Dimensional Analysis and Unit Conversions Flashcards

Scientific Units and Prefix Multipliers

Fundamental properties in chemistry are measured using specific units. While meters ($m$) are utilized for length, mass is often discussed in kilograms ($kg$). Although kilograms are considered a standard unit, the actual base unit in the prefix system is the gram ($g$). Chemists must be proficient in converting between different units that represent the same measurement to ensure data is represented in a convenient and readable format.

The Purpose of Prefix Multipliers

Prefix multipliers, or exponents, allow scientists to avoid writing long strings of leading zeros for very small numbers or trailing zeros for very large numbers. These prefixes represent powers of ten that scale a base unit to a more manageable size for the specific context of the measurement.

• Kilo ($k$): Represents $10^{3}$ or $1000$.

• Deci ($d$): Represents $10^{-1}$.

• Centi ($c$): Represents $10^{-2}$.

• Milli ($m$): Represents $10^{-3}$.

• Micro ($\mu$): Represents $10^{-6}$.

• Nano ($n$): Represents $10^{-9}$.

• Pico ($p$): Represents $10^{-12}$.

• Femto ($f$): Represents $10^{-15}$.

Application and Physical Context

Prefixes are chosen based on the physical scale of the object or phenomenon being studied:

• Bond Vibrations: The vibrations of hydrogen bonds to oxygen in water molecules occur at the scale of femtoseconds ($fs$). One femtosecond is equal to $10^{-15}$ seconds ($s$). This indicates that atoms move extremely fast in real life.

• DNA Width: DNA research often involves measurements of width around $3\,nm$. One nanometer is equal to $10^{-9}$ meters ($m$).

• Atomic Size: Angstroms ($\r{A}$) are frequently used to describe the size of atoms. One angstrom is defined as $10^{-10}\text{ meters}$.

Rules for Unit Conversion and Dimensional Analysis

Dimensional analysis treats units as mathematical entities. Units can be multiplied or canceled just like numbers. If a unit appears in both the numerator and the denominator of a calculation, they cancel each other out.

Creating Conversion Factors

A conversion factor is a fraction where the numerator and denominator represent the same amount in different units. When writing these factors based on the prefix multiplier table, follow these rules:

1. The number $1$ is always placed next to the unit with the prefix letter (e.g., $1\,ps$, $1\,km$).

2. The exponent ($10^{x}$) is always placed next to the base unit (the one without the prefix letter).

3. Strict Rule: Never write the $10^{x}$ value next to the prefix letter; the letter itself is the substitute for that value.

Example conversion factors for femtoseconds to seconds: $\frac{1\,fs}{10^{-15}\,s}$ or $\frac{10^{-15}\,s}{1\,fs}$

The Four-Step Problem-Solving Approach

1. Sort: Identify the given information and the goal of the problem. Identify the starting number and unit.

2. Strategize: Create a conceptual plan or a map for the conversion ($A \rightarrow B \rightarrow C$). Write out the required equivalencies.

3. Solve: Set up the fractions so units cancel. Multiply by numbers on top and divide by numbers on the bottom.

4. Check: Verify that units canceled correctly, the final unit matches the goal, the number of significant figures is correct, and the magnitude of the answer makes physical sense.

Multidimensional Conversions: Area and Volume

When converting units of area (squared units) or volume (cubed units), the exponent applies to both the numerical value and the unit label.

Volume Relationships

• A cube with sides of $1\,cm$ has a volume of $1\,cm \times 1\,cm \times 1\,cm = 1\,cm^3$.

• In the healthcare field, a cubic centimeter ($cm^3$) is often referred to as a "cc".

• $1\,cm^3 = 1\,cc = 1\,mL$.

• When converting between squared or cubed units, you must square or cube the entire conversion factor:

  • If $1\,cm = 10^{-2}\,m$, then $(1\,cm)^3 = (10^{-2}\,m)^3$, which means $1\,cm^3 = 10^{-6}\,m^3$.

Sample Problems and Detailed Solutions

Problem 1: Converting $0.001320\,dL$ to $\mu L$

• Equivalencies: $1\,dL = 10^{-1}\,L$; $1\,\mu L = 10^{-6}\,L$.

• Plan: $dL \rightarrow L \rightarrow \mu L$

• Calculation: $0.001320\,dL \times \frac{10^{-1}\,L}{1\,dL} \times \frac{1\,\mu L}{10^{-6}\,L} = 132.0\,\mu L$.

• Significant Figures: The starting number has four significant figures, so the result is written as $132.0$ to maintain precision.

Problem 2: Converting $2\,hours$ to Picoseconds ($ps$)

• Equivalencies: $1\,hr = 60\,min$; $1\,min = 60\,s$; $1\,ps = 10^{-12}\,s$.

• Plan: $hr \rightarrow min \rightarrow s \rightarrow ps$

• Calculation: $2\,hr \times \frac{60\,min}{1\,hr} \times \frac{60\,s}{1\,min} \times \frac{1\,ps}{10^{-12}\,s} = 7.2 \times 10^{15}\,ps$.

• Logic Check: Since picoseconds are tiny, a large positive exponent is expected for a multi-hour duration.

Problem 3: Converting $99\,pm$ to Angstroms ($\r{A}$)

• Equivalencies: $1\,pm = 10^{-12}\,m$; $1\,\r{A} = 10^{-10}\,m$.

• Plan: $pm \rightarrow m \rightarrow \r{A}$

• Calculation: $99\,pm \times \frac{10^{-12}\,m}{1\,pm} \times \frac{1\,\r{A}}{10^{-10}\,m} = 0.99\,\r{A}$.

Problem 4: Monolayer Area Conversion ($154\,\r{A}^2$ to $nm^2$)

• Equivalencies: $1\,\r{A} = 10^{-10}\,m$, so $1\,\r{A}^2 = 10^{-20}\,m^2$. $1\,nm = 10^{-9}\,m$, so $1\,nm^2 = 10^{-18}\,m^2$.

• Calculation: $154\,\r{A}^2 \times \frac{10^{-20}\,m^2}{1\,\r{A}^2} \times \frac{1\,nm^2}{10^{-18}\,m^2} = 1.54\,nm^2$.

Problem 5: Converting $1\,m^3$ to $mL$

• Equivalencies: $1\,cm = 10^{-2}\,m \rightarrow 1\,cm^3 = 10^{-6}\,m^3$; $1\,cm^3 = 1\,mL$.

• Calculation: $1\,m^3 \times \frac{1\,cm^3}{10^{-6}\,m^3} \times \frac{1\,mL}{1\,cm^3} = 1 \times 10^{6}\,mL$.

Questions & Discussion

Question: Why is a thousand milliliters in a liter written as $10^{3}$ if the table says milli is $10^{-3}$? Answer: One milliliter is equal to $10^{-3}$ liters. If you divide both sides by $10^{-3}$, you get $\frac{1}{10^{-3}}\,mL = 1\,L$. Since $\frac{1}{10^{-3}}$ is mathematically equivalent to $10^{3}$ (or $1000$), there are $1000\,mL$ in $1\,L$.

Question: Are all whole numbers in these problems considered exact for significant figures? Answer: No. Measurements of real-life objects, such as the $99\,pm$ measurement for an atom or the $154\,\r{A}^2$ for a monolayer, determine the number of significant figures. Conversion factors between metric units (like $1\,cm = 10^{-2}\,m$) are exact and are ignored when determining significant figures.

Question: Do I need to show the final unit conversion step if the value does not change (e.g., $cm^3$ to $mL$)? Answer: Yes. In introductory chemistry, it is necessary to write down the unit conversion (e.g., $1\,cm^3 = 1\,mL$) to demonstrate the logic of how units cancel and the final unit is arrived at, even if the numeric value remains the same.