Quantitative Chemistry: Counting Atoms and Calculating Yields
Welcome to Quantitative Chemistry! Don't let the long name scare you. "Quantitative" just means "measuring things." In this chapter, we move beyond simply describing chemical reactions and start calculating exactly how much substance reacts or is produced. This is essential for chemists, engineers, and manufacturers—it helps them save money and resources!
We will learn how to:
- Understand the golden rule of chemistry: Conservation of Mass.
- Calculate the "weight" of atoms and molecules using Relative Masses.
- Figure out the percentage of a useful element in a compound.
- Use balanced equations to predict masses.
1. The Golden Rule: Conservation of Mass
What Goes In Must Come Out
The most fundamental principle in chemistry is the Conservation of Mass.
The Principle: In any normal chemical reaction, no atoms are created or destroyed. The total mass of the reactants (the starting materials) must exactly equal the total mass of the products (what is made).
Analogy: Imagine you have a box containing 5 red LEGO bricks and 3 blue LEGO bricks. You build a spaceship out of them. The mass of the spaceship must be the same as the mass of the 8 original bricks. You didn't lose any bricks, you just rearranged them!
This rule is vital, and we apply it every time we write a chemical equation. If the mass changes, it usually means a gas has escaped or been taken in from the atmosphere.
Quick Review:
- Reactants (Start) = Products (End)
- Total Mass Before = Total Mass After
2. Relative Masses: The Atomic Weight
To do any calculations, we first need to know how "heavy" our atoms and molecules are. Since atoms are tiny, we use a concept called "relative mass," which compares everything to a standard atom (Carbon-12).
2.1 Relative Atomic Mass (\(A_r\))
The Relative Atomic Mass (\(A_r\)) is the mass of one atom of an element compared to the standard.
- The \(A_r\) is simply the mass number shown for each element on the Periodic Table (usually the larger number).
- Since Combined Science often ignores minor variations due to isotopes, we typically use the rounded whole number (e.g., Carbon = 12, Oxygen = 16, Hydrogen = 1).
Example: The \(A_r\) of Magnesium (Mg) is 24. The \(A_r\) of Chlorine (Cl) is 35.5.
2.2 Relative Formula Mass (\(M_r\))
The Relative Formula Mass (\(M_r\)) is the total mass of a compound. We calculate it by adding up the \(A_r\) of every single atom in the chemical formula.
Step-by-Step Guide to Calculating \(M_r\):
Example 1: Water (\(H_2O\))
- Identify the atoms and their quantities: 2 Hydrogen (H) and 1 Oxygen (O).
- Look up their \(A_r\) values: H = 1, O = 16.
- Calculate the total mass:
\(M_r\) of \(H_2O\) = (2 \times A_r \text{ of H}) + (1 \times A_r \text{ of O})\)
\(M_r\) of \(H_2O\) = \((2 \times 1) + (1 \times 16) = 2 + 16 = 18\)
Example 2: Magnesium Chloride (\(MgCl_2\))
- Atoms: 1 Magnesium (Mg), 2 Chlorine (Cl).
- \(A_r\) values: Mg = 24, Cl = 35.5.
- Calculate:
\(M_r\) of \(MgCl_2\) = (1 \times 24) + (2 \times 35.5)\)
\(M_r\) of \(MgCl_2\) = \(24 + 71 = 95\)
Remember the Bracket Rule: If a formula has brackets, like Calcium Nitrate, \(Ca(NO_3)_2\), the small number outside the bracket multiplies everything inside.
\(Ca(NO_3)_2\): 1 Ca, 2 N, (2 x 3) = 6 O.
3. Percentage Composition: Finding the Yield
Once we know the total mass (\(M_r\)) of a compound, we can calculate the percentage mass of any element inside it. This is really useful! If you mine an ore, you want to know what percentage of that rock is the valuable metal.
The Formula
\[\text{Percentage Mass of Element} = \frac{\text{Total } A_r \text{ of Element in Formula}}{\text{Total } M_r \text{ of Compound}} \times 100\]
Step-by-Step Example: What is the percentage of Carbon (C) in Carbon Dioxide (\(CO_2\))?
(Use \(A_r\): C=12, O=16)
- Calculate \(M_r\) of \(CO_2\):
C = 12 (1 atom)
O = 16 x 2 = 32 (2 atoms)
Total \(M_r\) = 12 + 32 = 44 - Identify the mass of the element we care about (Carbon):
Total \(A_r\) of C = 12 - Plug into the formula:
\[\text{Percentage C} = \frac{12}{44} \times 100\] - Calculate:
Percentage C \(\approx 27.3\%\)
Did you know? Knowing the percentage composition is how manufacturers determine the purity of ingredients. If a medicine is supposed to be 50% active ingredient by mass, chemists use these calculations to check it!
4. Balancing Chemical Equations
Balancing an equation is the way we formally show that mass is conserved. We must have the exact same number of atoms of each element on both sides of the arrow.
The Balancing Method (The Tally System)
Let's balance the reaction between Methane (\(CH_4\)) and Oxygen (\(O_2\)) to form Carbon Dioxide (\(CO_2\)) and Water (\(H_2O\)).
Unbalanced Equation:
\(CH_4 + O_2 \rightarrow CO_2 + H_2O\)
- Tally the atoms on both sides (Reactants vs. Products):
(C is balanced, H and O are not.)Element Reactants (LHS) Products (RHS) C 1 1 H 4 2 O 2 3 - Balance Hydrogen (H):
We have 4 H on the left and 2 H on the right (\(H_2O\)). We need two times more \(H_2O\). We place a coefficient (big number) 2 in front of \(H_2O\).
\(CH_4 + O_2 \rightarrow CO_2 + 2H_2O\)
- Re-Tally and Balance Oxygen (O):
Now let's count again:
Now we have 2 O on the left and 4 O on the right. We need to double the Oxygen on the left. Place a 2 in front of \(O_2\).Element Reactants (LHS) Products (RHS) C 1 1 H 4 (2 x 2) = 4 O 2 (2 in \(CO_2\)) + (2 in \(2H_2O\)) = 4 \(CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O\)
- Final Check:
C: 1 in, 1 out.
H: 4 in, 4 out.
O: 4 in, 4 out.
It is balanced!
Common Mistake to Avoid: You can only change the large numbers (coefficients) at the front of a formula. Never change the small subscript numbers (like the '4' in \(CH_4\)), as this would change the chemical identity of the substance!
5. Mass Calculations from Balanced Equations (Stoichiometry)
This is where we use all our skills! Since the numbers in a balanced equation represent the ratios of the particles, they also represent the ratio of their total relative formula masses (\(M_r\)).
In Combined Science, we focus on simple mass calculations, usually finding how much product is made from a known mass of reactant.
Key Principle: The Mass Ratio is Fixed
Consider the simple decomposition of Calcium Carbonate (\(CaCO_3\)) into Calcium Oxide (\(CaO\)) and Carbon Dioxide (\(CO_2\)):
\(CaCO_3 \rightarrow CaO + CO_2\) (It's already balanced 1:1:1)
Step A: Calculate the \(M_r\) for all substances involved.
(Use \(A_r\): Ca=40, C=12, O=16)
- \(M_r\) of \(CaCO_3\) = 40 + 12 + (3 x 16) = 100
- \(M_r\) of \(CaO\) = 40 + 16 = 56
- \(M_r\) of \(CO_2\) = 12 + (2 x 16) = 44
The mass ratio is 100 parts \(CaCO_3\) reacts to give 56 parts \(CaO\) and 44 parts \(CO_2\). (Notice: 56 + 44 = 100. Mass is conserved!)
Step B: Use the ratio to find the unknown mass.
Question: If 50 g of \(CaCO_3\) is decomposed, what mass of \(CaO\) is produced?
- Set up the relationship (Ratio):
Mass \(CaCO_3\) : Mass \(CaO\) is 100 : 56. - Find the scaling factor:
We started with 50 g of \(CaCO_3\).
We know that 100 g produces 56 g.
How many times smaller is 50 g than 100 g?
\(50 / 100 = 0.5\) (The reaction is running at half scale). - Apply the factor to the product:
Mass \(CaO\) produced = 56 g \(\times 0.5\)
Mass \(CaO\) produced = 28 g
Don't worry if this seems tricky at first! The method is always the same: Find the \(M_r\)s, establish the ratio, and then scale that ratio according to the mass given in the question.
Tip for success: Always write the calculated \(M_r\) beneath the balanced equation to keep track of your ratios.
Summary of Quantitative Chemistry
Quantitative chemistry links the numbers on the periodic table to the masses you measure in the lab. Remember these three key steps:
- Conservation: Mass never changes during a reaction.
- Calculation: Use \(A_r\) values to find the \(M_r\) of everything.
- Ratio: Use the \(M_r\) values and the coefficients in the balanced equation to find the fixed mass ratios for your calculation.