Measurements in Experiments Flashcards

SI Base Units and Standards

  • System International (SI): The standardized measurement system utilized universally across scientific disciplines.
  • Base Units: The SI system consists of seven fundamental base units, where each unit describes a single physical dimension:
    • Length: Meter (m\text{m}) — Measures basic distance.
    • Mass: Kilogram (kg\text{kg}) — Measures quantity of matter.
    • Time: Second (s\text{s}) — Measures duration of events.
    • Electric Current: Ampere (A\text{A}) — Measures flow of electric charge.
    • Temperature: Kelvin (K\text{K}) — Measures thermodynamic degree of hot or cold.
    • Amount of Substance: Mole (mol\text{mol}) — Measures total number of elementary particles (NA\text{N}_\text{A}).
    • Luminous Intensity: Candela (cd\text{cd}) — Measures perceived brightness of light.

Overview of 7 Base SI Units

  • Historical and Current Standards for SI Base Units:
    • Meter (length):
    • Original Standard: Defined as 110000000\frac{1}{10000000} of the distance from the equator to the North Pole.
    • Current Standard: Defined as the distance traveled by light in a vacuum during a time interval of 3.33564095×10−9 s3.33564095 \times 10^{-9}\,\text{s}.
    • Kilogram (mass):
    • Original Standard: Defined as the mass of 0.001 m30.001\,\text{m}^3 (1 dm31\,\text{dm}^3) of water.
    • Current Standard: Defined as the mass of a specific physical platinum-iridium alloy cylinder stored under international standard conditions.
    • Second (time):
    • Original Standard: Defined as (160)(160)(124)=0.000011574\left(\frac{1}{60}\right)\left(\frac{1}{60}\right)\left(\frac{1}{24}\right) = 0.000011574 average solar days.
    • Current Standard: Defined as 9 192 631 7709\,192\,631\,770 times the period of the radiation corresponding to the transition between two hyperfine levels of the ground state of a cesium-133 atom.

SI Base Standards

  • Derived Units:
    • Formed by algebraic combinations of the seven fundamental base units via multiplication or division.
    • Example: Speed is expressed in meters per second (m/s\text{m/s}).

Scientific Notation and Conversions

  • Scientific Notation Definition:

    • A shorthand mathematical notation used to write extremely large or extremely small numbers concisely.
    • Expressed in the form M×10kM \times 10^k, where MM is a coefficient greater than or equal to 11 and strictly less than 1010 (1≤M<101 \le M < 10), and kk is an integer exponent (positive or negative).

  • Physical Examples:

    • An ordinary copper penny contains approximately 2.0×10222.0 \times 10^{22} atoms (20 000 000 000 000 000 000 00020\,000\,000\,000\,000\,000\,000\,000 atoms in standard notation).
    • The average diameter of an atom is approximately 3.0×10−8 cm3.0 \times 10^{-8}\,\text{cm} (0.00000003 cm0.00000003\,\text{cm} in standard notation).

  • Converting Standard Notation to Scientific Notation:

    • Step 1: Shift the decimal point until the remaining number is between 11 and 1010.
    • Step 2: Count the total number of places the decimal point moved; this value determines the exponent kk on the base 1010.
    • Exponent Sign Rule:
    • If the original number is greater than 11, move the decimal point to the left and assign a positive exponent (k>0k > 0).
    • If the original number is less than 11 (between 00 and 11), move the decimal point to the right and assign a negative exponent (k<0k < 0).
    • Worked Conversion Examples:
    • 123 000 000 000 000=1.23×1014123\,000\,000\,000\,000 = 1.23 \times 10^{14}
    • 0.000000000236=2.36×10−100.000000000236 = 2.36 \times 10^{-10}
    • 7 080 000 000 000=7.08×10127\,080\,000\,000\,000 = 7.08 \times 10^{12}
    • 0.00000001256=1.256×10−80.00000001256 = 1.256 \times 10^{-8}
    • 57 000 000=5.7×10757\,000\,000 = 5.7 \times 10^7
    • 0.0053=5.3×10−30.0053 = 5.3 \times 10^{-3}

  • Converting Scientific Notation to Standard Notation:

    • The sign of exponent kk indicates the direction to shift the decimal point:
    • A positive exponent (+k+k) indicates moving the decimal point kk places to the right.
    • A negative exponent (−k-k) indicates moving the decimal point kk places to the left.
    • Worked Conversion Examples:
    • 1.35×105=135 0001.35 \times 10^5 = 135\,000 (move decimal 5 places right)
    • 2.7×10−3=0.00272.7 \times 10^{-3} = 0.0027 (move decimal 3 places left)
    • −2.01×104=−20 100-2.01 \times 10^4 = -20\,100 (move decimal 4 places right)
    • 2.87×109=2 870 000 0002.87 \times 10^9 = 2\,870\,000\,000 (move decimal 9 places right)
    • 1.9×10−5=0.0000191.9 \times 10^{-5} = 0.000019 (move decimal 5 places left)

Dimensions, Units, and Prefix Conversions

  • Dimensional Consistency:

    • Measurements of physical quantities must be expressed in units that correspond directly to their dimension.
    • Measurements combined in calculations must share identical units. For instance, calculating area (length×width\text{length} \times \text{width}) requires both factors to be measured in meters, yielding square meters (m2\text{m}^2).

  • SI Prefixes:

    • Prefixes added to base unit names represent standard powers of 10 to denote larger or smaller scales.

  • Prefix Conversion Exercises:

    • Microseconds in 2 seconds:2 s×106 μs/s=2×106 μs2\,\text{s} \times 10^6\,\mu\text{s/s} = 2 \times 10^6\,\mu\text{s}
    • Kilohertz in 750 megahertz:750 MHz=750×106 Hz=7.5×108 Hz=7.5×105 kHz750\,\text{MHz} = 750 \times 10^6\,\text{Hz} = 7.5 \times 10^8\,\text{Hz} = 7.5 \times 10^5\,\text{kHz}
    • Kilometers in 5 centimeters:5 cm=0.05 m=5×10−5 km5\,\text{cm} = 0.05\,\text{m} = 5 \times 10^{-5}\,\text{km}

  • Sample Unit Conversion Problem — Mass of a Bacterium:

    • Problem Statement: A typical bacterium has a mass of approximately 2.0 fg2.0\,\text{fg} (femtograms). Express this measurement in grams (g\text{g}) and kilograms (kg\text{kg}).
    • Given: mass=2.0 fg\text{mass} = 2.0\,\text{fg}
    • Conversion Factors: 1 fg=10−15 g1\,\text{fg} = 10^{-15}\,\text{g} and 1 kg=103 g1\,\text{kg} = 10^3\,\text{g}
    • Step 1: Convert femtograms to grams2.0 fg×10−15 g1 fg=2.0×10−15 g2.0\,\text{fg} \times \frac{10^{-15}\,\text{g}}{1\,\text{fg}} = 2.0 \times 10^{-15}\,\text{g}
    • Step 2: Convert grams to kilograms2.0×10−15 g×1 kg103 g=2.0×10−18 kg2.0 \times 10^{-15}\,\text{g} \times \frac{1\,\text{kg}}{10^3\,\text{g}} = 2.0 \times 10^{-18}\,\text{kg}

  • Light-Year Unit Conversion Problem:

    • Problem Statement: A light-year (ly\text{ly}) is defined as the distance light travels in one year, where 1 ly=9 500 000 000 000 km1\,\text{ly} = 9\,500\,000\,000\,000\,\text{km}. Convert 1 ly1\,\text{ly} into meters (m\text{m}).
    • Calculation:9.5×1012 km×103 m1 km=9.5×1015 m9.5 \times 10^{12}\,\text{km} \times \frac{10^3\,\text{m}}{1\,\text{km}} = 9.5 \times 10^{15}\,\text{m}

Accuracy, Precision, and Measurement Errors

  • Accuracy Definition:

    • A quantitative description of how close a experimental measurement is to the true, accepted, or target value of the quantity being measured.

  • Precision Definition:

    • The degree of exactness, agreement, or repeatability among multiple independent measurements obtained under identical conditions.

  • Uncertainty:

    • A quantitative measure of confidence in an experimental result or measurement. Lower numerical uncertainty indicates higher confidence.

  • Four Scenarios of Accuracy and Precision:

    • Precise and Accurate: Measurements cluster tightly together directly at the true expected value.
    • Precise, Not Accurate: Measurements cluster tightly together, but are displaced away from the true expected value (systematic bias).
    • Not Precise, Accurate: Measurements are widely scattered, but their mean value centers on the expected value.
    • Not Precise, Not Accurate: Measurements are widely scattered and off-center from the expected value.

Precise and Accurate Target and Plot

Precise, Not Accurate Target and Plot

Not Precise, Accurate Target and Plot

Not Precise, Not Accurate Target and Plot

  • Parallax Error:
    • Parallax is defined as an apparent shift in the position of an object caused by viewing it from different line-of-sight angles.
    • Failing to keep the eye perpendicular to a measurement scale introduces parallax error, reducing the accuracy of the reading.
    • Example (Reading a Liquid Meniscus in a Graduated Cylinder):
    • Line of sight too high looking down: Incorrect reading of 19.82 mL19.82\,\text{mL}.
    • Line of sight perpendicular (true eye level): Correct reading of 19.70 mL19.70\,\text{mL}.
    • Line of sight too low looking up: Incorrect reading of 19.62 mL19.62\,\text{mL}.

Parallax Error in Meniscus Reading

Rules for Significant Figures and Rounding

  • Significant Figures Definition:

    • Significant figures are all digits in a measured quantity that are known with absolute certainty plus the first estimated (uncertain) digit.
    • Reflects the numerical precision limit of the measuring instrument.

  • Rules for Determining Significant Zeros:

    • Rule 1 (Nonzero digits): All non-zero digits are always significant (e.g., 19951995 has 4 sig figs).
    • Rule 2 (Captive Zeros): Zeros positioned between other non-zero digits are significant (e.g., 50.3 m50.3\,\text{m} has 3 sig figs; 3.0025 s3.0025\,\text{s} has 5 sig figs).
    • Rule 3 (Leading Zeros): Zeros located in front of all non-zero digits are NOT significant; they serve only as decimal placeholders (e.g., 0.892 kg0.892\,\text{kg} has 3 sig figs; 0.0008 ms0.0008\,\text{ms} has 1 sig fig).
    • Rule 4 (Trailing Zeros with Decimal Point): Zeros at the end of a number and to the right of a decimal point are significant (e.g., 57.00 g57.00\,\text{g} has 4 sig figs; 2.000000 kg2.000000\,\text{kg} has 7 sig figs; 70000.070000.0 has 6 sig figs).
    • Rule 5 (Trailing Zeros without Decimal Point): Zeros at the end of a number without a visible decimal point are assumed non-significant (e.g., 1000 m1000\,\text{m} has 1 sig fig; 20 m20\,\text{m} has 1 sig fig; 7000070000 has 1 sig fig). Scientific notation is required to show additional precision.
    • Rule 6 (Exact Numbers): Pure counts or exact defined conversion ratios possess an infinite number of significant figures.

Rules for Determining Significant Zeros

  • Rules for Calculating with Significant Figures:
    • Addition and Subtraction: Round the calculated sum or difference to the column containing the leftmost estimated digit (the fewest decimal places).
    • Example:       97.3+5.85=103.15→103.297.3 + 5.85 = 103.15 \rightarrow 103.2
    • Multiplication and Division: Round the product or quotient to match the fewest number of significant figures present in any single factor.
    • Example:       123×5.35=658.05→658123 \times 5.35 = 658.05 \rightarrow 658

Rules for Calculating with Significant Figures

  • Rules for Rounding in Calculations:
    • Round Down:
    • When the digit following the last significant figure is 0,1,2,3, or 40, 1, 2, 3, \text{ or } 4 (e.g., 30.24→30.230.24 \rightarrow 30.2).
    • When the last significant figure is even and the next digit is exactly 55 followed only by zeros (e.g., 32.25→32.232.25 \rightarrow 32.2; 32.65000→32.632.65000 \rightarrow 32.6).
    • Round Up:
    • When the digit following the last significant figure is 6,7,8, or 96, 7, 8, \text{ or } 9 (e.g., 22.49→22.522.49 \rightarrow 22.5).
    • When the digit following the last significant figure is a 55 followed by any non-zero digit (e.g., 54.7511→54.854.7511 \rightarrow 54.8).
    • When the last significant figure is odd and the next digit is exactly 55 followed only by zeros (e.g., 54.75→54.854.75 \rightarrow 54.8; 79.3500→79.479.3500 \rightarrow 79.4).

Rules for Rounding in Calculations

Practice Exercises and Review Problems

  • Matching SI Base Quantities with Units:

    • Length =Meter (m)=\text{Meter (m)}
    • Mass =Kilogram (kg)=\text{Kilogram (kg)}
    • Time =Second (s)=\text{Second (s)}
    • Temperature =Kelvin (K)=\text{Kelvin (K)}
    • Amount of substance =Mole (mol)=\text{Mole (mol)}
    • Electric current =Ampere (A)=\text{Ampere (A)}
    • Luminous intensity =Candela (cd)=\text{Candela (cd)}

  • Multiple Choice Review Questions:

    • Question: Which of the following is NOT an SI base quantity?
    • Options: A) Time, B) Length, C) Pressure, D) Temperature
    • Answer: C) Pressure (Pressure is a derived quantity).
    • Question: The symbol for the SI unit of amount of substance is:
    • Options: A) K, B) A, C) mol, D) cd
    • Answer: C) mol
    • Question: What is the SI base unit for length?
    • Options: F) inch, G) foot, H) meter, J) kilometer
    • Answer: H) meter
    • Question: If you do not keep your line of sight directly over a length measurement, how will your measurement most likely be affected?
    • Options: F) Less precise, G) Less accurate, H) Fewer sig figs, J) Instrument error
    • Answer: G) Your measurement will be less accurate due to parallax error.
    • Question: A room is measured to be 3.6 m3.6\,\text{m} by 5.8 m5.8\,\text{m}. What is the area of the room using appropriate significant figures?
    • Options: F) 20.88 m220.88\,\text{m}^2, G) 2×101 m22 \times 10^1\,\text{m}^2, H) 2.0×101 m22.0 \times 10^1\,\text{m}^2, J) 21 m221\,\text{m}^2
    • Answer: J) 21 m221\,\text{m}^2 (3.6 m×5.8 m=20.88 m23.6\,\text{m} \times 5.8\,\text{m} = 20.88\,\text{m}^2, rounded to 2 sig figs).

  • Significant Figures Worksheet Practice Values:

    • 246.32→5246.32 \rightarrow 5 significant figures
    • 1.008→41.008 \rightarrow 4 significant figures
    • 700000→1700000 \rightarrow 1 significant figure
    • 107.854→6107.854 \rightarrow 6 significant figures
    • 0.00340→30.00340 \rightarrow 3 significant figures
    • 350.670→6350.670 \rightarrow 6 significant figures
    • 100.3→4100.3 \rightarrow 4 significant figures
    • 14.600→514.600 \rightarrow 5 significant figures
    • 1.0000→51.0000 \rightarrow 5 significant figures
    • 0.678→30.678 \rightarrow 3 significant figures
    • 0.0001→10.0001 \rightarrow 1 significant figure
    • 320001→6320001 \rightarrow 6 significant figures

  • Short Response Identification:

    • 0.0057 kg→20.0057\,\text{kg} \rightarrow 2 significant figures
    • 5.70 g→35.70\,\text{g} \rightarrow 3 significant figures
    • 6070 m→36070\,\text{m} \rightarrow 3 significant figures
    • 6.070×103 m→46.070 \times 10^3\,\text{m} \rightarrow 4 significant figures