gently agitate again and again until no further CO2 gas is generated.

While your gas-collecting cylinder is still submerged, adjust its height so that the water levels inside and outside of the cylinder are equal. Record this final volume. Also record the temperature of your water bath and the barometric pressure in the laboratory. The temperature of the water will be needed to calculate the partial pressure of water vapor in the cylinder, and the barometric pressure will be equalto the total pressure of your gaseous sample. We can then calculate the partial pressure of CO2 in your sample.

Carefully remove and dry your reaction vessel and remove the stopper. Record the new mass of the vessel and calculate the mass lost as CO2 gas.

Data: Here are some data we should have gathered empirically: - initial volume of water in cylinder- volume of water in cylinder after reaction- temperature of water- barometric pressure- original mass of reaction vessel with acid & sample- new mass of reaction vessel- total pressure of sample in cylinder (equal to atmospheric pressure once water levels inside and out are equal)

Conclusions: Here are some data we should be able to calculate:- volume of CO2 collected- partial pressure of H2O (at ambient temperature)- partial pressure of CO2

- mol CO2 generated- mass CaCO3 in original sample- percent of CaCO3 in original sample

What was the %-composition of calcium carbonate in your unknown sample? Can you show calculations to justify your conclusion? How do your measurements of %-composition compare when using the ideal gas law vs. the mass difference of the reaction vessel?

(Lab 4) Oxidation & Reduction: exchanging electrons as currency

The ionic charge and the oxidation state of an element in a chemical reaction are closely related, but they are not the same. Charge is a coulombic phenomenon, and is measurable empirically. Oxidation states are imaginary human constructs that merely help us organize reactivity patterns in our mind. Nevertheless, we find oxidation states helpful in deciding when a transfer of electrons has taken place. In this lab you will determine the relative reactivities of metals through experiment, and be able to identify the reactants and products of redox reactions.

LaboratoryHypothesis: We will be able to rank a series of elements according to their strength as reducing agents or oxidizing agents. In colloquial terms, we should be able to predict who will reduce whom.

Experiment:

For the first series of experiments we will have the following metals available: Cu, Zn, Mg, SnWe will also have the following metal ions available: Cu2+, Zn2+, Mg2+, Na+, H+, Sn2+

Our goal as a class is to pair up each of the metals with each of the cations listed and determine in each case whether a reaction occurs or not.

Your GSI will announce how they wish to assign all 24 combinations among your class groups. In general, use the metal cations as the provided 0.1 M nitrate solutions. The exception is for Sn2+, which you should use as SnCl2. For H+ use 1.0 M HNO3.

- Clean each of the metals you plan to use with sandpaper; this will remove any oxide coating on the surface.- Use 1 – 2 mL of each solution in a small test tube, vial, or well plate for observation.- Allow your mixtures to stand for 30 minutes before making a decision regarding whether or not a reaction took place.- If you are having difficulty distinguishing between a “no reaction” outcome and a sluggish reaction, you may increase the metal ion concentration by adding crystals of the solid to your 0.1 M solution.

The second series of experiments we will perform three reactions using the sodium iodide each time, but varying the metal cation present. For each of the following, determine if a redox reaction has takenplace.

Dissolve ca. 0.5 g of solid copper(II)nitrate in 20 mL deionized water. Dissolve ca. 0.5 g of sodium iodide in a separate 20 mL of deionized water. Pour the two solutions into a third beaker and record your observations.

Mix 2 mL of 0.1 M sodium iodide and 2 mL hexane. Add 2 mL of 0.1 M iron(III)chloride and shake well. Record your observations.

Mix 2 mL of 0.1 M sodium iodide and 2 mL hexane. Add 2 mL of 0.1 M tin(IV)chloride and shake well. Record your observations.

For next week:

You will need to prepare solutions of the following salts for your work next week, and store these solutions. One liter of each is recommended for your entire lab section to share.

0.1 M potassium nitrate from solid potassium nitrate 0.1 M zinc nitrate from solid zinc nitrate 0.1 M magnesium sulfate from solid magnesium sulfate 0.1 M iron(II) sulfate from solid iron(II) sulfate0.1 M copper(II) nitrate from solid copper(II) nitrate1.0 M copper sulfate from solid copper sulfate

Data: For the reactions you are responsible for testing, if a reaction occurs we gain insight into the relative reactivity of the participants. For example, if we were to observe:

Zn(s) + Cu2+(aq) → Zn2+

(aq) + Cu(s)

...we would conclude that Zn is more reactive than Cu (stronger reducing agent), and Cu2+ is more reactive than Zn2+ (stronger oxidizing agent) since the reaction proceeded to the right as written. If no reaction occurs, we would conclude the opposite.

Conclusions: Some ideas to consider for your post-lab write-up:- How does your class rank the reducing agent strength of the tested metals?- How does your class rank the oxidizing agent strength of the tested metal ions?- Compare the ranked reactivities of the tested metals with the ranked reactivities of the tested metal cations. Is there a pattern? Can you discern a predictable relationship?- Based on your data, can you predict the relative reducing agent strength of Na and H2?- For the three reactions you performed with sodium iodide and a varying metal, and based on your knowledge of likely spectator ions, who is reducing whom? Can you write a net ionic equation for the reaction you suspect is taking place? What about a balanced molecular equation for each? How sure are you about the oxidation state of the resulting metal cation? You may find a table of reduction potentials helpful.

(Lab 5) Galvanic Cells: electrons flowing spontaneously

IntroductionMost of our modern economy and civilization are based on moving electrons in time and space in a controlled manner. From the macroscale infrastructure of our electrical grid to the microscale of the circuitry in our mobile phones, we expect electrons to flow where we want them, when we want them.

We will investigate some fundamentals of electron flow in solution phase systems in this lab. Our emphasis will be on pairing a variety of half-cells to build circuits, and collecting data to characterize the behavior of our cells.

The term galvanic cell is used to describe a circuit involving the spontaneous transfer of electrons. In this context, “spontaneous” has the meaning that once we have mixed two or more chemical species, a reaction occurs without our further intervention; something in the reaction gains electrons and something else has lost electrons. Instead of mixing the oxidizing agent and reducing agent in the samesolution, we will prepare a separate solution of each and connect them with a wire. In this way we will exert some measure of control over where the electrons flow.

The electrode that is supplying the electrons (and hence being oxidized) is called the anode, while the electrode receiving the electrons (and is itself reduced) is called the cathode. Other components in our circuit will include a multimeter to measure the cell potential, and a salt bridge to neutralize the ionic charge that would otherwise build up at each terminal.

LaboratoryHypothesis: Through careful observation we can characterize the behavior of galvanic cells and determine the concentration of an unknown solution sample.

Experiments:

Polish two strips of copper, one strip of zinc, one of magnesium, and one of iron using sandpaper or steel wool. Rinse these five metal strips with dilute (1.0 M) HNO3 into the proper waste container (caution!), and again with deionized water. These polished metals will be your electrodes. These electrodes are reusable; please do not throw them away.

Fill one small with ca. 35 mL 0.1 M Zn(NO3)2, a second with ca. 35 mL 0.1 M CuSO4, a third with with ca. 35 mL 0.1 M MgSO4, and a fourth with with ca. 35 mL 0.1 M FeSO4. These solutions will be your half-cells. Place a copper electrode in the CuSO4 solution and the zinc electrode in the Zn(NO3)2 solution. Prepare a salt bridge by rolling a large piece of filter paper and saturating it with 0.1 M KNO3

solution. Place one end of the filter paper in each half cell. Connect two leads to the multimeter (one in the COMmon port, one in the V/Ω/mA port), then the probe end of each lead to the electrodes. If you obtain a negative potential, switch which electrode the leads are contacting. Set the multimeter to the appropriate range (~2000 mV is a good first guess). Record your measurement, and identify which electrode is the cathode and which is the anode. Repeat for the other five pairings of half-cells we can build using these four solutions and electrodes. Remember to prepare a fresh salt bridge for each new cell.

Construct a cell using your 0.1 M Zn(NO3)2 half-cell paired with the CuSO4 solution of unknown concentration, measure the cell potential and (assuming the reduction potential of the Zn2+(0.1 M)/Zn

half-cell is −0.76 V) calculate the reduction potential of this unknown Cu half-cell relative to the Zn half-cell.

Starting with the 1.0 M CuSO4 solution provided, prepare a series of 10-fold dilutions to obtain CuSO4

solutions of 0.1, 0.01, 0.001, and 0.0001 M concentrations. Portions of ca. 35 mL of each solution should suffice for all experiments.

Set up a galvanic cell using 1.0 M CuSO4 and 0.001 M CuSO4, and immerse a polished copper electrode in each. Measure the cell potential and determine the cathode and anode. Write an equation for the reaction occurring in each half-cell.

Set up four more galvanic cells, this time using the Zn/Zn2+ half-cell each time, and pairing it with the 0.1 M. 0.01 M, 0.001 M, and 0.0001 M CuSO4 half-cells in sequence. Make sure to prepare a new salt bridge each time you change the concentration. You may need to reset the multimeter to a lower range for these experiments. Record the potential difference for each pairing.

Data: -For the Cu/Zn, Mg/Zn, and Fe/Zn cells, assume the reduction potential of the Zn2+(0.1 M)/Zn half-cell is −0.76 V. Determine the reduction potential of the other three half-cells. Do these agree with the table of standard reduction potentials? How much uncertainty are you therefore dealing with in your measurements?

-For your Zn vs. Cu cells of varying [Cu2+], compare to the theoretical cell potential from a table of standard reduction potentials and the Nernst equation. How do things compare from your experimentalvalues vs. theory? Your instructor will be interested to see your calibration curve plot.

Conclusions: Can you write balanced equations for the oxidation/reduction reaction that took place in each cell you constructed? Can you identify the cathode and anode for each? What was the molar concentration of your unknown solution? For each cell you constructed, which direction did electrons flow? What about the direction of cations and anions flowing through the salt bridges? For the cell constructed using to copper electrodes, why did a current flow even though the electrodes were identical?