difference between experimental and theoretical values chemistry

Note:   Find out which of these versions your syllabus is likely to want you to know (even if they just call it "lattice enthalpy") and concentrate on that one, but be aware of the confusion!

Incidentally, if you are ever uncertain about which version is being used, you can tell from the sign of the enthalpy change being discussed. If the sign is positive, for example, it must refer to breaking bonds, and therefore to a lattice dissociation enthalpy.

Factors affecting lattice enthalpy

The two main factors affecting lattice enthalpy are the charges on the ions and the ionic radii (which affects the distance between the ions).

The charges on the ions

Sodium chloride and magnesium oxide have exactly the same arrangements of ions in the crystal lattice, but the lattice enthalpies are very different.

Note:   In this diagram, and similar diagrams below, I am not interested in whether the lattice enthalpy is defined as a positive or a negative number - I am just interested in their relative sizes. Strictly speaking, because I haven't added a sign to the vertical axis, the values are for lattice dissociation enthalpies. If you prefer lattice formation enthalpies, just mentally put a negative sign in front of each number.

You can see that the lattice enthalpy of magnesium oxide is much greater than that of sodium chloride. That's because in magnesium oxide, 2+ ions are attracting 2- ions; in sodium chloride, the attraction is only between 1+ and 1- ions.

The radius of the ions

The lattice enthalpy of magnesium oxide is also increased relative to sodium chloride because magnesium ions are smaller than sodium ions, and oxide ions are smaller than chloride ions.

That means that the ions are closer together in the lattice, and that increases the strength of the attractions.

You can also see this effect of ion size on lattice enthalpy as you go down a Group in the Periodic Table.

For example, as you go down Group 7 of the Periodic Table from fluorine to iodine, you would expect the lattice enthalpies of their sodium salts to fall as the negative ions get bigger - and that is the case:

Attractions are governed by the distances between the centres of the oppositely charged ions, and that distance is obviously greater as the negative ion gets bigger.

And you can see exactly the same effect as you go down Group 1. The next bar chart shows the lattice enthalpies of the Group 1 chlorides.

Note:   To save anyone the bother of getting in touch with me to point it out, it's not strictly fair to include caesium chloride in this list. Caesium chloride has a different packing arrangement of ions in its crystal, and that has a small effect on the lattice enthalpy. The effect is small enough that it doesn't actually affect the trend.

Calculating lattice enthalpy

It is impossible to measure the enthalpy change starting from a solid crystal and converting it into its scattered gaseous ions. It is even more difficult to imagine how you could do the reverse - start with scattered gaseous ions and measure the enthalpy change when these convert to a solid crystal.

Instead, lattice enthalpies always have to be calculated, and there are two entirely different ways in which this can be done.

You can can use a Hess's Law cycle (in this case called a Born-Haber cycle) involving enthalpy changes which can be measured. Lattice enthalpies calculated in this way are described as experimental values.

Or you can do physics-style calculations working out how much energy would be released, for example, when ions considered as point charges come together to make a lattice. These are described as theoretical values. In fact, in this case, what you are actually calculating are properly described as lattice energies .

Note:   If you aren't confident about Hess's Law cycles , it is essential that you follow this link before you go on.

Experimental values - Born-Haber cycles

Standard atomisation enthalpies

Before we start talking about Born-Haber cycles, there is an extra term which we need to define. That is atomisation enthalpy , ΔH° a .

You are always going to have to supply energy to break an element into its separate gaseous atoms.

All of the following equations represent changes involving atomisation enthalpy:

Notice particularly that the "mol -1 " is per mole of atoms formed - NOT per mole of element that you start with. You will quite commonly have to write fractions into the left-hand side of the equation. Getting this wrong is a common mistake.

Born-Haber cycles

I am going to start by drawing a Born-Haber cycle for sodium chloride, and then talk it through carefully afterwards. You will see that I have arbitrarily decided to draw this for lattice formation enthalpy. If you wanted to draw it for lattice dissociation enthalpy, the red arrow would be reversed - pointing upwards.

Focus to start with on the higher of the two thicker horizontal lines. We are starting here with the elements sodium and chlorine in their standard states. Notice that we only need half a mole of chlorine gas in order to end up with 1 mole of NaCl.

The arrow pointing down from this to the lower thick line represents the enthalpy change of formation of sodium chloride.

The Born-Haber cycle now imagines this formation of sodium chloride as happening in a whole set of small changes, most of which we know the enthalpy changes for - except, of course, for the lattice enthalpy that we want to calculate.

The +107 is the atomisation enthalpy of sodium. We have to produce gaseous atoms so that we can use the next stage in the cycle.

The +496 is the first ionisation energy of sodium. Remember that first ionisation energies go from gaseous atoms to gaseous singly charged positive ions.

The +122 is the atomisation enthalpy of chlorine. Again, we have to produce gaseous atoms so that we can use the next stage in the cycle.

The -349 is the first electron affinity of chlorine. Remember that first electron affinities go from gaseous atoms to gaseous singly charged negative ions.

And finally, we have the positive and negative gaseous ions that we can convert into the solid sodium chloride using the lattice formation enthalpy.

Note:   If you have forgotten about ionisation energies or electron affinities follow these links before you go on.

Now we can use Hess's Law and find two different routes around the diagram which we can equate.

As I have drawn it, the two routes are obvious. The diagram is set up to provide two different routes between the thick lines.

So, here is the cycle again, with the calculation directly underneath it . . .

-411 = +107 + 496 + 122 - 349 + LE

LE = -411 - 107 - 496 - 122 + 349

Note:   Notice that in the calculation, we aren't making any assumptions about the sign of the lattice enthalpy (despite the fact that it is obviously negative because the arrow is pointing downwards). In the first line of the calculation, I have just written "+ LE", and have left it to the calculation to work out that it is a negative answer.

How would this be different if you had drawn a lattice dissociation enthalpy in your diagram? (Perhaps because that is what your syllabus wants.)

Your diagram would now look like this:

The only difference in the diagram is the direction the lattice enthalpy arrow is pointing. It does, of course, mean that you have to find two new routes. You can't use the original one, because that would go against the flow of the lattice enthalpy arrow.

This time both routes would start from the elements in their standard states, and finish at the gaseous ions.

-411 + LE = +107 + 496 + 122 - 349

LE = +107 + 496 + 122 - 349 + 411

LE = +787 kJ mol -1

Once again, the cycle sorts out the sign of the lattice enthalpy for you.

Note:   You will find more examples of calculations involving Born-Haber cycles in my chemistry calculations book . This includes rather more complicated cycles involving, for example, oxides.

If you compare the figures in the book with the figures for NaCl above, you will find slight differences - the main culprit being the electron affinity of chlorine, although there are other small differences as well. Don't worry about this - the values in the book come from an older data source. In an exam, you will just use the values you are given, so it isn't a problem.

Theoretical values for lattice energy

Let's assume that a compound is fully ionic. Let's also assume that the ions are point charges - in other words that the charge is concentrated at the centre of the ion. By doing physics-style calculations, it is possible to calculate a theoretical value for what you would expect the lattice energy to be.

And no - I am not being careless about this! Calculations of this sort end up with values of lattice energy , and not lattice enthalpy . If you know how to do it, you can then fairly easily convert between the two.

There are several different equations, of various degrees of complication, for calculating lattice energy in this way. You won't be expected to be able to do these calculations at this level, but you might be expected to comment on the results of them.

There are two possibilities:

There is reasonable agreement between the experimental value (calculated from a Born-Haber cycle) and the theoretical value.

Sodium chloride is a case like this - the theoretical and experimental values agree to within a few percent. That means that for sodium chloride, the assumptions about the solid being ionic are fairly good.

The experimental and theoretical values don't agree.

A commonly quoted example of this is silver chloride, AgCl. Depending on where you get your data from, the theoretical value for lattice enthalpy for AgCl is anywhere from about 50 to 150 kJ mol -1 less than the value that comes from a Born-Haber cycle.

In other words, treating the AgCl as 100% ionic underestimates its lattice enthalpy by quite a lot.

The explanation is that silver chloride actually has a significant amount of covalent bonding between the silver and the chlorine, because there isn't enough electronegativity difference between the two to allow for complete transfer of an electron from the silver to the chlorine.

Comparing experimental (Born-Haber cycle) and theoretical values for lattice enthalpy is a good way of judging how purely ionic a crystal is.

Note:   If you have forgotten about electronegativity it might pay you to revise it now by following this link.

Why is magnesium chloride MgCl 2 ?

This section may well go beyond what your syllabus requires. Before you spend time on it, check your syllabus (and past exam papers as well if possible) to make sure.

The question arises as to why, from an energetics point of view, magnesium chloride is MgCl 2 rather than MgCl or MgCl 3 (or any other formula you might like to choose).

It turns out that MgCl 2 is the formula of the compound which has the most negative enthalpy change of formation - in other words, it is the most stable one relative to the elements magnesium and chlorine.

Let's look at this in terms of Born-Haber cycles.

In the cycles this time, we are interested in working out what the enthalpy change of formation would be for the imaginary compounds MgCl and MgCl 3 .

That means that we will have to use theoretical values of their lattice enthalpies. We can't use experimental ones, because these compounds obviously don't exist!

I'm taking theoretical values for lattice enthalpies for these compounds that I found on the web. I can't confirm these, but all the other values used by that source were accurate. The exact values don't matter too much anyway, because the results are so dramatically clear-cut.

We will start with the compound MgCl, because that cycle is just like the NaCl one we have already looked at.

The Born-Haber cycle for MgCl

Find two routes around this without going against the flow of any arrows. That's easy:

ΔH f = +148 + 738 + 122 - 349 - 753

ΔH f = -94 kJ mol -1

So the compound MgCl is definitely energetically more stable than its elements.

I have drawn this cycle very roughly to scale, but that is going to become more and more difficult as we look at the other two possible formulae. So I am going to rewrite it as a table.

You can see from the diagram that the enthalpy change of formation can be found just by adding up all the other numbers in the cycle, and we can do this just as well in a table.

The Born-Haber cycle for MgCl 2

The equation for the enthalpy change of formation this time is

So how does that change the numbers in the Born-Haber cycle?

You need to add in the second ionisation energy of magnesium, because you are making a 2+ ion.

You need to multiply the atomisation enthalpy of chlorine by 2, because you need 2 moles of gaseous chlorine atoms.

You need to multiply the electron affinity of chlorine by 2, because you are making 2 moles of chloride ions.

You obviously need a different value for lattice enthalpy.

You can see that much more energy is released when you make MgCl 2 than when you make MgCl. Why is that?

You need to put in more energy to ionise the magnesium to give a 2+ ion, but a lot more energy is released as lattice enthalpy. That is because there are stronger ionic attractions between 1- ions and 2+ ions than between the 1- and 1+ ions in MgCl.

So what about MgCl 3 ? The lattice energy here would be even greater.

The Born-Haber cycle for MgCl 3

So how does that change the numbers in the Born-Haber cycle this time?

You need to add in the third ionisation energy of magnesium, because you are making a 3+ ion.

You need to multiply the atomisation enthalpy of chlorine by 3, because you need 3 moles of gaseous chlorine atoms.

You need to multiply the electron affinity of chlorine by 3, because you are making 3 moles of chloride ions.

You again need a different value for lattice enthalpy.

This time, the compound is hugely energetically unstable, both with respect to its elements, and also to other compounds that could be formed. You would need to supply nearly 4000 kJ to get 1 mole of MgCl 3 to form!

Look carefully at the reason for this. The lattice enthalpy is the highest for all these possible compounds, but it isn't high enough to make up for the very large third ionisation energy of magnesium.

Why is the third ionisation energy so big? The first two electrons to be removed from magnesium come from the 3s level. The third one comes from the 2p. That is closer to the nucleus, and lacks a layer of screening as well - and so much more energy is needed to remove it.

The 3s electrons are screened from the nucleus by the 1 level and 2 level electrons. The 2p electrons are only screened by the 1 level (plus a bit of help from the 2s electrons).

Magnesium chloride is MgCl 2 because this is the combination of magnesium and chlorine which produces the most energetically stable compound - the one with the most negative enthalpy change of formation.

Where would you like to go now?

To the chemical energetics menu . . .

To the Physical Chemistry menu . . .

To Main Menu . . .

© Jim Clark 2010 (modified July 2013)

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Theoretical vs Experimental Lattice Energy

In this JC2 webinar we want to compare theoretical and experimental lattice energy.

Let Chemistry Guru, Singapore's renowned JC Chemistry tuition centre, guide you through this question.

We are asked to explain the differences between the experimental and theoretical values of lattice energy for AgF and AgI.

Experimental or actual lattice energy is determined from the Born-Haber cycle of an ionic compound.

Check out this video if you are interested to know how to draw Born-Haber cycle .

Theoretical or calculated lattice energy is based on the following formula assuming a highly ionic compound.

A highly ionic compound has only electrostatic attraction between cations and anions.

There is no distortion of electron clouds of the anions and no covalent character.

Therefore the difference between theoretical and experimental lattice energy tells us the extent of covalent character in that ionic compound.

If LE(experimental) is very close to LE(theoretical), the assumption that the ionic compound is highly ionic is true.

Therefore the ionic compound is highly ionic with little or no covalent character.

Ionic compounds that are highly ionic consist of: - cations with low charge density and polarising power - anions that are small and non-polarisable

If LE(experimental) is significantly different from LE(theoretical), the assumption that the ionic compound is highly ionic is false.

Therefore the ionic compound has significant covalent character.

Ionic compounds that have covalent character consist of: - cations with high charge density and polarising power - anions that are large and polarisable

Comparing AgF and AgI, larger iodide ion is more polarisable and there is a greater distortion of electron cloud of iodide.

AgI will have more significant covalent character than AgF hence a greater difference between theoretical and experimental lattice energy.

Topic: Energetics, Physical Chemistry, A Level Chemistry, Singapore

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Sources of Error in Science Experiments

Science labs usually ask you to compare your results against theoretical or known values. This helps you evaluate your results and compare them against other people’s values. The difference between your results and the expected or theoretical results is called error. The amount of error that is acceptable depends on the experiment, but a margin of error of 10% is generally considered acceptable. If there is a large margin of error, you’ll be asked to go over your procedure and identify any mistakes you may have made or places where error might have been introduced. So, you need to know the different types and sources of error and how to calculate them.

How to Calculate Absolute Error

One method of measuring error is by calculating absolute error , which is also called absolute uncertainty. This measure of accuracy is reported using the units of measurement. Absolute error is simply the difference between the measured value and either the true value or the average value of the data.

absolute error = measured value – true value

For example, if you measure gravity to be 9.6 m/s 2 and the true value is 9.8 m/s 2 , then the absolute error of the measurement is 0.2 m/s 2 . You could report the error with a sign, so the absolute error in this example could be -0.2 m/s 2 .

If you measure the length of a sample three times and get 1.1 cm, 1.5 cm, and 1.3 cm, then the absolute error is +/- 0.2 cm or you would say the length of the sample is 1.3 cm (the average) +/- 0.2 cm.

Some people consider absolute error to be a measure of how accurate your measuring instrument is. If you are using a ruler that reports length to the nearest millimeter, you might say the absolute error of any measurement taken with that ruler is to the nearest 1 mm or (if you feel confident you can see between one mark and the next) to the nearest 0.5 mm.

How to Calculate Relative Error

Relative error is based on the absolute error value. It compares how large the error is to the magnitude of the measurement. So, an error of 0.1 kg might be insignificant when weighing a person, but pretty terrible when weighing a apple. Relative error is a fraction, decimal value, or percent.

Relative Error = Absolute Error / Total Value

For example, if your speedometer says you are going 55 mph, when you’re really going 58 mph, the absolute error is 3 mph / 58 mph or 0.05, which you could multiple by 100% to give 5%. Relative error may be reported with a sign. In this case, the speedometer is off by -5% because the recorded value is lower than the true value.

Because the absolute error definition is ambiguous, most lab reports ask for percent error or percent difference.

How to Calculate Percent Error

The most common error calculation is percent error , which is used when comparing your results against a known, theoretical, or accepted value. As you probably guess from the name, percent error is expressed as a percentage. It is the absolute (no negative sign) difference between your value and the accepted value, divided by the accepted value, multiplied by 100% to give the percent:

% error = [accepted – experimental ] / accepted x 100%

How to Calculate Percent Difference

Another common error calculation is called percent difference . It is used when you are comparing one experimental result to another. In this case, no result is necessarily better than another, so the percent difference is the absolute value (no negative sign) of the difference between the values, divided by the average of the two numbers, multiplied by 100% to give a percentage:

% difference = [experimental value – other value] / average x 100%

Sources and Types of Error

Every experimental measurement, no matter how carefully you take it, contains some amount of uncertainty or error. You are measuring against a standard, using an instrument that can never perfectly duplicate the standard, plus you’re human, so you might introduce errors based on your technique. The three main categories of errors are systematic errors, random errors , and personal errors. Here’s what these types of errors are and common examples.

Systematic Errors

Systematic error affects all the measurements you take. All of these errors will be in the same direction (greater than or less than the true value) and you can’t compensate for them by taking additional data. Examples of Systematic Errors

• If you forget to calibrate a balance or you’re off a bit in the calibration, all mass measurements will be high/low by the same amount. Some instruments require periodic calibration throughout the course of an experiment , so it’s good to make a note in your lab notebook to see whether the calibrations appears to have affected the data.
• Another example is measuring volume by reading a meniscus (parallax). You likely read a meniscus exactly the same way each time, but it’s never perfectly correct. Another person taking the reading may take the same reading, but view the meniscus from a different angle, thus getting a different result. Parallax can occur in other types of optical measurements, such as those taken with a microscope or telescope.
• Instrument drift is a common source of error when using electronic instruments. As the instruments warm up, the measurements may change. Other common systematic errors include hysteresis or lag time, either relating to instrument response to a change in conditions or relating to fluctuations in an instrument that hasn’t reached equilibrium. Note some of these systematic errors are progressive, so data becomes better (or worse) over time, so it’s hard to compare data points taken at the beginning of an experiment with those taken at the end. This is why it’s a good idea to record data sequentially, so you can spot gradual trends if they occur. This is also why it’s good to take data starting with different specimens each time (if applicable), rather than always following the same sequence.
• Not accounting for a variable that turns out to be important is usually a systematic error, although it could be a random error or a confounding variable. If you find an influencing factor, it’s worth noting in a report and may lead to further experimentation after isolating and controlling this variable.

Random Errors

Random errors are due to fluctuations in the experimental or measurement conditions. Usually these errors are small. Taking more data tends to reduce the effect of random errors. Examples of Random Errors

• If your experiment requires stable conditions, but a large group of people stomp through the room during one data set, random error will be introduced. Drafts, temperature changes, light/dark differences, and electrical or magnetic noise are all examples of environmental factors that can introduce random errors.
• Physical errors may also occur, since a sample is never completely homogeneous. For this reason, it’s best to test using different locations of a sample or take multiple measurements to reduce the amount of error.
• Instrument resolution is also considered a type of random error because the measurement is equally likely higher or lower than the true value. An example of a resolution error is taking volume measurements with a beaker as opposed to a graduated cylinder. The beaker will have a greater amount of error than the cylinder.
• Incomplete definition can be a systematic or random error, depending on the circumstances. What incomplete definition means is that it can be hard for two people to define the point at which the measurement is complete. For example, if you’re measuring length with an elastic string, you’ll need to decide with your peers when the string is tight enough without stretching it. During a titration, if you’re looking for a color change, it can be hard to tell when it actually occurs.

Personal Errors

When writing a lab report, you shouldn’t cite “human error” as a source of error. Rather, you should attempt to identify a specific mistake or problem. One common personal error is going into an experiment with a bias about whether a hypothesis will be supported or rejects. Another common personal error is lack of experience with a piece of equipment, where your measurements may become more accurate and reliable after you know what you’re doing. Another type of personal error is a simple mistake, where you might have used an incorrect quantity of a chemical, timed an experiment inconsistently, or skipped a step in a protocol.

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