The value of Avogadro's number is $6.02214076 \times 10^{23}$ $mol^{-1}$. A student experimentally determines Avogadro's number to be $6.022141 \times 10^{23}$ $mol^{-1}$. When reporting this value with the correct number of significant figures reflecting the precision of the experimental result, the student should report:

## CORE INFORMATION\n✓ **Answer**: A\n## CONCEPT FRAMEWORK\n\t**Primary Concept**: Significant Figures\n\t**Related Topics**: Measurement, Precision, Accuracy\n\t**Prerequisites**: Understanding of significant figures rules.\n## SOLUTION PATHWAY\n**Given Information**:\n\t* Accepted Avogadro's Number: $6.02214076 \times 10^{23}$ mol⁻¹\n\t* Experimentally Determined Value: $6.022141 \times 10^{23}$ mol⁻¹\n**Method & Steps**:\n\t1. **Identify the number of significant figures in the experimental measurement.**\n\t\tReasoning: The experimental value dictates the precision of the reported result.\n\t\tThe experimentally determined value has seven significant figures (6.022141).\n\t2. **Report the value with the correct number of significant figures.**\n\t\tWorking: Since the experimental value has seven significant figures, the reported value should also have seven significant figures.\n\t\tKey Point: We do not consider the significant figures of the accepted value when reporting an experimental result.\n\t3. **Match the correct option.**\n\t\tResult: $6.022141 \times 10^{23}$ mol⁻¹\n\t\tVerification: This value has seven significant figures, matching the precision of the experimental measurement.\n## OPTION ANALYSIS\n[A] ✓: $6.022141 \times 10^{23}$ mol⁻¹ - This option correctly represents the experimental value with seven significant figures.\n[B] :x: $6.022 \times 10^{23}$ mol⁻¹ - This option has only four significant figures, which is less precise than the experimental measurement.\n[C] :x: $6.02214 \times 10^{23}$ mol⁻¹ - This option has six significant figures, which is less precise than the experimental measurement.\n[D] :x: $6.0221 \times 10^{23}$ mol⁻¹ - This option has five significant figures, which is less precise than the experimental measurement.\n## LEARNING AIDS\n\t**Common Mistakes**:\n\t\t1. Incorrectly rounding based on the accepted value: Remember to base the significant figures on the experimental measurement, not the theoretical or accepted value.\n\t\t2. Miscounting significant figures: Review the rules for determining significant figures, especially regarding zeros.\n\t**Quick Tips**:\n\t\t1. When performing calculations with experimental data, keep track of the least precise measurement to determine the significant figures in the final result.\n\t\t2. Practice identifying significant figures in various numbers to improve your accuracy.

The correct Option is "6.022141 × 10²³ mol⁻¹"

The mass of an electron is approximately $9.1093837 \times 10^{-31}$ kg. If this value is multiplied by the speed of light squared ($c^2 = (2.99792458 \times 10^8 m/s)^2$), what is the correct representation of the resulting energy in joules, considering significant figures?

## CORE INFORMATION\n✓ **Answer**: Option A\n## CONCEPT FRAMEWORK\n\t**Primary Concept**: Mass-Energy Equivalence\n\t**Related Topics**: Special Relativity, Significant Figures\n\t**Prerequisites**: Basic Algebra, Scientific Notation, Understanding of Significant Figures\n## SOLUTION PATHWAY\n**Given Information**:\n\tMass of electron (m) = $9.1093837 \times 10^{-31}$ kg\n\tSpeed of light (c) = $2.99792458 \times 10^8$ m/s\n\tEnergy (E) = $mc^2$\n**Method & Steps**:\n\t1. Calculate $c^2$:\n\t\tReasoning: We need to square the speed of light.\n\t\tFormula/Rule: $c^2 = (2.99792458 \times 10^8)^2$\n\t2. Calculate the energy E:\n\t\tWorking: $E = mc^2 = (9.1093837 \times 10^{-31} kg) * (2.99792458 \times 10^8 m/s)^2$\n\t\tKey Point: We must maintain the correct number of significant figures throughout the calculation. The mass has 8 significant figures, and the speed of light has 8 as well. Thus, the result should have 8 significant figures.\n\t3. Calculate the numerical value:\n Working: $E = 9.1093837 \times 10^{-31} \times (8.987551787368164 \times 10^{16}) = 8.1871057 \times 10^{-14}$ J\n\t4. Final Result:\n\t\tResult: $8.1871057 \times 10^{-14}$ J\n\t\tVerification: The units are correct (kg*(m/s)^2 = kg*m^2/s^2 = Joules). The number of significant figures is 8 as required.\n## OPTION ANALYSIS\n[A] ✓: The calculated result matches this option with correct number of significant figures.\n[B] :x: This option has more significant figures than allowed by the given data.\n[C] :x: This option has less significant figures than the original data.\n[D] :x: This option has less significant figures and is also not correctly rounded.\n## LEARNING AIDS\n\t**Common Mistakes**:\n\t\t1. [Error 1]: Incorrectly squaring the speed of light or miscalculating the exponents. [How to avoid]: Use a calculator carefully and check your work.\n\t\t2. [Error 2]: Rounding too early or not using the correct number of significant figures. [How to avoid]: Keep all digits in intermediate calculations and only round at the end based on the lowest number of significant figures in the initial values.\n\t**Quick Tips**:\n\t\t1. [Helpful shortcut]: When multiplying or dividing, the number of significant figures in the result should be the same as the number of significant figures in the value with the fewest significant figures.\n\t\t2. [Memory aid]: Remember the formula for mass-energy equivalence: $E=mc^2$.\n

The correct Option is "$8.1871057 \times 10^{-14}$ J"

The density of a substance is measured as $3.42 \times 10^{-2}$ kg/m$^3$. How many significant figures are in this measurement?

## CORE INFORMATION\n✓ **Answer**: [Option C]\n## CONCEPT FRAMEWORK\n\t**Primary Concept**: Significant Figures\n\t**Related Topics**: Measurement, Scientific Notation\n\t**Prerequisites**: Rules for identifying significant figures\n## SOLUTION PATHWAY\n**Given Information**:\n\t* Measured density: $3.42 \times 10^{-2}$ kg/m³\n**Method & Steps**:\n\t1. Identify the significant figures in the coefficient:\n\t\tReasoning: In scientific notation, all digits in the coefficient (the number before the power of 10) are considered significant.\n\t\tFormula/Rule: Rules for significant figures in scientific notation.\n\t2. Count the significant figures in the coefficient:\n\t\tWorking: The coefficient is 3.42, which has three digits.\n\t\tKey Point: The power of 10 (10^-2) does not affect the number of significant figures.\n\t3. Determine the total significant figures:\n\t\tResult: The number of significant figures in the measurement is 3.\n\t\tVerification: The number of significant digits are the number of digits in the mantissa.\n ## OPTION ANALYSIS\n[A] :x: [Incorrect. There are more than one significant figures.]\n[B] :x: [Incorrect. There are three significant figures, not two.]\n[C] ✓: [Correct. The number 3.42 has three significant digits. The exponent does not affect the number of significant figures.]\n[D] :x: [Incorrect. There are not four significant figures.]\n ## LEARNING AIDS\n\t**Common Mistakes**:\n\t\t1.[Error 1]: Counting the exponent as a significant figure. [Always remember that the exponent in scientific notation only indicates the magnitude and not the precision of the measurement.]\n\t\t2.[Error 2]: Not recognizing the rules for scientific notation. [Make sure you know the rules for significant figures in different formats.]\n\t**Quick Tips**:\n\t\t1.[Helpful shortcut]: The number of significant figures is equal to the number of digits in the coefficient of scientific notation.]\n\t\t2.[Memory aid]: Remember that leading zeros are not significant, but trailing zeros after a decimal point are significant. In scientific notation, all the digits in the coefficient are significant.

The measurement $4.00 \times 10^3$ m has how many significant figures?

## CORE INFORMATION\n✓ **Answer**: [Option C]\n## CONCEPT FRAMEWORK\n\t**Primary Concept**: Significant Figures\n\t**Related Topics**: Scientific Notation, Measurement\n\t**Prerequisites**: Understanding of significant figure rules\n## SOLUTION PATHWAY\n**Given Information**:\n\t\tMeasurement: $4.00 \times 10^3$ m\n\t\t\n**Method & Steps**:\n\t1.\tIdentify the non-zero digits: The non-zero digit is 4.\n\t\tReasoning: All non-zero digits are significant.\n\t\tFormula/Rule: N/A\n\t2.\tIdentify zeros between non-zero digits: There are no zeros between non-zero digits.\n\t\tWorking: N/A\n\t\tKey Point: N/A\n\t3.\tIdentify trailing zeros after a decimal point: There are two trailing zeros after the decimal point.\n\t\tReasoning: Trailing zeros after a decimal are significant.\n\t\tFormula/Rule: N/A\n\t4.\tCount the significant figures: Count all significant digits.\n\t\tResult: 3 significant figures\n\t\tVerification: 4.00 has 3 significant figures.\n ## OPTION ANALYSIS\n[A] :x: Incorrect. There are three significant figures, not one.\n[B] :x: Incorrect. There are three significant figures, not two.\n[C] ✓: Correct. The number 4.00 has three significant figures because the zeros after the decimal point are significant.\n[D] :x: Incorrect. There are three significant figures, not four.\n ## LEARNING AIDS\n\t**Common Mistakes**:\n\t\t1.[Error 1]: Forgetting that trailing zeros after a decimal point are significant. [How to avoid]: Always check the decimal point and trailing zeros.\n\t\t2.[Error 2]: Considering the power of 10 as a significant figure. [How to avoid]: The exponent part in scientific notation is not used to determine significant figures.\n\t**Quick Tips**:\n\t\t1.[Helpful shortcut]: Focus on the digits in the coefficient of the scientific notation, ignore the power of 10 when counting significant figures.\n\t\t2.[Memory aid]: Remember the rules for non-zero digits, zeros between non-zero digits, and trailing zeros after a decimal point.