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Molar Mass Calculations

Imagine you are working at a construction site and the engineer is a very precise man: he comes up with the estimate that you need to use 100 million grains of sand per batch of cement mixed. Now you might be tempted to start and actually count it out because you are paid an hourly wage, but if you need to do it efficiently there is another way.

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Jetzt kostenlos anmeldenImagine you are working at a construction site and the engineer is a very precise man: he comes up with the estimate that you need to use 100 million grains of sand per batch of cement mixed. Now you might be tempted to start and actually count it out because you are paid an hourly wage, but if you need to do it efficiently there is another way.

You can probably figure out how much an average grain of sand weighs by measuring 100 of them on a scale. Let's say you get 10 mg which means it's 1/10th of a mg per grain of sand. Then you can just calculate how many kilos of sand you need. In this case, you would need 10 kg of sand per batch of cement.A chemist faces this issue on a daily basis: we know how many atoms we need, but it is impossible for us to count them out one by one. So we invented something to deal with the issue. It's called the **atomic mass** (like the weight of an average grain of sand in the example above) and we use it to calculate how heavy a mole of those atoms is, i.e. how much they weigh. Then we can calculate how heavy a mole of molecules is by adding up atomic masses. This is called **molecular mass**. The units for both measures are atomic mass units.

**Atomic mass** is defined as the rest mass of an atom usually expressed in atomic units.

**Atomic weight **is what you would measure for the atomic mass near (or on the surface of ) a heavy object. Think really heavy like the earth. It is dependent on the mass of the earth, how far away you are from the earth's center, and the atomic mass.

When we do chemistry on planet Earth we talk about "atomic weight" but since we don't do chemistry on other planets or in space yet, we usually use the terms atomic** ***mass* and atomic** ***weight* interchangeably. From this point on **if we speak about molecular mass, we mean molecular weight**. This is also common in textbooks and as long as it is on Earth (as in most labs), you don't have to worry about this difference.

If this seems confusing to you, there are articles on our site to help you understand the basics before we can get to calculating: "Moles and Molar Mass" and "Avogadro's Number and the Mole" are good places to start. If you feel like you need to refresh your knowledge, go read them now and we will pick up from where we left off!

The following definitions come from the IUPAC gold book^{1}:

Molar mass is the ratio of the mass of the molecule compared to the unified atomic mass unit.

The atomic mass unit is a unit of mass (equal to the atomic mass constant), defined as** one-twelfth of the mass of a carbon-12 atom** in its ground state and used to express masses of atomic particles, u≈1.660 5402(10)×10^{−27 }kg.

In the periodic table, you will find the molar masses (or weights to be more precise) of the elements usually under their name. (see Fig. 1) If you then have a specific molecule consisting of more than one atom, like water (H_{2}O, three atoms) you can calculate how heavy a mole of it would be. Or even for a completely fictional compound you want to make e.g. StudySmarter-ium.

Calculating the molar mass of a compound is pretty straightforward, but I won't write it in the form of an equation (yet) because that makes it unnecessarily complicated and scary for most students. I will first give you a list of steps (kind of like an algorithm for a computer) to get to the final molar mass. If you repeat these steps , you will always be able to calculate the correct molar mass, no matter how complicated your compound is.

Water has the chemical formula H_{2}O which means it has 2 H atoms and 1 O atom.Similarly, amphetamine, an organic compound, has the chemical formula C_{9}H_{13}N meaning it has 9 C, 13 H and 1 N atom.

Let's do this one for the **ST**(i)**UDySmArTeR**(a) molecule (hint: look at the figure above). The numbers below the names of the elements are the average atomic masses, but in this case, we only have one of each, so there are no numbers in the compound formula.So in this case: 32.07 g/mol for Sulfur, 204.38 g/mol for Thallium, and so on until we get to 226.03 g/mol for Radium.

**Do it yourself for water and amphetamine, you will need it in the next step!**

**StudySmarter-ium has one of each element, so I just add up the atomic masses of all components** to get the molar mass of the compound: 1181.37 g/mol - ba heavy compound indeed!**Water is a bit trickier: **don't forget there are 2 hydrogens (H_{2}O)** **giving us 1.01 + 1.01 + 16.00 to get 18.02 g/mol as the molar mass. Or, simplified: 1.01*2 + 16.00 to get 18.02 g/mol as well.Lastly, our way too often abused pharmaceutical precursor, amphetamine. Consisting of 9 carbons, 13 hydrogens, and 1 nitrogen. Coming in with a molar mass of 9 * 12.01 + 13 * 1.01 + 1 * 14.01 = 135.23 g/mol.

Now you just rinse and repeat to calculate the molar mass of any compound. There will be a couple of examples of molar mass calculations at the end for you to try out and learn from.

There are actually a whole bunch of different types of weights and masses used to describe atoms and it can be important to know the difference between them. You might have even heard about some of them e.g. formula weight, monoisotopic mass, molecular mass, molecular weight, atomic weight, etc.Strictly speaking what we have been discussing (and what you need for your exam) is the **molar mass taking the actual formula of the compound** you are interested in and **adding up atomic masses**. Sometimes you build compounds with an unnatural isotope distribution, for example, a nuclear bomb. It contains Uranium which has an average weight closer to 235 g/mol. So, the way you make this... on second thought, I don't think I should teach you how to purify Uranium for your home atomic bomb shenanigans so I will omit that section here .The point is, in this case, you need to pay attention to which isotope of the compound you are using because each one would give a different final molar mass. In an extreme case, you will just have one isotope (such as in NMR solvents, at least very expensive ones) and then you would use the atomic weight of that specific isotope. For example, "heavy water" is water made up of Deuterium, which is just heavy hydrogen, but this tiny difference is a big deal when calculating the molar mass of heavy water. There are a lot of advanced applications for monoisotopic compounds (compounds with just one isotope in them), and you can learn more about these in the "Isotopes" article.

You want the formula, I understand, I also like formulas a lot. For a compound consisting of A_{Na}B_{Nb}C_{Nc}...._{ }atoms, you can use this formula to calculate the molar mass.

$\underset{}{\sum \mathrm{Ni}*\mathrm{Ar}\left(\mathrm{i}\right)}$$\sum _{i}{N}_{i}*A{r}_{i}$

where Ni is the number of i atoms in the formula of the compound and Ar(i) is the atomic weight of the atom i, for all atoms in the formula.

The $\sum _{}$ symbol just means the sum of everything included inside the symbol itself, in this case, the sum of the product of the number of atoms of each element in the compound and the atomic weight of said atom.

In other words, in the case of water:

${N}_{H}*Ar\left(H\right)+{N}_{O}*Ar\left(O\right)=2*1.01g/mol+1*16.00g/mol=18.02g/mol$

If this formula seems very complicated, just use the process described above, they are one and the same. The difference is this: at the top, it is described with words whilst here it is described with mathematical notation. Neither is better than the other!

So now you can calculate molar masses of compounds confidently, but what do you use this for? We can calculate the molar mass of any compound we need with a periodic table and we can use it to convert between grams of a substance and moles of a substance. **Chemistry works with moles but you can only measure grams or some related quantity, not moles.** Hence the importance of being able to change from one measure to the other.

Let me walk you through two examples of molar mass calculations to understand the relevance of molar masses in a chemist's daily life.

Let's say I know from a recent paper that my taste buds require 0.5 mol/L glucose (this measure is called concentration) to compensate for my bitter coffee in the morning. I also know that my coffee is on average 50 ml in volume, also known as 0.05 L. Multiplying the two together gives me the number of moles of sugar I need to put into my coffee in the morning, **r****oughly 0.025 mol** or 25 mmol.

How would you proceed? Multiplying by Avogadro's number (6.022x10^{23}) gives us 1.5055x10^{22} atoms of sugar molecules which is not too useful. What about the molar mass? The **formula of** **glucose is ****C**_{6}**H**_{12}**O**_{6}. Using a periodic table, we can calculate the molar mass of glucose. By using the steps or the formula above, we get to 180.16 g/mol for glucose:

**The molar mass comes out to approximately ****180.16 g/mol.**

The molar mass of 180.16 g/mol tells us that 1 mole of sugar weighs exactly 180.16 g, so can you figure out how much 0.025 mol weighs?

You are right, it *is* 180.16 g/mol x 0.025 mol, which is 4.5 g, or about 2 cubes of sugar.

And it works the other way around too! I just put 120 grams of sugar into a liter of coffee (it is exam period so don't judge me)! Will this taste sweet enough? Remember, we still are in need of 0.5 mol of sugar per liter of coffee.You can go about it this way: 120 g divided by the molar mass of glucose will give you 0.667 mol. Cool, what was our volume of coffee exactly? 1 L. So we basically added 0.667 mol/L of coffee. This will be sweet enough since it is more than 0.5 mol/L, which would be 90 g, actually.

You can notice an interesting phenomenon with the SI units in particular. They are designed in a way that you can do algebra with them. Simply treat gram, mole, liter, and so on as if they were variables like x, y, and z. Do you want to solve for grams? You can do it like this: g/mol * *something* = g. What could that *something *be?Naturally, you will find that to be moles. To get a clearer picture, substitute x for g and y for mol. Can you solve this equation?

x/y * something = x, yeah it is y and y is mol per definition.This is called dimensional analysis and can be very helpful if you know what kind of result ( for example grams, moles, liters, etc.) you are looking for.

I don't know about you, but for me what made chemistry interesting when I was a child were explosions. So let's calculate the molar mass of a couple of explosives just in case you like them too. Answers at the end of this section, but try to solve them yourself first.

There was once a show about a chemist with lung cancer cooking meth. It was pretty popular and in one of the iconic scenes, he used Mercury fulminate to demolish a building's upper floor. Let's calculate the molar mass though.

Now, Mercury fulminate can destroy a building but it doesn't work the way it was portrayed. If you just drop it, it won't explode on you.

Mercury fulminate has the chemical formula **Hg(CNO) _{2.}** This means that for every Mercury atom there are 2 carbon, 2 nitrogen, and 2 oxygen atoms. Can you calculate the molecular mass? And if I tell you that you need 2.5 mol of this stuff to force the bad guys to pay up for your meth, how many grams would you take with you?

Nitroglycerin is used to make dynamite by absorbing it on specific material. It is a very stable explosive. In its pure form, nitroglycerin is very easy to detonate, for example by accidentally knocking it over in the lab. A clumsy chemist just **knocked over 13 g** of it and we know that** 0.1 mol would definitely kill him**, but** **below that amount, he survives. The formula for nitroglycerin is **C**_{3}**H**_{5}**N**_{3}**O**_{9}, so should I call an ambulance?

Calculate the molar mass first!

There are more molar mass calculations on the flashcards if you feel like practicing some more.**Answers:**

**th****is is how much a mole of it weighs.**2.5 mol would then weigh 2.5 times as much, roughly 711.6 g.284.62 g/mol * 2.5 mol = 711.6 g- Nitroglycerin has a molar mass of 227.09 g/mol. Here you have two options:
- You can calculate how much 0.1 mol weighs: 0.1 mol would weigh 22.7 g, so he survives. 227.09g/mol * 0.1 mol = 22.7 g
- The other option is like this: How many moles are there in 13 g of nitroglycerin? You know that 1 mol is 227.09 g, so you can just divide 13 g by 227.09 g/moles to get the number of moles in that amount, it works both ways! You would get 0.057 moles, less than 0.1, so again, he survives. M(Nitroglycerin) = 227.09 g/mol , m(Nitroglycerin) = 13.0 g, n(Nitroglycen) = ? moln = m/M = (13.0 g) / (227.09 g/mol) = 0.057 mol

I hope you understand molar mass calculations now. I'll just go fetch that ambulance for my clumsy colleague. He might not die, but the burns are going to be nasty!

- Molar mass is important because counting up more than a couple hundred million atoms one by one would be very annoying and time-consuming. The molar mass
**tells you how many grams of atoms you need**instead of how many pieces of atoms. - Molar mass is calculated by adding up the sum of the atomic mass of each element in the compound.
- Multiplying a molar amount (number of atoms in a compound) by their molar mass gives you weight (grams). Similarly, by dividing a mass (grams) by the molar mass (g/mol), you get a molar amount.

- IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Oxford 1997; ISBN 0-9678550-9-8., https://doi.org/10.1351/goldbook.

** You only need to add up the atomic masses of all the atoms composing the molecule** to get the molar mass of the compound.

**Molar mass is usually used in stoichiometric calculations to convert between the units of moles and grams. **The__ chemical reactions__ __are__ fundamentally __dependent on moles__ but __in the lab__ we prefer to __use grams __since that is something __we can measure with a scale__, thus we need a conversion between the two.

_{2}S is as simple as adding up the atomic masses of the atoms from which the molecule is built. In this case, this means **two lithium and one sulfur atom** with atomic masses of *6.94* and *32.07 *respectively*.***Molar mass of Li _{2}S = 2 * 6.94 + 32.07 = 45.95 g/mol**

** You only need to add up the atomic masses of all the atoms composing the molecule** to get the molar mass of the compound.

_{2}S. It is done in the following way:

The process of calculating the molar mass of Li_{2}S is as simple as adding up the atomic masses of the atoms from which the molecule is built. In this case, this means **two lithium and one sulfur atom** with atomic masses of *6.94* and *32.07 *respectively*.***Molar mass of Li _{2}S = 2 * 6.94 + 32.07 = 45.95 g/mol**

C_{2}H_{5}N(CH_{2}CH_{2}Cl)_{2 }is a type of nitrogen mustard mainly which was used in World War I to kill a lot of soldiers. However, I use it in the lab to create catalysts. I forgot to write up the molar mass. Can you help me figure it out?

Well, first you have to count up all the elements in the molecule. Remember everything in the brackets is counted as many times as the subscript of the brackets. So, there are 6 carbons, 13 hydrogens, 2 chlorines and 1 nitrogen. This gives us a molar mass of 170.08 g/mol.

Calculate the molar mass of water H_{2}O and heavy water D_{2}O.

Hint: The atomic weight of Deuterium (D) is not in the periodic table. You can google it or treat it as 2.0 for this exercise.

Water consists of 2 hydrogens and one oxygen while heavy water consists of 2 deuteriums and one oxygen. Now we have to add up the atomic weights ( 1+1+16 for water and 2+2+16 for heavy water). So, you end up with 18 g/mol for water while for heavy water you get 20g/mol.

What is the molar mass of H_{2}?

It is 2 g/mole. You can get this by adding up the atomic mass 1 twice.

What is the atomic weight of Li_{2}S?

It is 45.95 g/mol. You can get this by adding up the atomic weights of 2 lithium atoms and 1 sulfur atom.

One of the most important proteins in Sars-CoV-2 (Covid-19 virus) has the composition of :

Carbon-1499 pieces, Hydrogen-2318 pieces, Nitrogen-402 pieces, Oxygen-445 pieces, Sulfur-22 pieces.

Yes, this is just one big molecule. Can you tell me the molar mass of this monstrosity?

Just like with any other compound, we can simply add up everything.

1499 * 12 + 2318 * 1 + 402 * 14 + 445 * 16 +22 * 32 = 33758 g/mol

If you consume 4 coffees at once, this is equivalent to 400mg of caffeine. You can survive up to 10 millimoles at once. **(fictional value, don't try this!) **

The molar mass of caffeine is 194.19 g/mol. Will you be okay?

Dividing 400 milligrams by 194.19 grams per mole gives us 0.0021 moles or 2.1 mmoles, so most likely you will be okay.

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