APPENDICES VERNIER SCALE BUOYANCY CORRECTIONS IN WEIGHING BAROMETER. ph METER WHEATSTONE BRIDGE VAPOR PRESSURES OF WATER AT VARIOUS TEMPERATURES

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1 APPENDICES VERNIER SCALE BUOYANCY CORRECTIONS IN WEIGHING BAROMETER ph METER WHEATSTONE BRIDGE VAPOR PRESSURES OF WATER AT VARIOUS TEMPERATURES DENSITIES OF WATER AT VARIOUS TEMPERATURES t-values FOR VARIOUS SAMPLE SIZES AND CONFIDENCE LEVELS ATOMIC WEIGHTS PERIODIC TABLE

This"is'often estimated by eye, but a more reliable way to get "the extra decimal" is to use a vernier scale, as illustrated in Fig. A-1.

If we had an object whose length, X, was less than 1 cm long we could measure it to the nearest mm with a cm rule and a vernier scale, as in (b); the scales are laid parallel, with a "0" at each end

With reference to the point at which both scales coincide, at figure "n", we can say : 1. On the cm scale, the distance (I) from the left end of X "n" is cm :, 1 = (n units) (1 -> = n cm umt 2.

3 VERNIER SCALE Scales vary.inin.quality, but with even the highest quality scale there comes the time when you must decide "just how far" between two of the smallest division maiks a given measurement lies. This"is'often estimated by eye, but a more reliable way to get "the extra decimal" is to use a vernier scale, as illustrated in Fig. A-1. The vernier scale shown in (a) is nine cm long and it is divided into ten equal parts; each unit is therefore 0.9 cm long. If we had an object whose length, X, was less than 1 cm long we could measure it to the nearest mm with a cm rule and a vernier scale, as in (b); the scales are laid parallel, with a "0" at each end of X. Let's assume that some number (n) between 0 and 10 exactly coincides on the two scales; it is "7" in (b). With reference to the point at which both scales coincide, at figure "n", we can say : 1. On the cm scale, the distance (I) from the left end of X "n" is cm :, 1 = (n units) (1 -> = n cm umt 2. On the vernier scale, the distance (I) from the-left end of X to "n" is cm 1 = (X cm) + (n units) (0.9-) = (X n) cm unit 3. Since the distance (I) is the same regardless of the scale that is used, we can set the expressions obtained in 1 and 2 equal to each other to give n = X + 0.9n The desired length of X is then seen to be In other words, if the two scales coincide at "7," as in (b), the length of X is 0.7 cm; if they coincide at "4," the length of X is 0.4 cm, etc. In actual practice, you might want to know how much past "3" a measurement lies on the cm scale, and you find that "10," not "7," on the cm scale coincides with "'7" on the vernier scale, Fig. A-1 (c) p. A-2. The figure in (b) is identical to that section of (c) which lies to the right of the 3-cm mark, so you quickly conclude that the object is 0.7 cm longer than 3 cm; i.e., it is 3.7 cm long. We also come to a general conclusion: the.number (n.) on the vernier scale that coincides with any number on the measuring scale is the number of tenths of divisions that should be added for interpolation on the measuring scale. The vernier principle is also applied to a mm scale in Fig. A-2. You can see by inspection that the length of the line AB lies somewhere between 7.2 and 7.3 cm, but the vernier scale shows that it is 7.23 cm. It doesn't make any difference what the unit of linear measure is, it can always be subdivided into tenths by a vernier scale whose units are just %o as large. A- 1

4 ' 8 Centimeter wale I I I I I I I I I I vernier scale (a) Vernier scale (each division = 0.9 cm) compared with centimeter scale. Vernier scale 7f- I-1 I Centimeter measuring scale I m I I I I I I I r I I I I I I I I I I I I I (b) Centimeter scale laid alongside vernier scale so that the figure 7 coincides on each scale. The object is 0.7 cm long. Vernier scale / Centimeter measuring scale I I I I I I I I I I. I I I I I ", (c) Centimeter scale laid alongside vernier scale so that the figure 7 on the vernier scale coincides with 10 on ttie centimeter-measuring scale. The object is 3.7 cm long. Figure A - 1 A LLLLLYl IIIIII~IIII Magnified section A-2 APPENDIXES Figure A - 2. Centimeter scale. Millimeter divisions laid alongside vernier scale with divisions that are 0.9 mm apart. The magnified section shows that the object is 7.23 cm long.

BUOYANCY CORRECTIONS IN WEIGHING Fundamental to a discussion of the effect of buoyancy on weighing is an understanding of the concepts of density and Archimedes' Principle.

5 BUOYANCY CORRECTIONS IN WEIGHING Fundamental to a discussion of the effect of buoyancy on weighing is an understanding of the concepts of density and Archimedes' Principle. The density, which is characteristic of a substance, is simply the ratio of its mass to its volume: ' The density is customarily expressed in the units grams per milliliter. Equation (1) indicates that if any two of the three quantities, mass, volume, and density, are known for a given sample, the third quantity can be calculated. It is a matter of experience that an object is b&oyed up or supported when it is immersed in a fluid. The more dense the fluid, the greater the buoyant force on an object immersed in it. This experience is summarized in the statement of Archimedes' Principle: an object immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced. Just as an object immersed in water is buoyed up by a force equal to the weight of water it displaces, any object in air is supported by a force equal to the weight of displaced air. Even though the density of air is.quite small, g/ml, the effect of buoyancy is often ~ i ~ c awhen n t large objects are weighed on the analytical balance. In these cases a buoyancy correkon can be applied to compensate for this effect. In weighing an object with the analytical balance the mass of the object is identified with the mass of the balance weights. However, if the object being weighed is larger in volume than the weights, it displaces more air and is buoyed up with a greater force. The net result is that it takes a lesser mass of weights to balance the object - the observed weight of the object is too small. The buoy~cy correction, which must be added to the apparent weight of the object, is simply the difference between the weight of air displaced by the object and that displaced by the weights. The weight of air displaced by the object is obtained by multiplying the density of air, d,, by the volume of the displaced air, which is equal to the volume, v,, of the object being weighed. Let this buoyant force be B,. The volume of the object is its weight divided by its density:

Similarly, the buoyancy of the balance weights, B,, is given by Note that w,, the weight of the balance weights, is the weight we would attribute to the object if we fail to take buoyancy into

The buoyancy correction, to be added to the observed'weight of the object, is 7 - Note that w,, the weight of the object, and w,, the weight of the balance weights, differ only by the small buoyancy

6 Similarly, the buoyancy of the balance weights, B,, is given by Note that w,, the weight of the balance weights, is the weight we would attribute to the object if we fail to take buoyancy into account. The buoyancy correction, to be added to the observed'weight of the object, is 7 - Note that w,, the weight of the object, and w,, the weight of the balance weights, differ only by the small buoyancy correction. The buoyancy correction expression can be simplified by substituting w, for w, Consideration of a typical example will demonstrate the validity of this substitution. We will assume that the object is about 25 rnl of water (do = 1.0 g/ml) and that the balance weights are of brass (d, = 8.4 g/ml). Assuming that the observed weight w,, is g, we can apply Eq. (3): The actual weight of the water, the weight that huld have been observed had it been weighed in a vacuum where the buoyancy is zero, is thus: = g. If we now use this as the weight of the object in Eq. (2), we find r -7 It is clear that the error introduced by using Eq. (3) is only g or 1 part in 250,000, which is insignificant when compared to the precision of the weight. volume, and other analytical measurements made in this course. The 'example can be used in a similar way to show that the approximate values of the densities of water and the weights do not introduce significant error. Buoyancy corrections need be applied only when objects of large volume are being weighed or when extremely high precision is required. An example of the first situation is calibration of volumetric glassware by weighing the water contained 'or delivered. The buoyancy correction in weighing small samples of chemicals is about 1 part in 5000, which is considerably smaller than the precision of most chemical analyses, and can be ignored. B-2 APPENDIXES

BAROMETER Because gas molecules possess mass they are attracted to the earth like all other. masses.

This pressure varies from place to place, primarily because of differences in the depth of the atmosphere at different altitudes.

Standard atmospheric pressure is 760 tom.

column of mercury at higher tempgatures.

C-2, is commonly used to measure the pressure of the atmosphere; The glass tube is evacuated and the open end immersed in mercury.

7 BAROMETER Because gas molecules possess mass they are attracted to the earth like all other. masses. The resulting weight of the atmosphere is transmitted to every surface with which the atmosphere comes in contact. It is most commonly measured as a force per unit area-an atmospheric pressure. This pressure varies from place to place, primarily because of differences in the depth of the atmosphere at different altitudes. For this reason a standard atmosphere has been defined as that pressure which will support a column of mercury cm high at 0 OC when the earth's gravitational constant is cm/sec2. Standard atmospheric pressure is 760 tom. One ton is exactly equal to the pressure exerted by a 1-mm column of mercury at 0 OC, but since the density of mercury decreases as the temperature increases, a given pressure will support a higher column of mercury at higher tempgatures. Furthermore, any scale by means of which the mercury column height is measured also expands with increasing temperature and for accurate work a correction is required. The barometer, Fig. C-1, p. C-2, is commonly used to measure the pressure of the atmosphere; The glass tube is evacuated and the open end immersed in mercury. The pressure of the ahnosphere forces mercury to rise in the tube until the force exerted by the atmosphere over the cross-sectianal area of the tube is balanced by the weight of the mercury column above the level of mercury in the reservoir. To use the barometer: 1. Adjust the mercury level in the reservoir so that the pointer just touches the surface. Take care to avoid parallax error by sighting horizontally across the mercury surface. Use the electric light to illuminate the reservoir and the white screen behind it. 2. Use the vernier adjustment knob to position the bottom of the vernier scale so that it appears to just touch the mercury meniscus. Sight horizontally, using the front and rear edges of the vernier to avoid parallax error. 3. Read the position of the bottom of the vernier scale on the centimeter scale to the nearest millimeter and use the vernier to read to the nearest tenth millimeter. See Appendix A on the use of the vernier scale. 4. Read the thermometer on the barometer and convert the height of the mercury column to pressure, using the equation PB(tom) = H(mm) [ 1 - at] where His the height of the column and t is the temperature of the barometer in "C. If the barometer has a brass scale the combined corrections for the density of mercury and the expansion of the scale give a the value 1.63 X lo4 per degree. If the scale is etched directly on the glass barometer tube the value of a is 1.72 X lo4 per degree. C-1

8 Vernier scale Centimeter scale - Glass ~ercury -f Reference point (ivory 1 ~eservoir level control knob (stainless steel) C-2 APPENDIXES. Figure C 1

ph METER ' The direct reading ph meter is a convenient instrum-ent which provides a very simple, rapid, and reliable 'means of deterinining the ph 'of a solution even though the solution be

Basically, it consists of (1) a glass electrode, (2) a calomel electrode, and (3) a digital voltmeter (DVM). The two electrodes are immersed in the solution who& ph is to be determined (Fie.

Shielded connecting cable Connecting wire Port for adding fresh KC1 solution Mixture of Hg and Hg2CJ2 Porous membrane Saturated KC1 solution Thin rnernbran Asbestos fiber for salt bridge Figure

9 ph METER ' The direct reading ph meter is a convenient instrum-ent which provides a very simple, rapid, and reliable 'means of deterinining the ph 'of a solution even though the solution be colored or contain strong oxidizing agents, reducing agents, or so-called. electrode poisons. It is also adaptable to very small samples. Basically, it consists of (1) a glass electrode, (2) a calomel electrode, and (3) a digital voltmeter (DVM). The two electrodes are immersed in the solution who& ph is to be determined (Fie. D-l), and the voltage of the galvanic cell that is thus formed is measured,with the DVMI. Shielded connecting cable Connecting wire Port for adding fresh KC1 solution Mixture of Hg and Hg2CJ2 Porous membrane Saturated KC1 solution Thin rnernbran Asbestos fiber for salt bridge Figure D - 1 The principles of galvanic cells are discussed in detail in your text and this paragraph, up to Eq. (I), will make sense to you only if you understand these principles. The calomel electrode involves the half-reaction 2 e- + Hg2C12 $ 2C Hg and the sliver-silver chloride electrode (inside the glass electrode) the half-reaction e- + AgCl 8 Cl- + Ag

10 Off-hand, there seems to be no reason at au why the voltage of this cell should depend on the H' concentration, because each of these electrodes is reversible only to C1-, which is kept constant at 0.1 M inside the glass electrode and at about /1(corresponding to saturated KC1) around the calomel electrode. At present the most generally accepted explanation attributes the sensitivity to H' concentration to the fact that the special glass at the end of the glass electrode has the unique property of exchanging some of the positive ions (such as Na" and K") in the e&emely thin gel layer at the glass surface for some of the H' ions in solution; as a result a "phase boundary potential (voltage)" is developed. The size of the potential that is created depends on the amount of Hf ions in solution. Ion exchange is discussed in your text. On one side of the glass surface the H' concentration remains constant (0.1 M HCl) and on the other side it varies with the Hf concentration in the beaker. The net result is that this thin glass membrane acts as though it were an electrode which is reversible to H', an electrode for which.we.could write the: usual halfcell potential equation for 25 "C. Ec = EE log [ H+] = EE ph This is a potential which we should add to that normally possessed by the Ag-AgC1 electrode inside the glass electrode. + EC I Ece11 = Edorne~ - [ E A~-A~CI EmIl = E LOmd - O,0591 log [ C E z,.a,cl log [ C1-Il - EE ph 1! Now, all of the EO values are constants, [C1-I2 3.5 and [C1-Il = 0.1, so we can lump all of these together in one constant, E O, and write The important conclusion to draw from this is that the voltage of the cell is a linear function of ph. Two major problems stand in the way of using this equation to de'termine the ph of the solution. The first of these concerns the fact that we do not really know the value of EO because it includes Ez which varies from electrode to electrode, and even for the same electrode it varies from day to day and depends on the kind of water soaking treatment it gets. Although it is somewhat of a nuisance, this difficulty is very simply overcome by immersing the electrode in a buffer solution of known ph and then making an adjustment on the ph meter (an internal resistance is varied) until the ph meter actually reads the ph value which you know that the buffer solution possesses; i.e., you calibrate the ph meter without ever really determining the value of EO. The meter scale is Linear so that if it reads correctly at one ph it will read correctly at the others as well, according to Eq. (1). The other problem is much more serious. The resistance of our cell is extremely high, that of the glass membrane itself being of the order of megohms (-2 X 10' to 2 X 10' ohms!). If we conriect this cell to a regular potentiometer, we will get no measurement at all because the current that comes from our cell wdd be so miniscule (of the order of 10-l2 amp) that the galvanometer would give no reading at any ph. This difficulty is surmounted by using a -~VM,'which is basically a voltage amplifier with an internal input resistance of megohms. This high input resistance allows the mexiurement of voltages by the DVM of high resistance E- 2 APPENDIXES

11 sources. After calibration as described in the previous paragraph, the meter can be made to read directly in ph. The glass electrode and its connecting wire are provided with unusually good shielding to prevent erratic and spurious readings due to stray currents and electrostatic charges; even the electrode itself has a silicone coating to minimize surface conduction along the glass. In order to get reliable readings at ph values of -10 or higher, or in solutions of high Na' concentration, it is necessary to use electrode glass containing Liz0 instead of Na20. The factor in Eq. (1) is specifically for 298 O K; we could make the equation more general by writing 1.98 X lo-'' T instead so that E,n = Ezll + (1.98 X (T) (ph) The ph meter has a temperature compensation knob which varies still another internal resistm'ce that enables the Dm 'to read the ph directly regardless of the temperature of the solution. One of the nice features of the ph meter is that it can also be used as a D'VM to read voltages directly from other sources. All you have to do is switch out the glass electrode and use the voltage scale directly. Because the DVM i has such high resistance and draws such a small current, it can be satisfactorily used in place of the potentiometer in the majority of applications. ph METER D-3

WHEATSTONE BRIDGE A Wheatstone bridge is a simple device.that will..measure the electrical resistance of an instrument.

in such a way that, when no current flows through the galvanometer, the resistance of the instrument is very simply related to the values of the other three resistances.

E- 1, a battery whose voltage is E is connected to X and Y, the terminals of the resistance R. When the switch is closed, current (i) flows through R according to Ohm's Law.

It is important to realize that the same current (i) flows through all parts of the circuit, through the copper wire connections with essentially no resistance as well as through R.

13 WHEATSTONE BRIDGE A Wheatstone bridge is a simple device.that will..measure the electrical resistance of an instrument. The resistance of the instrument in question is connected in a loop with three other known resistances (at least one of which can be varied), and combined with a sensitive galvanometer and a battery in such a way that, when no current flows through the galvanometer, the resistance of the instrument is very simply related to the values of the other three resistances. The principles of its operation are given below. 1. The important relationship to know is Ohm's Law. In Fig. E- 1, a battery whose voltage is E is connected to X and Y, the terminals of the resistance R. When the switch is closed, current (i) flows through R according to Ohm's Law. i i A i R Battery Switch - i -i - - Rm R" - i 111- i li Figure E - 1 Figure E-2 2. It is important to realize that the same current (i) flows through all parts of the circuit, through the copper wire connections with essentially no resistance as well as through R. It's like water flowing through a pipe; no matter what the various diameters (or resistances) of the pipes along the way, the same number of gallons per minute must go out the bottom end or pass any given point along the way as are put in at the top end. Even though two different separate resistances are placed between X and Y, as in Fig. E-2, the current through each is the same; Rm and R, are said to be connected in series. If we apply Ohm's Law to each of the resistances separately, we have Em =ir, and En = ir, If we apply Ohm's Law to the two together, E = i (R, + R,)

3. Instead of connecting two resistances in series between X and Y as in Fig. E-2, we might have connected two resistances "in parallel" as are R, and R, in Fig. E -3; i.e., we provide two different parallel paths by which current might flow between X and Y.

14 3. Instead of connecting two resistances in series between X and Y as in Fig. E-2, we might have connected two resistances "in parallel" as are R, and R, in Fig. E -3; i.e., we provide two different parallel paths by which current might flow between X and Y. Here, the total current flowing between X and Y must still be i; it must be the sum of the currents i, and i, flowing in the two branches. Figure E The W heatstone bridge combines the principl&s of 1-2, and 3 in an ingenious way, as in Fig. E-4; R, and R1 are connected in series, as are Re and RJ; R, and R1 are in parallel with Rz and RJ. Furthermore, a galvanometer (G) is connected at A and B. R, is the component whose unknown resistance is to be measured. The values of resistances R1, Re, and R3 are known. Figure E - 4. Wheatstone bridge. 5. When the switch is closed, the variable resistance (R2) is adjusted so that no current flows through the galvanometer, as indicated by no movement of the galvanometer needle. It is this recorded value of R2 which is used in the calculations below.

The consequences of no current flowing between A and B are as follows: (a) The voltages between X and A and between X and B must be the same; i.e., E, = El.

When Ohm's Law is applied to each of these resistances, these equalities lead to a very simple, useful relationship, as follows: R, = G2 = resistance of the component 3 6.

It makes no difference what the voltage of the battery is; it is often between 1.5 and G'volts.

15 The consequences of no current flowing between A and B are as follows: (a) The voltages between X and A and between X and B must be the same; i.e., E, = El. (b) The voltages between Y and A and between Y and B must be the same; i.e., El = Ed. (c) i, = il, and i2 = i3. When Ohm's Law is applied to each of these resistances, these equalities lead to a very simple, useful relationship, as follows: R, = G2 = resistance of the component 3 6. Note that the unit of resistance (R) is the'"ohm, the unit of potential (E) is the volt, and the unit of current (i) is the ampere. It makes no difference what the voltage of the battery is; it is often between 1.5 and G'volts. The accuracy of the measurement depends on the accuracy of the known resistances and the sensitivity of the galvanometer. The bridge described here is for direct current; it uses a battery. If alternating current is used, the resistances must be "non-inductively wound," provision must be made for balancing the capacitance of the circuit, and a more suitable null indicator than a galvanometer should be used. WHEATSTONE BRIDGE E-3

17 VAPOR PRESSURES OF WATER AT VARIOUS TEMPERATURES t, O C p, torr t, O C p, torr t, OC p, torr i I

+a~un AT 3.98" C. BY TEE ~ ~~~YATIONAL BUUAU or WEIC~TS AXD ME~BUX~S (1910).

19 DENSITIES OF WATER AT VARIOUS TEMPERATURES Dmrn ur GRAMS PER CWIC C E ~ ~ COMPUTED R, FROM Tgll hlatlte VALUES BY -EN SCHEEL AND HOEST (1900), rm, TEE AFJSOLOTE +a~un AT 3.98" C. BY TEE ~ ~~~YATIONAL BUUAU or WEIC~TS AXD ME~BUX~S (1910). SOURCE: Handbook of Chemistry and Physics, 52nd ed., The Chemical Rubber Co., Cleveland, Ohio.

21 t - VALUES FOR VARIOUS SAMPLE SIZES AND CONFIDENCE LEVELS DEGREES OF FREEDOM (f) Percenta~e Confidence Level OOO , , d f equals the sample size, n, minus the number of parameters. For example, if an average is determined from the data f = n - 1. If a slope and an intercept are determined by drawing the best line through the data, f = n - 2.

23 ATOMIC WEIGHTS OF THE ELEMENTS 1995 WAC Commission on Atomic Weights and lsotopic Abundances. Pure Appl. Chem., 68, (1996); 69, (1997). World Wide Web version of atomic weight data prepared by G. P. Moss, originally from a file provided by D. R Lide. ( SY mbol Ac A1 Am Sb Ar As At Ba Bk Be Bi Bh B Br Cd Cs Ca C f C Ce C1 Cr Co Cu Cm Db Dy Es Er Eu Fm F Fr Gd Ga Ge Name Actinium Aluminium Americium Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Bohrium Boron Bromine Cadmium Cesium Calcium Californium Carbon Cerium Chlorine Chromium Cobalt Copper Curium Dubnium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Atomic Wt [ (1) (1) (2) [ (7) ( (3) (2) (7) (1) (8) (2) (4) [ (8) (1) (9) (6) (9) (3) [2471 [ (3) [ (3) (1) [ (5) [ (3) (1) 72.6 l(2)

24 Name Gold Hafnium Hassium Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Meitnerium Mendelevium Mercury Molybdenum Neodymium Neon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Rutherfordium Samarium Scandium Seaborgium Selenium Atomic Wt (2) (2) (2) (2) (7) ' (3) ' (3) (3) (2) 83.80(1) (2) [ (1) 6.941(2) (1) (6) (9) [2661 [ (2) 95.94(1) (3) (6) [ (2) (2) (7) [ (3) (3) (1) (2) (2) [2441 ~ (1) (2) (2) [2261 [ (1) (2) (3) (2) [26 1 I (3) (8) [263 I 78.96(3)

25 Svmbol Si Ag Na Sr S Ta Tc 'Te - Tb TI Th Tm Sn Ti W u v Uun Uuu Xe Yb Y Zn Zr Name Silicon Silver Sodium Strontium Sulfur Tantalum Technetium Tellunum " Terbium Thallium Thorium Thulium Tin Titanium Tungsten Uranium Vanadium Ununnilium Unununium Xenon Ytterbium Yttrium Zinc Zirconium Atomic Wt (3) (2) (2) 87.62(1) (6) (1) [981 ' (3) (2) (2) (1) (2) (7) (1) (1) (1) (1) (2691 ( (2) (3) (2) 65.39(2) (2)

27 PERIODIC TABLE

This resource contains three different versions of the periodic table, including a blank one for colouring!

Teaching notes This resource contains three different versions of the periodic table, including a blank one for colouring! It also contains tables of the Group 0, 1 and 7 elements with a few columns for