Scaling vs. Summing
Solving a Common Significant Figure Puzzle in Chemistry
As a general chemistry student, you have learned the two fundamental rules for significant figures: one for addition/subtraction and one for multiplication/division. A common and confusing paradox arises when a single operation can be viewed in two different ways.
Consider calculating the total mass of three identical samples, each weighing 10.1 g. Should you add (10.1 + 10.1 + 10.1) or multiply (3 × 10.1)? The two rules give different answers.
10.1 + 10.1 + 10.1 = 30.3. By the addition rule (rounding to the tenths place), the answer is 30.3 g.3 × 10.1 = 30.3. By the multiplication rule (rounding to 3 sig figs), the answer is 30.3 g.
In this case, the answers happen to be the same. But what if the mass was 50.1 g?
50.1 + 50.1 + 50.1 = 150.3. By the addition rule, the answer is 150.3 g.3 × 50.1 = 150.3. By the multiplication rule, the answer must be rounded to 3 sig figs: 150. g or 1.50 × 102 g.
These answers are not the same. Which one is correct? The solution lies in distinguishing between inexact (measured) numbers and exact numbers.
The Core Principle: Inexact vs. Exact Numbers
This is the single most important concept for resolving significant figure ambiguity.
Inexact Numbers
Inexact numbers have uncertainty and typically come from a measurement. Their significant figures indicate the precision of the value. Examples include a mass from a balance (23.5 g) or a volume from a graduated cylinder (150 mL).
Exact Numbers
Exact numbers have infinite precision and zero uncertainty. They do not limit the precision of a calculation. They arise from two sources:
- Counting:
4hydrogen atoms in CH4;3trials in an experiment. - Definitions:
1000in 1000 g = 1 kg;12in 12 inches = 1 foot.
The Guiding Question: Are You Scaling or Summing?
With the Golden Rule in mind, you can solve any problem by asking a simple question:
“Am I summing separate, independent inexact numbers, or am I scaling one inexact number with an exact number?”
- Summing (Use Addition/Subtraction Rule): This applies when combining two or more separate, independent measurements. The result is rounded to the last place value that is significant in all of the numbers being added or subtracted.
- Scaling (Use Multiplication/Division Rule): This applies when you take a single measurement and multiply or divide it by a count or a defined ratio (an exact number). The precision is limited by the significant figures of your inexact numbers.
Case Studies in Action
Let’s apply this principle to common chemistry scenarios where the choice between summing and scaling is critical.
Case Study 1: Molar Mass of Sucrose (C12H22O11)
Calculating molar mass is a multi-step process. The correct method follows the order of operations, applying the relevant sig fig rule at each step. A common error is to misapply one rule to the entire calculation.
We will use the following atomic masses: C = 12.01 g/mol, H = 1.01 g/mol, O = 16.00 g/mol.
Correct Method: Scaling, then Summing
The subscripts (12, 22, 11) are exact counts. We scale each atomic mass first, then sum the results.
- Scale (Multiplication):
- C:
12 × 12.01 = 144.12 - H:
22 × 1.01 = 22.22 - O:
11 × 16.00 = 176.00
- C:
- Sum (Addition): The least precise place value in all three scaled results is the hundredths place.
144.12 + 22.22 + 176.00 = 342.34
- Apply Addition Rule: The answer must be rounded to the hundredths place.
- Final Correct Answer: 342.3 g/mol
Incorrect Method: Global “Fewest Sig Figs” Rule
A frequent mistake is to find the input with the fewest sig figs and apply that rule to the final answer, ignoring the order of operations.
- Analyze Initial Sig Figs:
- C (
12.01): 4 sig figs - H (
1.01): 3 sig figs (The fewest) - O (
16.00): 4 sig figs
- C (
- Perform Calculation:
- The raw sum is still
342.34.
- The raw sum is still
- Apply Flawed Logic: The student incorrectly rounds the final answer to 3 significant figures because of hydrogen.
- Final Incorrect Answer: 34 g/mol
This error incorrectly discards known precision by misapplying the rules.
Case Study 2: Enthalpy Changes (Summing vs. Scaling)
This case study provides a direct comparison where adding two numbers versus multiplying by two gives different results, and both are correct in their specific contexts.
Scenario A: Hess’s Law (Summing)
You perform two separate experiments to measure the enthalpy of two different reactions. By chance, they both yield the same value.
- Rxn 1: ΔH = +50.1 kJ
- Rxn 2: ΔH = +50.1 kJ
You combine these reactions using Hess’s Law.
Analysis: You are summing two independent, inexact measurements. Apply the addition rule. The least precise place in both numbers is the tenths place.
Calculation:
50.1 + 50.1 = 100.2The answer must be rounded to the tenths place.
Final Answer: 100. kJ
Scenario B: Doubling Stoichiometry (Scaling)
You take one reaction with a measured enthalpy of ΔH = +50.1 kJ and double its coefficients (A → B becomes 2 A → 2 B).
Analysis: You are scaling a single measurement (50.1, 3 sig figs) by an exact count of 2. Apply the multiplication rule.
Calculation:
2 × 50.1 = 100.2The answer must be rounded to 3 significant figures.
Final Answer: 1.00 × 102 kJ
This direct comparison shows that context is everything. The same numbers can lead to different, correct answers depending on whether you are summing independent measurements or scaling a single measurement by an exact number.
Quick Reference Guide
| Scenario | Primary Operation | Rationale | Sig Fig Rule to Apply |
|---|---|---|---|
| Molar Mass | Scaling, then Summing | Subscripts are exact counts. | Follow multi-step rules: determine precision after multiplication, then apply addition rule to the sum. |
| Average Atomic Mass | Weighted Averaging | Isotopic mass and abundance are both measured values. | Follow multi-step rules: Multiply mass by abundance for each isotope (mult. rule), then sum the results (add. rule). |
| Stoichiometry (e.g., doubling a reaction) | Scaling | Mole ratios and coefficients are exact definitions. | Multiplication/Division |
| Hess’s Law | Summing | You are combining separate, measured reaction enthalpies. | Addition/Subtraction |
| Temperature Change (ΔT) | Subtraction | You are finding the difference between two measurements. | Addition/Subtraction |
| Mass by Difference | Subtraction | You are finding the difference between two measurements. | Addition/Subtraction |
| Averaging Multiple Trials | Summing, then Scaling | You sum separate measurements, then divide by an exact count. | Follow multi-step rules: Sum first (add. rule), then divide that result (mult. rule). |
The Take-Home Message
The apparent conflict between significant figure rules is not a conflict at all. It is a call to think like a scientist. Before you apply a rule, look at your numbers. Are they fallible, inexact numbers from a lab instrument, or are they perfect, unchanging counts and definitions?
When you encounter a calculation, always ask yourself the guiding question:
Am I combining separate measurements (summing), or am I scaling one measurement by a perfect count (scaling)?
Answering that question will tell you exactly which rule to use, every time.