Measuring Mass, Volume, Temperature and Density, and Knowing How Wrong You Are

A number written down without its uncertainty is not a measurement. It is a claim, and usually a larger claim than the instrument can support. “2 g of metol” from a balance that reads whole grams means somewhere between 1 g and 3 g; the same words from a balance reading to a milligram mean something a thousand times tighter. The words are identical and the two solutions are different developers.

By the end of this page you should be able to pick the right vessel for a quantity, read it without lying to yourself, say how wrong each of your instruments is and how you know, carry that uncertainty through a mixing sequence, and decide which of your errors is worth spending money on.

The instruments, and what each one is actually for

Instrument	What it is for	What it is not for
Balance	every solid, and any liquid you would rather weigh than measure	nothing; a balance is the most trustworthy instrument you will own
Beaker or jug	holding, stirring, dissolving	measuring. Its printed graduations are a rough guide
Graduated cylinder (a graduate)	measuring out a volume of liquid you are going to use	holding a reaction, or storing anything
Volumetric flask	making a solution up to one exact final volume	measuring out portions; it has one mark and one purpose
Pipette or syringe	small volumes, 0.5 to 50 ml, where a graduate’s error would dominate	anything that must be sterile or that you then re-use across families
Thermometer	the temperature of a solution, at the moment you read it	the temperature of the room, or of a tank you measured ten minutes ago
Magnetic stirrer	dissolving without splashing, and holding a solution uniform while you read its temperature	speed; it will not make an insoluble thing dissolve
Storage bottle	keeping a made-up solution	measuring, and never for anything unlabelled

Two entries in that table deserve their own paragraph.

A beaker is not a measuring device. Its graduations are moulded or printed on a wide, straight cylinder, and the reason they cannot be precise is geometry rather than manufacturing. Take a beaker of 70 mm internal diameter: its cross-section is π × 35² ≈ 3,850 mm², so 1 mm of height is 3.85 ml. A graduate of 26 mm internal diameter has a cross-section of about 530 mm², so the same 1 mm of height is 0.53 ml. Reading either one to the nearest millimetre — which is about as well as anyone reads a scale — gives an error seven times larger in the beaker, purely because the liquid is spread out. That is why the same volume is measured in the narrowest vessel that will hold it, and why the last stage of making up a solution happens in a volumetric flask, whose neck is narrower still.

ILFORD says the same thing from the other end, in its introduction to film process control: use appropriately sized and calibrated measuring cylinders and vessels for the volume being dispensed, because it is easier and more accurate to measure 100 ml into a 100 ml measuring cylinder than into a 1000 ml one. Same liquid, same operator, different vessel, different answer.

Why the same reading error is a different volume in each vessel

1. 1Beaker, about 70 mm across — 1 mm of height is about 3.85 ml — a holding vessel, not a measuring one
2. 2Graduate, about 26 mm across — 1 mm of height is about 0.53 ml — the working measuring vessel
3. 3Volumetric flask, neck about 10 mm across — 1 mm of height is about 0.08 ml — one mark, one volume, made up to

Diameters are stated as assumptions and the volumes follow from them arithmetically; measure your own vessels and the ranking will not change.

!What this course does not tell you about tolerance classes

Volumetric glassware is manufactured to tolerance classes defined in international standards, and a class A flask promises a tighter tolerance than a class B one. This course has not read those standards — they are sold rather than published — and so it quotes no tolerance figure for any class or size, and no “class B 100 ml flask is ±0.3 ml” table.

What it tells you to do instead is better anyway. Look at the vessel: good glassware carries its nominal volume, a reference temperature, a marking that says whether it is calibrated to contain or to deliver, and often the tolerance itself. If your vessel carries none of that, you do not know its tolerance from the vessel — so measure it. The commissioning lab weighs the water your own graduate delivers, which tests the vessel you actually own rather than the class somebody printed on a box.

To contain, and to deliver

A vessel calibrated to contain a volume holds that volume when filled to the mark. A vessel calibrated to deliver discharges that volume when emptied in the specified way. They are not the same number, and the reason is not subtle: liquid wets glass and plastic, and a film of it stays behind when you pour. Fill a volumetric flask to the mark with 250 ml and pour it out, and rather less than 250 ml arrives. Fill a pipette to its mark and let it drain, and the design has already allowed for the film.

The practical rule follows without any standard being quoted. Make solutions up to volume in the vessel; measure portions out with something designed to deliver them. And when you transfer something you weighed — a solid dissolved in a beaker, say — rinse the beaker into the flask with part of the water you were going to add anyway. ILFORD’s own mixing instructions do exactly this: measure the concentrate, use some of the dilution water to rinse out the measuring cylinder into the mixing vessel, then add the rest of the water to make up to the final working volume.

Reading a meniscus

Water in glass climbs the wall, so its surface is a curve. The reading is taken at the bottom of the curve, with your eye level with it.

Reading a meniscus, and the two ways of getting it wrong

1. 1Eye above: the meniscus looks higher than it is — the volume is read too large — and filling to a mark, you stop short
2. 2Eye level with the bottom of the curve: correct — the front and back of the graduation ring line up as a single mark
3. 3Eye below: the meniscus looks lower than it is — the volume is read too small — and filling to a mark, you overfill

Bring the graduate to your eye level, not your eye to the graduate. Standing over a cylinder on a bench is the commonest reason two people reading the same vessel disagree.

A general chemistry text puts the resolution question well: a numerical scale generally permits a reading to one-tenth of its smallest division. A graduate marked in 1 ml divisions is therefore read to the nearest 0.1 ml, with the tenths digit estimated — and it would be pointless to attempt a hundredths digit, because the tenths are already uncertain. That single rule settles most arguments about how many figures to write down.

Balances

Three properties, and people use one word for all three.

• Resolution (or readability) is the size of the smallest change the display can show. A balance that reads to 0.01 g has a resolution of 0.01 g. That is a fact about the display.
• Repeatability (a kind of precision) is how closely repeated weighings of the same object agree. You can measure this yourself in five minutes.
• Accuracy (or trueness) is how close the reading is to the real mass. You cannot measure it at all without a mass you already trust.

A general chemistry text draws precision and accuracy with arrows in a target: arrows tightly grouped and on the bull’s eye are accurate and precise; tightly grouped and off to one side are precise but not accurate; scattered are neither. The photographic versions of those three targets are worth naming, because you will produce all of them.

Four ways a measurement can be wrong, and the darkroom version of each

1. 1Accurate and precise — the batch matches the formula and matches last time
2. 2Precise, not accurate — systematic error — a balance reading heavy every time: repeatable, and repeatably wrong
3. 3Accurate on average, not precise — random error — sloppy volume reading: no two batches alike, average about right
4. 4Neither — no calibration and no discipline; nothing can be diagnosed

Systematic error is found by comparison with a reference and can be corrected. Random error is reduced by technique, or averaged down by repeating. Only one of the two is fixed by buying a better instrument.

Why a 0.01 g balance is not enough for 0.2 g of phenidone

A general chemistry text’s convention for a balance reading is that the last displayed digit is uncertain, so a reading of 6.72 g carries a nominal uncertainty of ±0.01 g. Apply that to two weighings a darkroom actually makes:

🧮The same balance, two very different measurements

A balance with a resolution of 0.01 g, taking the display alone as ±0.01 g:

• Weighing 100 g of sodium sulfite: 0.01 / 100 = 0.01 per cent. Irrelevant. Your spatula carries away more than that.
• Weighing 2.00 g of metol: 0.01 / 2.00 = 0.5 per cent. Comfortable.
• Weighing 0.20 g of phenidone: 0.01 / 0.20 = 5 per cent — and that is the floor, from the display alone, before repeatability, drift, draughts or static are counted.

The same argument disposes of the kitchen scale. A kitchen scale that reads to 1 g, used to weigh 0.5 g, gives 1 / 0.5 = 200 per cent. That is not a measurement with a large error in it. It is not a measurement.

Phenidone is used at a few tenths of a gram per litre, and a developing agent at 5 per cent out changes the developer. That is the argument for a 0.001 g balance for the small ingredients, or for the alternative the concentration page gives: weigh a larger mass of the substance into a stock solution and measure out a portion of the solution, moving the precision from your balance to your pipette.

!Minimum weighable mass, and what this course does not quote

Laboratory practice expresses this as a minimum weighable quantity: the smallest mass you may weigh on a given balance if the result is to be within some stated percentage, derived from the balance’s repeatability rather than from its display. This course could not read a manufacturer’s repeatability or linearity specification for the inexpensive balances sold to photographers, and therefore quotes no minimum weighable mass and no rule of thumb. The arithmetic above is a floor: your balance is at least this uncertain, and probably worse. Measure your own repeatability — the recipe is in the commissioning lab — and you will have a number that is about your balance rather than about balances in general.

Weighing done properly

1. Level, still surface. Not on the drainer, not on a fridge, not next to an open window. A balance responds to draughts and to a leaning elbow.
2. Let it settle, then zero it. Then place the empty vessel and tare. Taring is subtracting the vessel’s mass, which is why you never weigh onto the pan.
3. Add the solid slowly. You cannot take it back, and scraping some out again puts it into the air.
4. Weigh by difference for anything hygroscopic or expensive. Weigh the source container, tip some out, weigh it again; the difference is what left, and the substance never sat exposed on a pan. This also defeats static, which makes light powders creep.
5. Transfer quantitatively. Rinse the weighing vessel into the mixing vessel with some of the water the recipe already calls for. What stays in the beaker is missing from the solution.
6. Write the reading down before you do anything else — the actual reading, not the number you were aiming at. “2.03 g” and “2 g” are different records, and only one of them lets you explain a result later.

Temperature

Which instrument. ILFORD’s process-control guide recommends a good-quality liquid-in-glass thermometer to check the calibration of any built-in temperature sensor, and gives the reason plainly: the advantage of a thermometer over an electronic probe is that very little can go wrong with it. A digital probe is faster and easier to read in dim light; a glass thermometer is the reference you check the probe against. Own both if you can, and know which is which.

Where and when you read it. Four habits, each defeating a specific error:

• Immersion depth. A thermometer measures the temperature of the part of it that is in the liquid, plus some of the stem in the air. Put it in as far as its design intends and keep that depth the same every time, so that whatever error remains is at least the same error.
• Thermal lag. A thermometer reads its own temperature, not the liquid’s, until the two have equalised. Wait, watch the reading stop moving, and only then read it. A cold probe dropped into 20 °C developer reads low for the first half-minute.
• Stir before you read. An unstirred tank is stratified: warmer at the top, colder against the wall it is leaning on. The number you want is the average the film sees, and stirring is how you get it.
• Read the solution, not the room. A tray of developer standing in a 17 °C room is not at 20 °C ten minutes after you poured it, whatever the bottle was at.

Checking against a fixed point. A thermometer that reads consistently is precise; a thermometer that reads correctly is accurate; you find out which you have by putting it somewhere whose temperature is fixed by physics rather than by another thermometer. A general chemistry text gives the two fixed points: water freezes at 0 °C and boils at 100 °C.

• The ice point. A slurry of crushed ice and water, stirred, sits at the melting point of ice as long as both phases are present. Crushed rather than cubed, so the water and ice are in contact everywhere; stirred, so there is no warm layer; and the bulb held in the slurry rather than touching the vessel. This is the check to trust, and the commissioning lab performs it.
• The steam point. Boiling water is a fixed point only at a stated pressure, and the pressure where you live is neither standard nor constant. This course has not verified an altitude or pressure correction table and therefore does not give one. Use the boiling point as a coarse check that the thermometer’s span is not badly wrong — a thermometer reading 0 °C in ice and 93 °C in boiling water is faulty, not merely uncorrected — and never as a calibration. It also introduces the only real hazard on that page, which is a scald.

📓What a calibration entry looks like

Every check produces one line, and the line is worth more than the check.

Date	Instrument	Check	Reading	Reference	Deviation	Action
2026-09-04	Digital probe, kitchen drawer	ice point, stirred slurry	+0.6 °C	0.0 °C	+0.6 °C	subtract 0.6 from readings; re-check in 3 months

The Action column is the one people leave out and the one that matters. A deviation you recorded and did not act on is a deviation you will meet again in a year, in a negative you cannot explain.

Density and specific gravity

Density is mass per unit volume. Specific gravity is the same thing expressed as a ratio to water, and ILFORD’s process-control guide defines it for photographic use in one sentence: the weight of the solution compared with the same volume of water, so that since 1 litre of water weighs 1000 g, 1 litre of a solution of specific gravity 1.08 to 1.09 weighs 1080 to 1090 g. That is its worked example, for a rapid fixer at 1+4.

Two things follow.

First, a formula stated by volume of a concentrate is not the same instruction as by mass. Glacial acetic acid is the standing example, because it is appreciably denser than water: CAMEO gives its density as 8.8 lb per US gallon, which is about 1.05 g per millilitre. So 100 ml of it is about 105 g, and a formula that says “10 ml of glacial acetic acid” and a formula that says “10 g” differ by 5 per cent before you have made a single reading error. Whenever a source gives a liquid by volume, write down that it was by volume.

Second, specific gravity is a free check on a dilution. ILFORD publishes it for its own solutions, and the published figures behave exactly as the arithmetic predicts:

Developer	Stock	1+1	1+3
PERCEPTOL	1.110	1.058	1.028
ID-11	1.090	1.047	1.022
MICROPHEN	1.095	1.050	1.024

Take ID-11: the stock’s excess over water is 0.090. Halve it, as diluting one volume of stock with one of water should, and you predict 1.045 against a published 1.047. Quarter it, for 1+3, and you predict 1.0225 against a published 1.022. A hydrometer costing very little — ILFORD suggests one covering 1.000 to 1.200 — will tell you whether the bottle in your hand really is the dilution the label says. This is the only instrument in this part that checks a solution rather than an instrument.

Units, and the traps in them

• Grams are not millilitres. They coincide for water and for nothing else. 100 ml of glacial acetic acid is about 105 g; 100 ml of a strong sulfite solution is more than 100 g.
• Degrees Celsius only. Fahrenheit appears in this course only inside a quotation from a source that used it.
• 1 g/L and 0.1 % w/v are the same concentration, because per cent w/v means grams per 100 ml. The trap is the factor of ten: 1 % w/v is 10 g/L, not 1 g/L. The concentration page owns this arithmetic; it is listed here because it is a measurement error as often as a calculation one.
• Teaspoons never appear in this course. A spoon measures volume; a formula specifies mass; and the bulk density of a powder depends on its crystal size, its moisture and whether the spoon is level. Two spoonfuls of the same sulfite from two different tubs are not the same mass, and this course has no source that would let it tell you by how much.

🕰The unit trap is older than you are

Wall’s 1912 dictionary gives the British system a photographer of the day had to hold in his head: formulae written in apothecaries’ weight, where an ounce is 480 grains; chemicals sold by avoirdupois weight, where an ounce is 437½ grains; and liquids measured in fluid ounces of 480 minims. Three ounces, one word. Its metric conversion table gives 1 gramme as 15.432 grains and 1 litre as 35 ounces and 94 minims — from which, since a fluid ounce is 480 minims, the imperial fluid ounce works out at 28.4 ml and the grain at 64.8 mg. Every one of those numbers is a chance to be 10 per cent wrong, and the concentration page shows a case where a period source was.

Uncertainty, and how it travels

Significant figures are the crude version, and they are enough most of the time

A general chemistry text’s rules are the ones to use: when adding or subtracting, round to the same number of decimal places as the least precise term; when multiplying or dividing, round to the same number of significant figures as the least precise term. A result calculated from a measurement is at least as uncertain as the measurement.

That last sentence is why the third digit is so often a lie. Weigh 2.0 g on a balance reading to 0.1 g, make it up to 100 ml in a graduate you have never checked, and your calculator will offer 2.0000 % w/v. You are entitled to 2 % w/v, and you should write it that way, because the extra digits claim a precision the graduate cannot deliver.

Where the uncertainty actually goes

For anything made by multiplying and dividing — which is every concentration — it is the relative uncertainties that matter, and in the worst case they add.

C = m / V, so ΔC/C ≤ Δm/m + ΔV/V

Worst-case relative uncertainty of a concentration

C is the concentration, m the mass you weighed and V the final volume; Δm and ΔV are the uncertainties in those two, and ΔC is the resulting uncertainty in the concentration. The sign is “less than or equal to” because this is a bound: it is the answer you get if every error happens to fall the same way. Independent errors partly cancel, so the true combined uncertainty is smaller. This course teaches the bound rather than the statistical combination because the bound is arithmetic anyone can check, and because a proper treatment of how independent uncertainties combine belongs to a metrology text this course has not read.

🧮Carrying the uncertainty through a real sequence

Make a 10 % w/v stock and dilute it to a 2 % working solution.

Step 1, the stock. Weigh 10.0 g on a balance reading to 0.1 g: Δm/m = 0.1/10.0 = 1 %. Make up to 100 ml in a graduate you have checked to about ±1 ml: ΔV/V = 1/100 = 1 %. The stock’s concentration is uncertain by at most 1 + 1 = 2 %. So it is 10.0 % w/v ± 0.2, and writing “10.00 %” was never available to you.

Step 2, the dilution. Take 50 ml of stock with the same graduate (±1 ml, 2 %) and make up to 250 ml in a 250 ml graduate (±2 ml, 0.8 %). The dilution contributes at most 2.8 per cent.

Step 3, the total. The working solution’s concentration inherits the stock’s uncertainty and adds its own: at most 2 + 2.8 = 4.8 per cent. Nominally 2.00 % w/v; honestly 2.0 % w/v, give or take about 0.1.

Notice which step dominates. It is not the balance. It is the 50 ml measured in a 100 ml graduate, and the fix costs nothing: measure 50 ml in a 50 ml graduate, or measure the stock by weight instead. Uncertainty analysis earns its place by telling you which of your habits to change.

Which error is actually worth chasing

Three errors compete for your attention in every processing run: concentration, temperature and time. They are not equal, and there is published data for one of them.

Development time against temperature, for two nominal times at 20 °C

• Nominal 12 minutes at 20 °C
• Nominal 8 minutes at 20 °C

Show the numbers behind this plot

Two falling curves read from ILFORD's published film development time and temperature compensation chart. The lower curve is a development that takes 8 minutes at 20 degrees Celsius; the chart gives 9 minutes 45 seconds at 18 degrees, 8 minutes 45 at 19, 8 minutes at 20, 7 minutes 15 at 21, 6 minutes 30 at 22, 5 minutes 30 at 24, 5 minutes at 25 and 4 minutes 15 at 27. The upper curve is a development that takes 12 minutes at 20 degrees: 14 minutes 45 at 18, 13 minutes 15 at 19, 12 minutes at 20, 10 minutes 45 at 21, 9 minutes 45 at 22, 8 minutes 15 at 24, 7 minutes 30 at 25 and 6 minutes 30 at 27. Both curves fall steeply and by similar proportions, roughly a tenth of the time for each degree near 20 degrees. The single most useful reading is around the 20 degree mark on the lower curve: one degree warmer requires 45 seconds less development and one degree cooler requires 45 seconds more, on a nominal 8 minutes, which is a change of about 9 per cent in either direction.

Series	Processing temperature, °C	Development time required, minutes
Nominal 12 minutes at 20 °C	18.00	14.75
Nominal 12 minutes at 20 °C	19.00	13.25
Nominal 12 minutes at 20 °C	20.00	12.00
Nominal 12 minutes at 20 °C	21.00	10.75
Nominal 12 minutes at 20 °C	22.00	9.75
Nominal 12 minutes at 20 °C	24.00	8.25
Nominal 12 minutes at 20 °C	25.00	7.50
Nominal 12 minutes at 20 °C	27.00	6.50
Nominal 8 minutes at 20 °C	18.00	9.75
Nominal 8 minutes at 20 °C	19.00	8.75
Nominal 8 minutes at 20 °C	20.00	8.00
Nominal 8 minutes at 20 °C	21.00	7.25
Nominal 8 minutes at 20 °C	22.00	6.50
Nominal 8 minutes at 20 °C	24.00	5.50
Nominal 8 minutes at 20 °C	25.00	5.00
Nominal 8 minutes at 20 °C	27.00	4.25

Points are read from ILFORD's published chart, not modelled: the chart's own worked example, 8 minutes at 20 °C becoming 5 minutes 30 seconds at 24 °C, is the sixth point on the lower curve. The chart is offered by ILFORD as a guide for all film and developer combinations, so treat it as the shape of the relation rather than as a specification for your film.

Read the lower curve at 20 °C. A development nominally of 8 minutes requires 8 minutes 45 seconds at 19 °C and 7 minutes 15 seconds at 21 °C. So:

• A 1 °C temperature error is worth about 9 per cent of development time. If you do not compensate, that is the size of the development error you have made.
• A 5 per cent timing error is worth 5 per cent — smaller than one degree, and much easier to control, since a phone will time to the second.
• A 2 per cent concentration error cannot be put on this axis, and this course will not invent a conversion. It found no manufacturer statement of an acceptable concentration tolerance for a working solution, and no published measurement of the density change a small concentration error produces. Part IX measures it, with the course’s own step wedge and densitometer, and until then the honest statement is: temperature is the one with a published number against it, and it is the largest of the three for the smallest-sounding cause.

The practical conclusion is the one that surprises people. The best value in a darkroom is not a better balance; it is a thermometer you have checked and a water bath. A degree is worth nine times what a second is.

The three checks you can actually perform

Every one of these produces a line in the calibration log, and all three are done in the commissioning lab.

1. The balance, against a known mass. Place a mass you trust; read; record the deviation. This tests trueness. If you have no calibration mass, you can still test repeatability: weigh one object ten times, removing and replacing it each time, and record the spread. Those are different tests and they answer different questions — repeatability without a reference tells you the balance is consistent, and says nothing about whether it is right.

✕A coin is not a calibration mass, and this course does not publish a coin table

The obvious substitute is a coin of known nominal mass. This course could not verify current United Kingdom coin masses or their manufacturing tolerances from the issuing authority, and therefore publishes no coin table. Two further objections stand whatever the nominal figure is: circulating coins wear, and they carry grease and dirt. A coin is a serviceable repeatability object — it is the same mass each of the ten times you weigh it — and a poor accuracy reference. Calibration masses are inexpensive; the planner lists them.

2. The thermometer, at the ice point. Crushed ice, water, stirred, bulb immersed and not touching the vessel. Wait for the reading to stop moving. Record the reading, the reference (0 °C) and the deviation. If the deviation is stable, it is a systematic error and you can subtract it.

3. The graduate, gravimetrically. Fill to a mark, weigh what comes out, and use ILFORD’s own figure — 1 litre of water weighs 1000 g — to convert. A graduate that says 100 ml and delivers 96 g of water is delivering about 96 ml, and now you know. This one check tests the vessel you actually own, in the way you actually use it, and it subsumes every question about tolerance classes that this page declined to answer.

Recap

• A measurement without an uncertainty is a claim. Write the reading you got, not the number you wanted, and let the instrument decide how many digits you are entitled to.
• Resolution is what the display shows, repeatability is how consistent it is, accuracy is whether it is right. You can measure the second yourself; the third needs a reference.
• The same 1 mm reading error is about 3.85 ml in a beaker, 0.53 ml in a graduate and 0.08 ml in a volumetric flask’s neck. Measure in the narrowest vessel that will hold it.
• To contain is not to deliver, because a film of liquid stays behind. Make up to volume in the vessel; rinse a weighing vessel into the batch with water the recipe already calls for.
• A 0.01 g balance weighing 0.2 g is 5 per cent uncertain from the display alone; a 1 g kitchen scale weighing 0.5 g is not making a measurement.
• Read a meniscus at its lowest point, with the vessel at your eye level. Read a scale to a tenth of its smallest division and no further.
• Relative uncertainties add through a mixing sequence, in the worst case. The step that dominates is usually a volume measured in a vessel far larger than it.
• ILFORD’s compensation chart makes a 1 °C error worth about 9 per cent of development time, against 5 per cent for a 5 per cent timing error. Buy the thermometer.
• Three checks, three log lines: known mass on the balance, ice point for the thermometer, gravimetric check for the graduate.

Check your understanding

Question 1. A balance reading to 0.01 g is used to weigh 2.00 g of metol and 100.00 g of sodium sulfite for the same litre of developer. Taking the display alone as the uncertainty, which weighing matters more, and why?

• The sulfite, because it is the larger mass and therefore the larger absolute error
• The metol, because 0.01 g on 2.00 g is 0.5 per cent while 0.01 g on 100 g is 0.01 per cent, and it is the relative error that changes the developer
• Neither; both are within the balance's specification
• The sulfite, because it is the preservative and the developer fails without it

Show the answer and why

Answer: The metol, because 0.01 g on 2.00 g is 0.5 per cent while 0.01 g on 100 g is 0.01 per cent, and it is the relative error that changes the developer

The absolute uncertainty is the same 0.01 g in both cases - that is what a fixed resolution means - so the two weighings differ only in what that 0.01 g is a fraction of. Fifty times more of it lands on the metol. This is the general rule for a formula: the small ingredients set the precision you need from your balance, which is why a developing agent used at a few tenths of a gram is the one that decides whether a 0.001 g balance is worth buying. Note also that "within specification" is not an answer to the question asked: a specification tells you the instrument is behaving, not that the result is precise enough for the use.

Question 2. Your graduate is marked in 2 ml divisions. To how many figures should you record a volume from it, and why?

• To the nearest 2 ml, because you cannot read between the marks
• To the nearest 0.2 ml, estimating a tenth of the smallest division, with that last digit understood to be uncertain
• To the nearest 0.01 ml, because your calculator will use that many figures anyway
• To whatever precision the recipe asks for

Show the answer and why

Answer: To the nearest 0.2 ml, estimating a tenth of the smallest division, with that last digit understood to be uncertain

The convention a general chemistry text gives is that a numerical scale permits a reading to about one-tenth of its smallest division, with that final digit estimated and understood to be uncertain. On 2 ml divisions that is 0.2 ml. Two of the wrong answers are the two directions of the same mistake: refusing to estimate throws away information you really do have, and writing 0.01 ml claims information you do not. The last option inverts the logic - a recipe cannot grant your vessel a precision it does not have; if the recipe needs more, you need a different vessel.

Question 3. A student weighs 10.0 g on a balance reading to 0.1 g and makes it up to 100 ml in a graduate they have checked to about a millilitre, then takes 50 ml of that with the same 100 ml graduate and makes it up to 250 ml. Which single change most reduces the uncertainty in the final concentration?

• Buying a balance that reads to 0.001 g
• Measuring the 50 ml portion in a 50 ml graduate rather than in the 100 ml one
• Using distilled water instead of tap water
• Making 500 ml instead of 250 ml

Show the answer and why

Answer: Measuring the 50 ml portion in a 50 ml graduate rather than in the 100 ml one

Work in relative terms. The balance contributes 0.1 in 10.0, or 1 per cent. The 100 ml make-up contributes about 1 per cent. The 50 ml portion, measured in a vessel whose uncertainty is about a millilitre, contributes 1 in 50, or 2 per cent - the largest single term, and it comes from using a vessel twice the size of the volume being measured. Improving the balance attacks a 1 per cent term and costs money; changing the graduate attacks a 2 per cent term and costs nothing. Distilled water is a contamination question rather than an uncertainty one, and scaling the batch up changes every term in proportion, so it changes nothing at all.

Question 4. A thermometer reads +0.6 °C in a stirred slurry of crushed ice and water on three separate occasions. What kind of error is this, and what do you do?

• A random error; average more readings until it goes away
• A systematic error; record the deviation in the calibration log and subtract 0.6 °C from readings until the next check
• The thermometer is broken and should be replaced
• Nothing: 0.6 °C is too small to matter in photographic processing

Show the answer and why

Answer: A systematic error; record the deviation in the calibration log and subtract 0.6 °C from readings until the next check

Repeating the same offset three times is the signature of a systematic error, and a systematic error is the good kind: it is found by comparison with a reference and then corrected arithmetically. Averaging cannot remove it, because it is present in every reading. The last option is the one worth arguing with: ILFORD's compensation chart makes 1 °C worth about 9 per cent of development time, so 0.6 °C is worth roughly 5 per cent - the same size as a timing error most people would take trouble over. Record it, subtract it, and re-check on a date you write down.

Question 5. A formula calls for 15 ml of glacial acetic acid. You have a balance but no small graduate, so you weigh 15 g instead. What have you done?

• Nothing; grams and millilitres are interchangeable for liquids
• Under-dosed the acid by about 5 per cent, because glacial acetic acid has a density near 1.05 g/ml, so 15 ml would have weighed about 15.8 g
• Over-dosed the acid by about 5 per cent
• Made an error too small to have any effect

Show the answer and why

Answer: Under-dosed the acid by about 5 per cent, because glacial acetic acid has a density near 1.05 g/ml, so 15 ml would have weighed about 15.8 g

CAMEO gives the density of glacial acetic acid as 8.8 lb per US gallon, about 1.05 g per millilitre, so a millilitre of it weighs more than a millilitre of water. Weighing out the same number of grams as the recipe asked for in millilitres therefore delivers less acid, by about the 5 per cent that the density exceeds 1. The size of the error is not the interesting part - the habit is. Whenever a source specifies a liquid by volume, record that it was by volume, and convert with a density you can cite rather than assuming the liquid behaves like water.

Question 6. ILFORD publishes ID-11 developer with a specific gravity of 1.090 as stock and 1.047 at 1+1. Why is that pair of numbers a useful check on your own dilutions?

• Because specific gravity is a direct measure of developer activity
• Because the excess of the stock over water, 0.090, halves to 0.045 when equal volumes of stock and water are mixed, and the published 1.047 confirms it, so a hydrometer reading tells you whether the bottle really is the dilution its label claims
• Because specific gravity changes with temperature and can be used as a thermometer
• Because a hydrometer measures concentration in per cent w/v directly

Show the answer and why

Answer: Because the excess of the stock over water, 0.090, halves to 0.045 when equal volumes of stock and water are mixed, and the published 1.047 confirms it, so a hydrometer reading tells you whether the bottle really is the dilution its label claims

The check works because the density excess over water is very nearly proportional to how much solute is present, so halving the solute should halve the excess - and ILFORD's own published figures do behave that way, within about two parts in a thousand, across all three of its powder developers at both 1+1 and 1+3. That makes a cheap hydrometer an independent test of an arithmetic step you might have got wrong. It is not a measure of activity: an oxidised developer of the right composition has the same density and does not work. And it reads specific gravity, not per cent w/v; converting between them needs the same relation you have just used.

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