measurement

MEASUREMENT

Numbers play a major role in chemistry.
All scientific phenomena are described in small units that represent other quantities.
Concepts of numbers in science:
- Units of measurement
- Quantities that are measured and calculated
- Uncertainty in measurement
- Significant figures

Système International d’Unités (International System of Units).
- Fundamental units from which all others are derived.
- A different base unit is used for each quantity.

Measurement	Unit	Abbreviation	Value in SI Units
Length	angstrom	Å	1 Å = 0.1 nm = 10^{-10} m
Mass	atomic mass unit	u (amu)	1 u = 1.66054 × 10^{-27} kg (rounded to six digits)
	metric ton	t	1 t = 10^{3} kg
Time	minute	min	1 min = 60 s
	hour	h	1 h = 60 min = 3600 s
Temperature	degree Celsius	°C	TK = t°C + 273.15
Volume	liter	L	1 L = 1000 cm³

Measured with thermometer; three common scales: A. Fahrenheit scale:
- Common in US
- Water freezes at 32 °F and boils at 212 °F
- 180 degree units between melting and boiling points of water
  B. Celsius scale:
- Most common for use in science
- Water freezes at 0 °C
- Water boils at 100 °C
- 100 degree units between melting and boiling points of water
  C. Kelvin scale:
- SI unit of temperature is kelvin (K) (no degree symbol in front of K)
- Water freezes at 273.15 K and boils at 373.15 K
- 100 degree units between melting and boiling points
- Only difference between Kelvin and Celsius scales is zero point (Absolute Zero) which corresponds to nature’s lowest possible temperature

How to convert between °F and °C:
(tF = (tC × rac{9}{5}) + 32)
Example: 100 °C = ? °F
- To convert: (t_F = (100 °C × rac{9}{5}) + 32) = 212 °F
Common laboratory thermometers are marked in Celsius scale. Must convert to Kelvin scale:
- (TK = tC + 273.15)
- Example: What is the Kelvin temperature of a solution at 25 °C?
- (T_K = 25 °C + 273.15 = 298 K)

Convert 121 °F to the Celsius scale:
- Method: (tC = (tF - 32) × rac{5}{9})
- (t_C = (121 - 32) × rac{5}{9} = 49 °C)
Convert 121 °F to the Kelvin scale:
- We already have it in °C, so:
- (T_K = 49 °C + 273.15 = 322 K)

Different measuring devices have different uses and different degrees of accuracy, which deliver variable volumes.
All measured numbers have some degree of inaccuracy:
- Uncertainty must always be communicated.
Example: A good measurement is meaningless without knowing the uncertainty.
Instruments vary in precision:
- Analog instruments: ½ of the smallest increment
- Digital instruments: smallest scale division for absolute uncertainty calculation.

Systematic errors (Determinate errors):
- Caused by instruments, methods, or analyst practices. These errors can often be corrected when identified.
  - Examples: badly worn out instruments, uncalibrated devices, poor technique.
Random errors (Indeterminate errors):
- Unpredictable errors which arise by chance, associated with limitations in measuring instruments. They cannot be eliminated but can be reduced.
- Follow no regular pattern. Example: reproducibility (precision) vs. accuracy.

Accuracy: Refers to how closely a measured value aligns with the accepted true value.
Precision: Refers to the reproducibility of measurements; multiple measurements can be close to each other but not necessarily accurate.

Definition: Digits that are meaningful in terms of accuracy of a measurement.
Rules for Significant Figures:
1. All non-zero numbers are significant (e.g., 3.456 has 4 sig. figs).
2. Zeros between non-zero numbers are significant (e.g., 20,089 has 5 sig. figs).
3. Trailing zeros are significant if there is a decimal point (e.g., 500. has 3 sig. figs).
4. Final zeros without a decimal point are not significant (e.g., 104,956,000 has 6 sig. figs).
5. Leading zeros are never significant (e.g., 0.00012 has 2 sig. figs).

Scientific notation indicates significant figures clearly, focusing only on numbers between 1 and 10.
Example:
- Estimated populations can be reported with uncertainty, indicating complete precision.

Addition/Subtraction:
- Result rounded to the least number of decimal places.
- Example: 20.42 + 1.322 + 83.1 = 104.8 (rounded to one decimal place)
Multiplication/Division:
- Result rounded to the least number of significant figures present in any measured number involved in the calculation.
- Example: 10.54 × 31.4 × 16.987 = (4 sig. figs)(3 sig. figs)(5 sig. figs) = 3 sig. figs

Molarity (C): Number of moles of a substance per liter of solution.
- Formula: C = \frac{\text{mol of substance}}{\text{L of solution}}
- Example: 1 µM = 10^{-6} M
Percent Composition:
- Weight percent: \text{wt}\% = \frac{\text{mass of solute}}{\text{mass of total solution}} \times 100
- Density: \rho = \frac{\text{mass of solute}}{\text{volume of total solution}} (unit: g/mL)
- Specific gravity: \frac{\text{Density of substance}}{\text{Density of water at 4 °C}} (dimensionless)

Example Problem:
1. Calculate the volume of a box with dimensions 15.5 cm, 27.3 cm, and 5.4 cm, ensuring correct number of significant figures.
2. Determine the density of a gas by weighing a container before and after filling and calculating the difference.
Analyzing known compositions helps in distinguishing error types and ensuring accuracy and precision.

Mean: Sum of measured values divided by number of measurements. Formula: \bar{x} = \frac{ΣX_i}{n}
Standard deviation (s): Indicates the width of a distribution.
- Formula: s = \sqrt{\frac{Σ(x_i - \bar{x})^2}{n-1}}
Variance: Square of standard deviation, quantifies how data points differ from the mean.

Data should be analyzed to find mean, standard deviations, and other metrics to assess the spread of data.
Example Data Set 1: (3, 5, 7, 10, 10) vs. Data Set 2: (7, 7, 7, 7, 7)
- Mean and Median: Both have the same but demonstrate differences in variance.

Ensure all calculations respect significant figures rules to maintain accuracy in scientific measurements.
Understand how changing measurement techniques affects the overall error and confidence in results.
Final Exercises: Include calculating means, medians, variances, and undertaking several measurements to understand fluctuation and precision.