# 9.3: Mole-Mass and Mass-Mass Calculations

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Learning Objectives

- From a given number of moles of a substance, calculate the mass of another substance involved using the balanced chemical equation.
- From a given mass of a substance, calculate the moles of another substance involved using the balanced chemical equation.
- From a given mass of a substance, calculate the mass of another substance involved using the balanced chemical equation.

Mole-mole calculations are not the only type of calculations that can be performed using balanced chemical equations. Recall that the molar mass can be determined from a chemical formula and used as a conversion factor. We can add that conversion factor as another step in a calculation to make a **mole-mass calculation**, where we start with a given number of moles of a substance and calculate the mass of another substance involved in the chemical equation, or vice versa.

For example, suppose we have the balanced chemical equation

\[2Al + 3Cl_2 → 2AlCl_3\nonumber \]

Suppose we know we have 123.2 g of Cl_{2}. How can we determine how many moles of AlCl_{3} we will get when the reaction is complete? First and foremost, *chemical equations are not balanced in terms of grams; they are balanced in terms of moles*. So to use the balanced chemical equation to relate an amount of Cl_{2} to an amount of AlCl_{3}, we need to convert the given amount of Cl_{2} into moles. We know how to do this by simply using the molar mass of Cl_{2} as a conversion factor. The molar mass of Cl_{2} (which we get from the atomic mass of Cl from the periodic table) is 70.90 g/mol. We must invert this fraction so that the units cancel properly:

\[123.2\, \cancel{g\, Cl_{2}}\times \frac{1\, mol\, Cl_{2}}{70.90\cancel{g\, Cl_{2}}}=1.738\, \, mol\, Cl_{2}\nonumber \]

Now that we have the quantity in moles, we can use the balanced chemical equation to construct a conversion factor that relates the number of moles of Cl_{2} to the number of moles of AlCl_{3}. The numbers in the conversion factor come from the coefficients in the balanced chemical equation:

\[\frac{2\, mol\, AlCl_{3}}{3\, mol\, Cl_{2}}\nonumber \]

Using this conversion factor with the molar quantity we calculated above, we get

\[1.738\, \cancel{mol\, Cl_{2}}\times \frac{2\, mol\, AlCl_{3}}{3\, \cancel{mol\, Cl_{2}}}=1.159\, mol\, AlCl_{3}\nonumber \]

So, we will get 1.159 mol of AlCl_{3} if we react 123.2 g of Cl_{2}.

In this last example, we did the calculation in two steps. However, it is mathematically equivalent to perform the two calculations sequentially on one line:

\[123.2\, \cancel{g\, Cl_{2}}\times \frac{1\, mol\, Cl_{2}}{70.90\cancel{g\, Cl_{2}}}\times \frac{2\, mol\, AlCl_{3}}{3\, \cancel{mol\, Cl_{2}}}=1.159\, mol\, AlCl_{3}\nonumber \]

The units still cancel appropriately, and we get the same numerical answer in the end. Sometimes the answer may be slightly different from doing it one step at a time because of rounding of the intermediate answers, but the final answers should be effectively the same.

Example \(\PageIndex{1}\):

How many moles of HCl will be produced when 249 g of AlCl_{3} are reacted according to this chemical equation?

\[2AlCl_3 + 3H_2O(ℓ) → Al_2O_3 + 6HCl(g)\nonumber \]

**Solution**

We will do this in two steps: convert the mass of AlCl_{3} to moles and then use the balanced chemical equation to find the number of moles of HCl formed. The molar mass of AlCl_{3} is 133.33 g/mol, which we have to invert to get the appropriate conversion factor:

\[249\, \cancel{g\, AlCl_{3}}\times \frac{1\, mol\, AlCl_{3}}{133.33\, \cancel{g\, AlCl_{3}}}=1.87\, mol\, AlCl_{3}\nonumber \]

Now we can use this quantity to determine the number of moles of HCl that will form. From the balanced chemical equation, we construct a conversion factor between the number of moles of AlCl_{3} and the number of moles of HCl:

\[frac{6\, mol\, HCl}{2\, \cancel{mol\, AlCl_{3}}}\nonumber \]

Applying this conversion factor to the quantity of AlCl_{3}, we get

\[1.87\, \cancel{mol\, AlCl_{3}}\times \frac{6\, mol\, HCl}{2\, \cancel{mol\, AlCl_{3}}}=5.61\, mol\, HCl\nonumber \]

Alternatively, we could have done this in one line:

\[249\, \cancel{g\, AlCl_{3}}\times \frac{1\, \cancel{mol\, AlCl_{3}}}{133.33\, \cancel{g\, AlCl_{3}}}\times \frac{6\, mol\, HCl}{2\, \cancel{mol\, AlCl_{3}}}=5.60\, mol\, HCl\nonumber \]

The last digit in our final answer is slightly different because of rounding differences, but the answer is essentially the same.

Exercise \(\PageIndex{1}\)

How many moles of Al_{2}O_{3} will be produced when 23.9 g of H_{2}O are reacted according to this chemical equation?

\[2AlCl_3 + 3H_2O(ℓ) → Al_2O_3 + 6HCl(g)\nonumber \]

**Answer**-
0.442 mol

A variation of the mole-mass calculation is to start with an amount in moles and then determine an amount of another substance in grams. The steps are the same but are performed in reverse order.

Example \(\PageIndex{2}\)

How many grams of NH_{3} will be produced when 33.9 mol of H_{2} are reacted according to this chemical equation?

\[N_2(g) + 3H_2(g) → 2NH_3(g)\nonumber \]

**Solution**

The conversions are the same, but they are applied in a different order. Start by using the balanced chemical equation to convert to moles of another substance and then use its molar mass to determine the mass of the final substance. In two steps, we have

\[33.9\cancel{mol\, H_{2}}\times \frac{2\, mol\, NH_{3}}{3\cancel{mol\, H_{2}}}=22.6\, mol\, NH_{3}\nonumber \]

Now, using the molar mass of NH_{3}, which is 17.03 g/mol, we get

\[22.6\cancel{mol\, NH_{3}}\times \frac{17.03\, g\, NH_{3}}{1\cancel{mol\, NH_{3}}}=385\, g\, NH_{3}\nonumber \]

Exercise \(\PageIndex{2}\)

How many grams of N_{2} are needed to produce 2.17 mol of NH_{3} when reacted according to this chemical equation?

\[\ce{N_2(g) + 3H_2(g) → 2NH_3(g)} \nonumber \]

**Answer**-
30.4 g (Note: here we go from a product to a reactant, showing that mole-mass problems can begin and end with any substance in the chemical equation.)

It should be a trivial task now to extend the calculations to **mass-mass calculations**, in which we start with a mass of some substance and end with the mass of another substance in the chemical reaction. For this type of calculation, the molar masses of two different substances must be used—be sure to keep track of which is which. Again, however, it is important to emphasize that before the balanced chemical reaction is used, the mass quantity must first be converted to moles. Then the coefficients of the balanced chemical reaction can be used to convert to moles of another substance, which can then be converted to a mass.

For example, let us determine the number of grams of SO_{3} that can be produced by the reaction of 45.3 g of SO_{2} and O_{2}:

\[ 2SO_2(g) + O_2(g) → 2SO_3(g)\nonumber \]

First, we convert the given amount, 45.3 g of SO_{2}, to moles of SO_{2} using its molar mass (64.06 g/mol):

\[45.3\cancel{g\, SO_{2}}\times \frac{1\, mol\, SO_{2}}{64.06\cancel{g\, SO_{2}}}=0.707\, mol\, SO_{2}\nonumber \]

Second, we use the balanced chemical reaction to convert from moles of SO_{2} to moles of SO_{3}:

\[0.707\cancel{mol\, SO_{2}}\times \frac{2\, mol\, SO_{3}}{2\cancel{mol\, SO_{2}}}=0.707\, mol\, SO_{3}\nonumber \]

Finally, we use the molar mass of SO_{3} (80.06 g/mol) to convert to the mass of SO_{3}:

\[0.707\cancel{mol\, SO_{3}}\times \frac{80.06\, g\, SO_{3}}{1\cancel{mol\, SO_{3}}}=56.6\, g\, SO_{3}\nonumber \]

We can also perform all three steps sequentially, writing them on one line as

\[45.3\cancel{g\, SO_{2}}\times \frac{1\, mol\, SO_{2}}{64.06\cancel{g\, SO_{2}}}\times \frac{2\, mol\, SO_{3}}{2\cancel{mol\, SO_{2}}}\times \frac{80.06\, g\, SO_{3}}{1\cancel{mol\, SO_{3}}}=56.6\, g\, SO_{3}\nonumber \]

We get the same answer. Note how the initial and all the intermediate units cancel, leaving grams of SO_{3}, which is what we are looking for, as our final answer.

Example \(\PageIndex{3}\):

What mass of Mg will be produced when 86.4 g of K are reacted?

\[\ce{MgCl2(s) + 2K(s) → Mg(s) + 2KCl(s)}\nonumber \]

**Solution**

We will simply follow the steps

mass K → mol K → mol Mg → mass Mg

In addition to the balanced chemical equation, we need the molar masses of K (39.09 g/mol) and Mg (24.31 g/mol). In one line,

\[86.4\cancel{g\, K}\times \frac{1\, mol\, K}{39.09\cancel{g\, K}}\times \frac{1\, \cancel{mol\, Mg}}{2\cancel{mol\, K}}\times \frac{24.31\, g\, Mg}{1\cancel{mol\, Mg}}=26.87\, g\, Mg\nonumber \]

Exercise \(\PageIndex{3}\)

What mass of H_{2} will be produced when 122 g of Zn are reacted?

\[\ce{Zn(s) + 2HCl(aq) → ZnCl_2(aq) + H_2(g)} \nonumber \]

**Answer**-
3.77 g

## Summary

- Mole quantities of one substance can be related to mass quantities using a balanced chemical equation.
- Mass quantities of one substance can be related to mass quantities using a balanced chemical equation.
- In all cases, quantities of a substance must be converted to moles before the balanced chemical equation can be used to convert to moles of another substance.