4: Determination of Avogadro's Number (Experiment)

Hazard Overview

Chemical Hazards

• Corrosive sulfuric acid

 

Mechanical Hazards

• Electrical hazard

 

Dispensary Provided Items

• 0.5 M potassium sulfate solution
• 100 mL beaker
• Universal indicator
• Nichrome electrodes

 

Introduction

Several international agreements are important in relating the mass and amount of chemical substances. These involve the atomic mass scale and the definitions of Avogadro's number and the mole. The atomic mass scale is based on the agreement that the relative atomic mass of one atom of \( \ce{^{12}C} \) shall be exactly 12. A statement that
the atomic mass of \( \ce{^1H} \) is 1.00780 then means that one \( \ce{^1H} \) atom has a mass that is 1.00780/12 times the mass of one \( \ce{^{12}C} \) atom. This leads to the tabulated atomic masses of elements being dimensionless, abundance-weighted averages of the atomic masses of the various isotopes that make up a naturally occurring macroscopic sample of the element. Avogadro's number, \( N_0 \), is defined as the number of atoms in exactly 12 g of isotopically pure \( \ce{^{12}C} \). A mole (mol) of any substance is the amount that contains \( N_0 \) units of the substance. It follows that the mass in grams of \( N_0 \) atoms (one mole) of any element is numerically equal to the relative atomic mass of the element. The same conclusion applies to molecules.

An atomic mass unit (amu or u) is defined as exactly 1/12 of the mass of a single atom of \( \ce{^{12}C} \). Since 1 mol of \( \ce{^{12}C} \) has a mass of exactly 12 g, one atom of \( \ce{^{12}C} \) has a mass of \( 12/{N_0} \) g. Hence, 1 u is numerically equal to the inverse of Avogadro's number. This means that a substance's atomic or molecular mass has the same numerical value, both in u and in g/mol. Another unit of mass that is equivalent to an atomic mass unit is the Dalton (Da). This is used frequently for biological molecules with very large molecular masses (kDa).

The numerical value of Avogadro's number cannot be deduced from purely theoretical considerations but must be determined experimentally by somehow counting the number of entities in one mole. You will do this from data obtained in the electrolysis of a water solution. Electrolysis of \( \ce{H2O} \) causes it to decompose into the gases \( \ce{H2} \) and \( \ce{O2} \), with the amounts produced dependent on the number of electrons transferred during the electrolysis. Avogadro's number can be found from the simultaneous determination of the number of moles \( \ce{H2} \) and/or \( \ce{O2} \) formed and the total number of electrons transferred.

The number of moles of \( \ce{H2} \) or \( \ce{O2} \) is found from the ideal gas law by collecting the gas and measuring its pressure, volume and temperature. The number of electrons transferred is determined from the measurement of the total charge or number of coulombs that flowed during the electrolysis and the known charge of the electron. One ampere of current flow corresponds to the passage of one coulomb of electrical charge through a circuit in a one-second time interval. Hence, if the current, \( I \), is constant and known, the number of coulombs, \( Q \), can be determined by measuring the time, \( t \), during which the current flows using:

\[ Q = I \cdot t \]

If the current is not constant, then the integral of \( I \cdot t \)t must be evaluated to determine \( Q \). To avoid such difficulties, you will perform the electrolysis at a constant current using a power supply.

The electrolysis is carried out by placing two inert electrodes in a \( \ce{H2O} \) solution and connecting the electrodes to the output terminals of a power supply. When the power supply is turned on and warmed up, a constant current will flow around the circuit. The current is carried in the external part of the circuit (wires) by the movement of electrons and in the \( \ce{H2O} \) solution by the movement of ions. The negative pole of the power supply delivers electrons to the solution, while the positive pole accepts electrons from the solution. Hence, reduction of the \( \ce{H2O} \) occurs at the electrode connected to the negative pole, and oxidation of the \( \ce{H2O} \) occurs at the electrode connected to the positive pole. We can postulate that the electrode processes are

\[ \ce{2 H2O_{(l)} + 2 e^{-} \rightarrow H2_{(g)} + 2 OH^{-}_{(aq)}} \]

at the negative pole (cathode, reduction occurs), and

\[ \ce{H2O_{(l)} \rightarrow 2 H^{+}_{(aq)} + 1/2 O2_{(g)} + 2 e^{-}} \]

at the positive pole (anode, oxidation occurs). The overall reaction is

\[ \ce{3 H2O_{(l)} \rightarrow 2 H^{+}_{(aq)} + 2 OH^{-}_{(aq)} + H2_{(g)} + 1/2 O2_{(g)}} \]

The postulated reactions indicate that \( \ce{OH^{-}_{(aq)}} \) will accumulate around the cathode, making the solution around the cathode basic, while \( \ce{H^{+}_{(aq)}} \) will accumulate around the anode, making the solution around it acidic. However, if the solution is stirred, the \( \ce{H^{+}_{(aq)}} \) and \( \ce{OH^{-}_{(aq)}} \) that form initially at the anode and cathode, respectively, will combine with each other to again make \( \ce{H2O _{(l)}} \). Hence, the net reaction of the electrolysis in a well-stirred solution becomes

\[ \ce{H2O_{(l)} \rightarrow H2_{(g)} + 1/2 O2_{(g)}} \]

Note that one mole of \( \ce{H2O} \) is decomposed for every two moles of electrons transferred around the circuit as a current. This follows from the preceding discussion, but it also follows from the net reaction. The oxidation number of each \( \ce{H} \) decreases from +1 in \( \ce{H2O} \) to 0 in \( \ce{H2} \) for a net gain of 2 electrons per \( \ce{H2O} \). Concomitantly, the oxidation number of \( \ce{O} \) increases from -2 in \( \ce{H2O} \) to 0 in \( \ce{O2} \) for a net loss of 2 electrons per \( \ce{H2O} \). Thus, the net reaction also shows that two moles of electrons are involved for every mole of \( \ce{H2O} \) decomposed.

Several requirements exist for water electrolysis to proceed at a reasonable rate. Pure liquid water has a very small conductivity since the concentrations of ions, \( \ce{H^{+}_{(aq)}} \) and \( \ce{OH^{-}_{(aq)}} \) from its self-dissociation, are very small (\( \times 10^{-7} M \)). The small conductivity limits the current flow, so an impossibly long time is required for any observable electrolysis to occur. To have significant conductivity and thereby obtain a reasonable current and rate of decomposition, an aqueous ionic solution must be used for the electrolysis. The added ions are chosen to not interfere with the electrode processes. They are present simply to carry current through the solution.

There are also requirements for the voltage that must be applied across the electrodes. This must be sufficient to overcome any energy requirement that may exist for the decomposition to occur. Once this minimum voltage is exceeded, electrolysis occurs. Its rate and the observed current subsequently increase linearly with the applied voltage if there are no complications. Hence, larger values of the applied voltage give larger rates of electrolysis and decomposition. The various parameters are adjusted to get the desired amount of decomposition in a given time. However, care must be taken to avoid excessive heating of the solution from the current flow since this can introduce errors in the data analysis. The details are discussed later in the section that describes the calculations.

You will verify several aspects of the above discussion, along with determining Avogadro's number, in the present experiment. First, you will make some qualitative observations that establish the postulated electrode reactions. Second, you will make quantitative measurements to determine Avogadro's number. Third, you will determine how the rate of decomposition varies with the current.

 

Experimental Procedure

Qualitative Observations

Figure \(\PageIndex{1}\) shows schematically the apparatus that will be used for qualitative observation of the electrolysis of \( \ce{H2O} \). Your TA will operate this apparatus (there will be one for the whole class), and you will record your observations. 0.5 M \( \ce{K2SO4} \) solution will be used to observe the electrolysis of \( \ce{H2O} \) at a reasonable rate. This solution contains a sufficient concentration of ions, \( \ce{K^{+}_{(aq)}} \) and \( \ce{SO4^{2-}_{(aq)}} \), to allow this. Several drops of "universal indicator" are added to the \( \ce{K2SO4} \) solution in order to give it a fairly intense color. This indicator changes color depending on the acid/base content of the solution. The changes occur in the same order as the colors in a rainbow (red, orange, yellow, green, blue, indigo, violet) as the solution changes its acid/base concentration by powers of ten from \( \times 10^{-4} M \) acid (red) through \( \times 10^{-7} M \) acid and/or base (a "neutral" solution) to \( \times 10^{-10} M \) base (violet). If you are familiar with the terminology, the indicator color changes like the rainbow as the solution pH goes from 4 to 10.

• Verify that the power supply is set to about 10 V and the current set to a maximum of 0.4 A. Let it warm up for about 5 minutes.
• Add about 50 mL of \( \ce{K2SO4} \) to a beaker, and mix with several drops of universal indicator.
• Fill the funnels with the colored solution. Tap the connecting tube to remove all bubbles from the line.
• Immerse the nichrome electrodes into each funnel and connect the other end to an electrical lead with an alligator clip.
• Connect the leads to the power supply, turn it on (if it's not already on) and observe the changes in the solutions over time.

Figure \(\PageIndex{1}\): Schematic of Experimental Apparatus for Qualitative Electrolysis of
\( \ce{H2O} \). (CC BY-NC-SA;??????)

Record the color of the \( \ce{K2SO4} \) solution with the added indicator prior to turning on the power supply. Record and interpret what you see happening in each of the two funnels as electrolysis occurs. Make sure to correlate your observations with the polarity of the electrode in the funnel (the polarity of the power supply terminal attached to the electrode). You should be able to ascertain the identity of the gas that is evolved at each electrode from your observations and the postulated electrode reactions given earlier. Periodically check on the solution throughout the lab period.

Quantitative Measurements

For the quantitative measurements, you will collect both the \( \ce{H2} \) and \( \ce{O2} \) gases that evolve during the electrolysis in inverted burettes. Figure \(\PageIndex{2}\) shows the apparatus at the start and during the electrolysis schematically. The volumes of the uncalibrated portion of the two burettes between the 50 mL mark and the stopcock, called the "Head Space" in Figure \(\PageIndex{2}\), need to be known (you measured them in Experiment 1, and should use the value you calculated then).

Figure \(\PageIndex{2}\): Schematic of Experimental Apparatus for Quantitative Electrolysis
of \( \ce{H2O} \). (The image on the left is before the experiment; the image on the right
is what you can expect after electrolysis.) (CC BY-NC-SA;??????)

Measure the pressure with a mercury barometer and record the result. You should check this again at the end of the period and use the average over the period if the value changes. Consult with your TA for further instructions.

• Wear gloves and obtain about 400 mL of 0.50 M \( \ce{H2SO4} \) solution in a 600 mL beaker.
• The burette apparatus should already be set up for you at your lab stations.
• Check to make sure that all connections are good. There are two inverted burettes clamped to a support rod. Add a magnetic stir bar into the 600 mL beaker containing the \( \ce{H2SO4} \) solution.
• The power supply should be on and set to the specification: 10 V and 0.4 A.
• Remove the hotplate and place the beaker under the burettes. Place the hotplate back where it was.
• Open the burette stopcock, and carefully draw the dilute \( \ce{H2SO4} \) solution up into the burette using a pipette bulb over the burette tip to near but not above the 50 mL mark. Close the stopcock to keep the solution level stable. Fill both of the burettes.
• Record the starting volume of the solution in each burette, estimating to 0.02 mL and paying attention to the fact that you are reading the burette "upside down."
• Measure with a meter stick to the nearest 1 mm the height of the solution column (the meniscus) from the top of the liquid in the beaker to the top of the liquid in each burette.
• Measure and record the temperatures of the \( \ce{H2SO4} \) in the beaker and the air near the top of the burettes.
• Wait 1-2 minutes and again read the starting volume of the solution in each burette. Make sure that your burette is not leaking.
• Turn on the magnetic stirrer and set the speed of the stirrer to its minimum. Do not spin it rapidly since air bubbles can form and enter the burettes, and gas bubbles from the electrolysis can be pulled out of the burettes. This would introduce errors in your results.
• Under the Experiment dropdown menu on the Logger Pro software, select "Data collection." Set duration to 30 minutes and seconds per sample to 1 sample/min.
• Plug the red wire into the positive terminal and simultaneously start Logger Pro data collection by pressing the "Collect" button. Bubbles should start appearing at the electrodes.
• Run until about 30 mL of gas has been collected in the burette with the greater volume of evolved gas. Unplug the red wire from the positive terminal and record the ending time. Leave the power supply on.
• Measure the temperatures of the solution in the beaker and of the air near the mid-height of the burettes. If the electrolysis has caused a significant rise in the temperature of the solution, wait a few (3 to 5) minutes for the temperature of the solution to cool to within 0.5°C of the air temperature.
• Record the final burette readings, and measure the height of the liquid level in the burettes.

Repeat the electrolysis at 6 V, and if time permits, perform a third measurement at 8 V. These measurements will take longer, so you may need to adjust the duration of the electrolysis.

When you are done, drain each burette into the beaker and remove the beaker of solution the same way you put it on. Put the used acid solution back into the original container in the hood.

Thoroughly rinse off the parts of the apparatus and the outside portions of the burettes that were submerged in the acid solution with a deionized water wash bottle into the rinse beaker. Rinse off the magnetic stir bar and return it.

Calculations

You will calculate Avogadro's number on the basis of both the amounts of \( \ce{H2} \) and \( \ce{O2} \) that were evolved. To do this, you need to compute the number of moles of evolved \( \ce{H2} \) and \( \ce{O2} \), and the number of electrons that were transferred while the gases were being evolved.

First, determine the average value of the current in ampere (amp, A) and its standard deviation for each run. (Note that even if the observed current did not appear to change over a run, it would nevertheless have a standard deviation corresponding to the minimum current difference that can be read from the ammeter.) Next, multiply the average current by the total electrolysis time in seconds for each run to obtain the corresponding number of coulombs passed through the solution. The resulting product will have the same relative standard deviation as the current, assuming that there is no significant error in the time measurement. Alternatively, you can get the total number of coulombs by integrating the area under the curve recorded by Logger Pro.

Dividing the number of coulombs transferred by the charge of an electron, 1.6021773 \( \times 10^{-19} \) C, yields the number of electrons transferred for each run.

An accurate calculation of the moles of \( \ce{H2} \) and \( \ce{O2} \) that are produced involves a Dalton's Law calculation with several partial pressures. At the start of a run, the gas in the burette volume above the liquid column is a mixture of air and \( \ce{H2O} \) vapor. The gas volume equals the volume of the uncalibrated region of the burette plus the (positive) difference between the initial burette volume reading and 50.00 mL. The total pressure of this gas mixture is:

\[ P_{total\,in\,burette} = P_{T} = P_{atm} - P_{column\,height} = P_{air\,initial} \]

At the end of a run, the now-larger gas volume in the burette above the liquid column contains air, the evolved \( \ce{H2} \) or \( \ce{O2} \) gas, and \( \ce{H2O} \) vapor. The gas volume is equal to the volume of the uncalibrated region and the volume difference between the final burette reading and 50.00 mL. The total pressure of the gas mixture at the end is given by atmospheric pressure minus the pressure corresponding to the new, smaller height of the solution column.

Table \(\PageIndex{1}\): Vapor Pressure of Pure Water

Temperature / °C	Pressure / Torr
19	16.5
20.0	17.5
21.0	18.6
22.0	19.8
23.0	21.1
24.0	22.4

The pressure exerted by the height of the solution column can be calculated from the product of the measured height, the acceleration due to gravity, and the density of the
solution.

However, it is simpler to obtain this pressure by comparison with a \( \ce{Hg} \) column at 0°C. An aqueous column with height \( h \) and density \( \rho \) is related to a \( \ce{Hg} \) column at 0°C with height \( h_0 \) and density \( \rho_0 \) by:

\[ P_{column\,height} = h_0 = h\left(\frac{\rho}{\rho_0}\right) \]

The density of \( \ce{Hg} \) at 0°C, \( \rho_0 \), is 13.5951 g/mL. The density of 0.50 M \( \ce{H2SO4} \) at 20°C is about 1.030 g/mL; this changes only slightly near room temperature.

The partial pressure of \( \ce{H2O} \) in the gas mixture is the vapor pressure of \( \ce{H2O} \) above the solution at the temperature of the solution. Table \(\PageIndex{1}\) gives the vapor pressure at several temperatures around room temperature for pure water. The correct temperature is somewhat uncertain in the present experiment because of the heating of the solution in the beaker, caused by the current flow, and the minimal mixing of the solution in the burette with the solution in the beaker. You will use the air temperature adjacent to the burette to estimate the temperature of the solution in the burette (and of the gas in the burette). As evident from the data in Table \(\PageIndex{1}\), if this estimate is accurate to \( \pm \)1.0 °C, then the water vapor pressure can be accurate to about \( \pm \)1 Torr.

The second concern is the composition of the aqueous solution. Over a 0.50 M \( \ce{H2SO4} \) solution, the vapor pressure of \( \ce{H2O} \) will be slightly less than that of pure water. The \( \ce{H2SO4} \) dissociates in aqueous solution into \( \ce{H^{+}_{(aq)}} \), \( \ce{HSO4^{-}_{(aq)}} \), and \( \ce{SO4^{2-}_{(aq)}} \) ions. These species occupy some fraction of the surface, and thereby reduce the concentration of \( \ce{H2O} \) molecules at the surface that can escape into the vapor. The vapor pressure of the \( \ce{H2O} \) is given approximately by the mole fraction of \( \ce{H2O} \) in the solution times the vapor pressure of pure \( \ce{H2O} \) (Raoult's Law).

The mole fraction in Raoult's Law is based on the total moles of solute species. \( \ce{H2SO4} \) is dissociated mainly into \( \ce{H^{+}_{(aq)}} \) and \( \ce{HSO4^{-}_{(aq)}} \) ions in water solution. Although partial dissociation of \( \ce{HSO4^{-}_{(aq)}} \) into \( \ce{H^{+}_{(aq)}} \) and \( \ce{SO4^{2-}_{(aq)}} \) ions also occurs, the amount is small and can be neglected in the present case. The total moles of solute species can be taken as two times the moles of dissolved \( \ce{H2SO4} \).

Therefore, the mole fraction of \( \ce{H2O} \) is:

\[ X_{\ce{H2O}} = \frac{n_{\ce{H2O}}}{n_{\ce{H2O}}+2n_{\ce{H2SO4}}} \]

\[ P_{\ce{H2O}} = X_{\ce{H2O}} \cdot P_{vap\ce{H2O}}^{*} \]

The mole fraction of \( \ce{H2O} \) in a 0.50 M \( \ce{H2SO4} \) solution with a density of 1.030 g/mL is 0.982 on this basis. The vapor pressure of \( \ce{H2O} \) above the 0.50 M \( \ce{H2SO4} \) solution is then 0.982 times the vapor pressure of pure water. The partial pressure of air changes between the start and end of a run. Since the number of moles of air in the burette is constant, the pressure of air at the end can be calculated from the pressure of air at the start, both the ending and the starting gas volumes, and both the ending and the starting gas temperatures:

\[ P_{air} = P_2 = P_1 \cdot \frac{V_1}{V_2} \cdot \frac{T_2}{T_1} \]

If the gas temperature is constant, then a simple Boyle's law calculation suffices (\( P_1V_1 = P_2V_2 \)).

The partial pressure of \( \ce{H2} \) and \( \ce{O2} \) in the respective burette at the end is given by:

\[ P_{\ce{x2}} = P_T - (P_{\ce{H2O}} + P_{air}) \]

The moles of \( \ce{H2} \) and \( \ce{O2} \) that are produced by the electrolysis are calculated for each run from the partial pressure of the evolved gas, the gas volume at the end and the air temperature near the mid-height of the burette at the end.

The partial pressure of \( \ce{H2} \) and \( \ce{O2} \) in the respective burette at the end is given by the total pressure minus the sum of the vapor pressure of \( \ce{H2O} \) and the partial pressure of air. The moles of \( \ce{H2} \) and \( \ce{O2} \) that are produced by the electrolysis are calculated for each run from the partial pressure of the evolved gas, the gas volume at the end and the air temperature near the mid-height of the burette at the end.

 

Lab Report

Be sure to calculate the following for your lab report:

• Calculate the moles of \( \ce{H2} \) and \( \ce{O2} \) that are produced in each run.
• Calculate the ratio of moles of \( \ce{H2} \) to moles of \( \ce{O2} \) for each run, the mean value of the ratio, the standard deviation, and the 95% confidence limits of the mean.
• Obtain Avogadro's number from the data for both \( \ce{H2} \) and \( \ce{O2} \) for each run, the mean value, the standard deviation, and the 95% confidence limits of the mean. Remember that two moles of electrons are required per mole of \( \ce{H2O} \) that decomposes.

Examine the results for the three different currents by calculating the moles of \( \ce{H2} \) and \( \ce{O2} \) that are produced per second at each current. Compare these production rates with the corresponding values for the average current. Discuss your observations.

Also, include the answers to the following questions in your lab report.

1. Calculate the volume in mL of \( \ce{O2} \) gas at STP that is generated by quantitative decomposition of 1.000 mL of liquid \( \ce{H2O} \) at 19.0°C and 25.0°C.
2. Calculate the mole fraction of \( \ce{H2O} \) in 0.6350 M \( \ce{H2SO4} \), which has a density of 1.0385 g/mL, based on the total moles of solute species and the complete dissociation of the acid into \( \ce{H^{+}_{(aq)}} \) and \( \ce{HSO4^{-}_{(aq)}} \) ions.
3. The electrolysis of an aqueous solution generates 50.0 mL of \( \ce{O2} \) gas at STP. Assume that the error in the volume is \( \pm \)0.3 mL, with no error in the temperature and pressure. What is the number of moles of evolved \( \ce{O2} \) and its uncertainty?
4. Assume that the relative error in measuring the volume of the evolved \( \ce{H2} \) or \( \ce{O2} \) gas is 1.0%, the relative error in the number of electrons transferred is 0.4%, and the relative errors in all other relevant quantities are negligible. What is the relative error in the calculated value of Avogadro's number?
5. Discuss the direction of the error in the calculated value of Avogadro's number that would result in this experiment for the following cases.
6. Estimate the magnitudes of the errors for the following three cases:
   • The vapor pressure of water is entirely neglected.
   • An incorrect density of 1.50 g/mL is used for the \( \ce{H2SO4} \) solution.
   • The thermometer used to measure the temperature of the gas reads low by 1.0°C.

 

Data Sheet

Outline of Intermediate Calculations

For Each Run

Electrolysis Time (second)
Average Current (amp)
Electrons Transferred
Initial Temperature (°C)
Final Temperature (°C)

	\( \ce{H2} \) Burette	\( \ce{O2} \) Burette
Initial Total Gas Pressure (torr)
Initial Gas Volume (mL)
Initial Air Pressure (torr)
Final Total Gas Pressure (torr)
Final Gas Volume (mL)
Final Air Pressure (torr)
Final \( \ce{H2} \) or \( \ce{O2} \) Pressure (torr)
Moles of \( \ce{H2} \) or \( \ce{O2} \) Produced (mol)
Avogadro's Number

Determination of the Avogadro's Number

Barometric Pressure (torr)

Volume of Head Space (mL)

\( \ce{H2} \) Burette:

\( \ce{O2} \) Burette:

	Run 1		Run 2		Run 3
	\( \ce{H2} \) Burette	\( \ce{O2} \) Burette	\( \ce{H2} \) Burette	\( \ce{O2} \) Burette	\( \ce{H2} \) Burette	\( \ce{O2} \) Burette
Average Temperature (°C)
Average Current (A)
Electrolysis Time (s)
Initial Height of Solution (mm)
Initial Burette Reading (mL)
Final Height of Solution (mm)
Final Burette Reading (mL)
Moles of \( \ce{H2} \) or \( \ce{O2} \)

Ratio of Moles of \( \ce{H2} \) to \( \ce{O2} \)

Average Value
Standard Deviation
95% Confidence Limits

Avogadro's Number in \(10^{23}\)

Average Value
Standard Deviation
95% Confidence Limits

Rate of \( \ce{H2} \) or \( \ce{O2} \) (mol/s)
Rate / Current (mol/A⋅s)