Chem notes 3

Averages and Significant Figures

• Finding the average of a group of densities.

• Example Calculation: Assume densities are provided, the average is calculated.

• Calculated Average: 4.7292.

• To find an average, add all numbers and divide by the count of numbers.

• Rule for Addition: The answer should have the least number of decimal places as the numbers being added.

  • If some numbers go four places past the decimal and others go three, the answer should go three places past the decimal.

• Identify the last significant digit based on the decimal place rule but don't round yet for further calculations.

• For a division problem, consider significant figures.

  • Example: Dividing by 5, which is an exact number, thus having unlimited significant figures, so do not consider when determining significant figures for the answer.

• Four significant figures in 4.729 after identifying that the number legitimately ends at '9'.

• Dividing 4.7292 by 5 results in 0.94584.

• Rule for Division: The answer should have the least number of significant figures as the numbers being divided.

• Rounding 0.94584 to four significant figures gives 0.9458 grams per milliliter (g/mL).

• In numbers with a decimal, start counting significant figures from the first non-zero number.

Graphing

• When working with graphs, especially with accurate densities, avoid overstating accuracy.

• Accuracy on a graph is limited by the divisions on the axes.

• If a graph is divided into ones, you can accurately estimate to the tenths place.

• When noting a point on a graph, provide both X and Y values.

• Example:

  • X value on the line for 11 should be written as 11.0 to indicate accuracy to the tenths place.

  • Y value slightly above 11, estimate to 11.3.

• When finding the slope of a line, use a best-fit line, not a connect-the-dots approach.

• Pick two points on the line that are further apart for better accuracy when determining slope.

• Graphs are generally not precise beyond the tenths place.

Percent Error

• Percent Error Calculation: \frac{{\text{{Experimental Value}} - \text{{Theoretical Value}}}}{{\text{{Theoretical Value}}}} \times 100 \%.

• Units cancel out in percent error calculations, leaving only the percent sign (%).

• When subtracting, the number of decimal places should match the least precise measurement.

• When dividing, the result should have the same number of significant figures as the number with the least significant figures.

• Example:

  • \frac{{3.5}}{{1.1}} = 3.1818 \ldots , rounded to two significant figures because 1.1 has two significant figures, gives 3.2%.

  • Do not round prematurely as it reduces accuracy.

Metric System & Conversions

• The metric system is based on powers of 10, making conversions straightforward.

• SI Units (Standard Units):

  • Length: meter (m)

  • Mass: kilogram (kg)

  • Amount: mole (mol)

  • Time: second (s)

  • Volume: cubic meters (m^3) , but liters (L) is more practical.

• The standard kilogram is stored in a vault in France and is compared with sister copies every 50 years to calibrate instruments worldwide.

• Some countries, including the USA, Liberia, and Myanmar, have not fully switched to the metric system.

• Metric system prefixes combined with base units:

  • Example: centimeter (centi- + meter).

• Constants Chart in Pearson homework has conversions, equalities, and constants.

• Memorize common metric conversions such as:

  • 1000 millimeters (mm) = 1 meter (m)

  • These relationships work for grams, joules, and liters as well.

• Prefixes and Base Units:

  • From one box to the next implies a difference of 10.

• Mnemonic for remembering metric prefixes: "King Henry Died By Drinking Chocolate Milk".

  • Kilo, Hecto, Deca, Base, Deci, Centi, Milli.

• Converting Between Metric Units (Ladder Method):

  • Converting 1000 milligrams (mg) to grams (g): move three steps to the left \rightarrow 1 g.

  • Converting 160 centimeters (cm) to millimeters (mm): move one step to the right \rightarrow 1600 mm.

  • Converting 0.19 grams to kilograms: move three steps to the left \rightarrow 0.00019 kg.

Dimensional Analysis (Factor Label Method)

• Dimensional analysis is used for conversions, especially between metric and English systems.

• Steps:

  1. Identify the given (starting number).

  2. Include units in all steps because units guide the conversion process.

  3. Multiply the given by a conversion factor to obtain the desired unit.

• A conversion factor is an equivalent value equal to one.

• Conversion Factors:

  • Example: \frac{{1000 \text{{ grams}}}}{{1 \text{{ kilogram}}}} = 1 .

  • If converting 32 kg to grams, use \frac{{1000 \text{{ grams}}}}{{1 \text{{ kilogram}}}} . Kilograms cancel, resulting in grams.

• Example: Converting 23.80 grams to kilograms.

  • Use the conversion factor \frac{{1 \text{{ kilogram}}}}{{1000 \text{{ grams}}}} .

  • 23.80 \text{{ grams}} \times \frac{{1 \text{{ kilogram}}}}{{1000 \text{{ grams}}}} = 0.02380 \text{{ kilograms}} .

• Memorize the conversion: 1 inch = 2.54 cm.

• Example: Convert 3.6 inches to centimeters.

  • 3.6 \text{{ inches}} \times \frac{{2.54 \text{{ cm}}}}{{1 \text{{ inch}}}} = 9.1 \text{{ cm}} .

• Quiz one will cover metric stuff, including significant figures.

Multi-Step Conversions

• Sometimes more than one step is needed to get from the starting unit to the desired unit.

• Example: Convert 430 cm to feet, using 1 inch = 2.54 cm and 12 inches = 1 foot.

  • 430 \text{{ cm}} \times \frac{{1 \text{{ inch}}}}{{2.54 \text{{ cm}}}} \times \frac{{1 \text{{ foot}}}}{{12 \text{{ inches}}}} = 14 \text{{ feet}} .

Complex Units

• For units with two parts, such as grams per milliliter (g/mL), both units may need to be converted.

• Example: Convert 8.3 milligrams per liter (mg/L) to grams per milliliter (g/mL).

  • \frac{{8.3 \text{{ mg}}}}{{1 \text{{ L}}}} \times \frac{{1 \text{{ g}}}}{{1000 \text{{ mg}}}} \times \frac{{1 \text{{ L}}}}{{1000 \text{{ mL}}}} = 8.3 \times 10^{{-6}} \text{{ g/mL}} .

Units to a Power

• When dealing with square or cubic units, remember to apply the square or cube to both the number and the unit.

• Example 1: 1 kilometer squared (\text{{km}}^2) = how many meters squared (\text{{m}}^2)?

  • Start with: 1 km = 1000 m.

  • Square both sides: (1 \text{{ km}})^2 = (1000 \text{{ m}})^2.

  • 1 \text{{ km}}^2 = 1,000,000 \text{{ m}}^2.

• Example 2: 1 meter cubed (\text{{m}}^3) = how many millimeters cubed (\text{{mm}}^3)?

  • Start with: 1 m = 1000 mm.

  • Cube both sides: (1 \text{{ m}})^3 = (1000 \text{{ mm}})^3.

  • 1 \text{{ m}}^3 = 1,000,000,000 \text{{ mm}}^3.

• Example 3: Cubic inch to centimeter cubed, cube everything on both sides with the equality 1 inch = 2.54 cm.

  • Start with: 1 in = 2.54 cm

  • Cube both sides: (1 \text{{ in}})^3 = (2.54 \text{{ cm}})^3

  • 1 \text{{ in}}^3 = 16.387 \text{{ cm}}^3.

• Example 4: Cubic foot to inch cubed, cube everything on both sides with the equality 1 foot = 12 inches.

  • Start with: 1 ft = 12 in

  • Cube both sides: (1 \text{{ ft}})^3 = (12 \text{{ in}})^3

  • 1 \text{{ ft}}^3 = 1728 \text{{ in}}^3