Step 1: Determine the Molar Mass of Sodium Sulphate ($Na_{2}SO_{4}$)

To calculate the mass percent of each element, we first determine the molar mass of the compound by summing the atomic masses of all constituent atoms [Per the Law of Definite Proportions].

Atomic masses used (in $g/mol$):

• $Na = 22.99$
• $S = 32.06$
• $O = 16.00$

Calculation:

$Molar\ Mass = (2 \times 22.99) + (1 \times 32.06) + (4 \times 16.00)$

$Molar\ Mass = 45.98 + 32.06 + 64.00 = 142.04\ g/mol$

Step 2: Define the Mass Percent Formula

The mass percent of an element in a compound is defined by the ratio of the total mass of that element in one mole of the compound to the total molar mass of the compound, multiplied by $100$.

Formula: $Mass\ \%\ of\ Element = \left( \frac{Total\ Mass\ of\ Element}{Molar\ Mass\ of\ Compound} \right) \times 100$

Step 3: Calculating Individual Mass Percentages

Step 4: Verification of Results

To verify the accuracy of the calculation, sum the individual mass percentages. The sum must equal $100\%$ [Per the Principle of Conservation of Mass].

$32.37\% + 22.57\% + 45.06\% = 100.00\%$

Final Solution: The mass percentages of elements in $Na_{2}SO_{4}$ are: Sodium ($Na$) = $32.37\%$, Sulphur ($S$) = $22.57\%$, and Oxygen ($O$) = $45.06\%$.

Initial Value Analysis

The given number is $6.0012$. Scientific notation follows the standard form $N \times 10^n$, where $N$ is a number between $1$ and $10$ (i.e., $1 \leq N < 10$) and $n$ is an integer representing the power of $10$.

Step 1: Identifying the Significant Coefficient

To convert $6.0012$ into scientific notation, the decimal point must be placed such that the resulting coefficient is greater than or equal to $1$ and less than $10$. In the value $6.0012$, the decimal point is already positioned such that the first non-zero digit ($6$) is to the left of the decimal point. [Per the rules of significant figures and standard scientific notation format].

Step 2: Determining the Power of Ten

Since the decimal point does not require any movement to satisfy the condition $1 \leq N < 10$, the decimal point remains in its current position. Consequently, the value is effectively multiplied by $10^0$.

$\text{Calculation: } 6.0012 = 6.0012 \times 10^0$

Analytical Verification

Visual Representation of Decimal Placement

Final Solution: 6.0012 \times 10^0

Step 1: Establishing the Stoichiometric Foundation

The balanced chemical equation for the reaction is given as:

$A + B_2 \rightarrow AB_2$

From the balanced equation, the stoichiometric coefficients indicate the molar ratio required for the reaction [Per the Law of Definite Proportions]:

• $1 \text{ mole of } A$ reacts with $1 \text{ mole of } B_2$.

Step 2: Analyzing the Given Molar Quantities

We are provided with the following initial amounts:

• Initial moles of $A = 2.5 \text{ mol}$
• Initial moles of $B_2 = 5.0 \text{ mol}$

Step 3: Calculating the Required Ratio for Consumption

To determine the limiting reagent, we compare the actual molar ratio with the stoichiometric ratio. The limiting reagent is the reactant that is entirely consumed, thereby limiting the extent of the reaction [Per the Limiting Reagent Concept].

Step 4: Identification of the Limiting Reagent

Since the value for $A$ ($2.5$) is less than the value for $B_2$ ($5.0$), reactant $A$ will be consumed completely before $B_2$.

Specifically, $2.5 \text{ mol}$ of $A$ will react with exactly $2.5 \text{ mol}$ of $B_2$ to produce $2.5 \text{ mol}$ of $AB_2$. This leaves an excess of $(5.0 - 2.5) = 2.5 \text{ mol}$ of $B_2$ remaining.

Visual Representation of Reaction Stoichiometry

Final Solution: The limiting reagent is A, because it is present in a quantity smaller than the stoichiometric requirement to react with the total amount of $B_2$.

Step 1: Identification of Input Values and Significant Figures

To determine the number of significant figures in the result of multiplication and division, the rule is to round the final answer to the same number of significant figures as the number with the fewest significant figures in the calculation [Per the Rules for Significant Figures in Multiplication/Division].

Step 2: Determining the Limiting Factor

We compare the significant figures identified in Step 1:

• $0.02856 \rightarrow 4 \text{ s.f.}$
• $298.15 \rightarrow 5 \text{ s.f.}$
• $0.112 \rightarrow 3 \text{ s.f.}$
• $0.5785 \rightarrow 4 \text{ s.f.}$

Step 3: Logical Application of the Rule

The precision of the final product/quotient is limited by the least precise measurement. Since the least precise term ($0.112$) has 3 significant figures, the final result must be rounded to 3 significant figures to maintain scientific integrity [Per the Convention of Significant Figures].

Final Solution: The answer should be reported with 3 significant figures.

Step 1: Establishing the Metric Conversion Factors

To convert from kilometers ($km$) to millimeters ($mm$) and picometers ($pm$), we must utilize the standard decimal prefixes of the International System of Units (SI). The following base relationships are defined relative to the meter ($m$):

• $1 \text{ km} = 10^3 \text{ m}$ [Definition of kilo prefix]
• $1 \text{ mm} = 10^{-3} \text{ m} \implies 1 \text{ m} = 10^3 \text{ mm}$ [Definition of milli prefix]
• $1 \text{ pm} = 10^{-12} \text{ m} \implies 1 \text{ m} = 10^{12} \text{ pm}$ [Definition of pico prefix]

Step 2: Conversion to Millimeters (mm)

To find the value of $1 \text{ km}$ in millimeters, we apply the chain-link conversion method [Dimensional Analysis Method]:

$1 \text{ km} = 1 \text{ km} \times \left( \frac{10^3 \text{ m}}{1 \text{ km}} \right) \times \left( \frac{10^3 \text{ mm}}{1 \text{ m}} \right)$

Applying the laws of exponents ($a^m \times a^n = a^{m+n}$):

$1 \text{ km} = 1 \times 10^{(3+3)} \text{ mm} = 10^6 \text{ mm}$

Step 3: Conversion to Picometers (pm)

Similarly, to find the value of $1 \text{ km}$ in picometers, we perform the following conversion:

$1 \text{ km} = 1 \text{ km} \times \left( \frac{10^3 \text{ m}}{1 \text{ km}} \right) \times \left( \frac{10^{12} \text{ pm}}{1 \text{ m}} \right)$

Applying the laws of exponents:

$1 \text{ km} = 1 \times 10^{(3+12)} \text{ pm} = 10^{15} \text{ pm}$

Visual Representation of Scale

Final Solution: 1 km = $10^6$ mm = $10^{15}$ pm

Step 1: Theoretical Foundation of Significant Figures

Significant figures are defined as the meaningful digits in a measured or calculated quantity. They represent the precision of the measurement, including all certain digits plus the first uncertain digit [Per IUPAC Guidelines for Chemical Nomenclature and Data].

Step 2: Analysis of Trailing Zeros in Whole Numbers

The number under consideration is $126,000$. According to the established rules of scientific measurement:

• Non-zero digits: The digits $1, 2,$ and $6$ are always significant [Rule: All non-zero digits are significant].
• Trailing zeros: In a whole number that does not contain a decimal point, trailing zeros are considered ambiguous unless specified by scientific notation or a decimal point [Per standard conventions of precision].

Step 3: Evaluating Ambiguity

Without additional information or a decimal point at the end, the trailing zeros in $126,000$ are generally treated as non-significant placeholders indicating the magnitude of the value rather than the precision of the measurement. Thus, only the non-zero digits are counted.

Step 4: Precision Visual Representation

Step 5: Final Determination

Based on the standard protocol for whole numbers ending in zeros, the significant digits are $1, 2,$ and $6$. The three trailing zeros serve only to place the decimal point and are not significant.

Final Solution: The number 126,000 has 3 significant figures.

Step 1: Determination of Mass of Carbon and Hydrogen in the Sample

The combustion reaction is: $C_xH_y + O_2 \rightarrow CO_2 + H_2O$. We utilize the law of conservation of mass to determine the mass of carbon and hydrogen contained within the initial fuel sample [Per the Law of Conservation of Mass].

Mass of Carbon in $CO_2$:

Molar mass of $CO_2 = 12.01 + (2 \times 16.00) = 44.01 \, \text{g/mol}$.

Mass of Carbon = $\left( \frac{12.01 \, \text{g/mol}}{44.01 \, \text{g/mol}} \right) \times 3.38 \, \text{g} = 0.9226 \, \text{g}$

Mass of Hydrogen in $H_2O$:

Molar mass of $H_2O = (2 \times 1.008) + 16.00 = 18.016 \, \text{g/mol}$.

Mass of Hydrogen = $\left( \frac{2 \times 1.008 \, \text{g/mol}}{18.016 \, \text{g/mol}} \right) \times 0.690 \, \text{g} = 0.0772 \, \text{g}$

Step 2: Conversion to Molar Amounts

Now, we convert the mass of each element into its respective moles to determine the atomic ratio.

Step 3: Determination of the Simplest Molar Ratio

Divide each mole value by the smallest value ($0.0766$) to find the integer ratio [Per Dalton's Atomic Theory].

• Ratio for Carbon: $\frac{0.0768}{0.0766} \approx 1$
• Ratio for Hydrogen: $\frac{0.0766}{0.0766} = 1$

Step 4: Construction of Empirical Formula

Since the ratio of $C:H$ is $1:1$, the empirical formula represents the simplest whole-number ratio of atoms in the compound.

Final Solution: The empirical formula of the welding fuel gas is $\text{CH}$.

Step 1: Theoretical Foundation

To determine the number of molecules in a given quantity of a substance, we employ Avogadro’s Constant ($N_A$). By definition, one mole of any substance contains exactly $6.022 \times 10^{23}$ elementary entities (atoms, molecules, or ions).

The relationship is defined by the formula:
$N = n \times N_A$
Where:

• $N$ = Total number of molecules
• $n$ = Amount of substance in moles ($mol$)
• $N_A$ = Avogadro's constant ($6.022 \times 10^{23} \, mol^{-1}$)

Step 2: Identifying Given Parameters

Step 3: Algebraic Calculation

Substituting the given values into the established formula:

$N = 3 \, mol \times 6.022 \times 10^{23} \, molecules/mol$

Performing the arithmetic multiplication:

$N = 18.066 \times 10^{23} \, molecules$

Step 4: Scientific Notation Normalization

To adhere to the standard format for scientific notation ($a \times 10^n$, where $1 \le a < 10$):

$N = 1.8066 \times 10^1 \times 10^{23}$

$N = 1.8066 \times 10^{24} \, molecules$

Visual Representation of Mole-to-Molecule Conversion

Final Solution: The number of molecules in 3 moles of ethane ($C_2H_6$) is $1.8066 \times 10^{24}$ molecules.

Step 1: Identification of Given Quantities and Constants

• Given mass of Helium ($m$): $52\text{ g}$
• Molar mass of Helium ($M$): $4.0026\text{ g/mol}$ (Standard atomic weight)
• Avogadro's constant ($N_A$): $6.022 \times 10^{23}\text{ atoms/mol}$ [Per the definition of the mole]

Step 2: Calculation of Moles of Helium

To determine the number of particles, we first calculate the amount of substance ($n$) in moles using the relation between mass and molar mass:

Substituting the given values:

Step 3: Calculation of Total Number of Atoms

The total number of atoms ($N$) is the product of the number of moles ($n$) and the Avogadro constant ($N_A$):

Step 4: Scientific Notation Adjustment

Adjusting the value to standard scientific notation:

Visual Representation of Mole-Atom Conversion

Following the rules of significant figures, since the input "$52$" has two significant figures, the result should be rounded accordingly.

Final Solution: The number of atoms in 52 g of Helium is $7.82 \times 10^{24}$ atoms.

Step 1: Establishing Fundamental Equivalence

To perform the required unit conversions, we rely on the defined relationship between the cubic centimeter ($cm^3$), the milliliter ($mL$), the liter ($L$), and the cubic decimeter ($dm^3$).

• The primary equivalence is defined as: $1 \text{ mL} = 1 \text{ cm}^3$ [By standard definition in the SI system].
• The relationship between liters and milliliters is: $1 \text{ L} = 1000 \text{ mL}$ [By definition of the metric prefix 'milli-' where $1 \text{ m} = 10^{-3}$ units].
• The relationship between liters and cubic decimeters is: $1 \text{ L} = 1 \text{ dm}^3$ [By geometric derivation: $1 \text{ dm} = 10 \text{ cm}$, therefore $(1 \text{ dm})^3 = (10 \text{ cm})^3 = 1000 \text{ cm}^3 = 1 \text{ L}$].

Step 2: Calculation of Volume in Liters (L)

To convert from $mL$ to $L$, apply the conversion factor $1 \text{ L} / 1000 \text{ mL}$:

$1 \text{ mL} \times \left( \frac{1 \text{ L}}{1000 \text{ mL}} \right) = 0.001 \text{ L}$

In scientific notation, this is expressed as $1 \times 10^{-3} \text{ L}$.

Step 3: Calculation of Volume in Cubic Decimeters ($dm^3$)

Since $1 \text{ L} = 1 \text{ dm}^3$, the value in cubic decimeters must be numerically equivalent to the value in liters.

$0.001 \text{ L} \times \left( \frac{1 \text{ dm}^3}{1 \text{ L}} \right) = 0.001 \text{ dm}^3$

In scientific notation, this is expressed as $1 \times 10^{-3} \text{ dm}^3$.

Visual Representation of Volume Scaling

Conversion Summary Table

Final Solution: 1 mL = 0.001 L = 0.001 $dm^3$ (or $1 \times 10^{-3} \text{ L} = 1 \times 10^{-3} \text{ dm}^3$)

Step 1: Understanding Scientific Notation Requirements

Scientific notation represents a number in the form $N \times 10^n$, where $N$ is a coefficient such that $1 \le |N| < 10$ and $n$ is an integer exponent [Per the International System of Units (SI) standards for numerical representation].

Step 2: Identifying the Significant Figures

In the integer $8008$, all digits are significant. The zeros are trapped between non-zero digits, which characterizes them as significant according to the rules of significant figures [Rule: Zeros between non-zero digits are always significant].

Step 3: Calculating the Decimal Shift

To express $8008$ in the form $N \times 10^n$, we must place the decimal point after the first non-zero digit.
Original number: $8008.0$
Target placement: $8.008$
To move the decimal point three places to the left, we divide by $10^3$: $8008 \div 1000 = 8.008$ [By the Law of Indices, to maintain equality, we must multiply the result by $10^3$].

Step 4: Final Formalization

Combining the coefficient and the power of ten: $8.008 \times 10^3$

Final Solution: 8.008 \times 10^3

Step 1: Identifying Fundamental Atomic Properties

To determine the number of atoms in a given mass expressed in atomic mass units ($u$), we must first identify the atomic mass of the element Helium ($\text{He}$).

• Atomic mass of one atom of $\text{He} = 4.0026 \, u \approx 4 \, u$. [Per the definition of the atomic mass scale, where $1 \, u$ is defined as $1/12$th of the mass of a carbon-12 atom].

Step 2: Establishing the Conversion Factor

The relationship between the mass in atomic mass units and the number of atoms is derived from the definition of the atomic mass unit itself:

$\text{Number of atoms} = \dfrac{\text{Total mass of the sample in } u}{\text{Mass of one atom in } u}$

Step 3: Calculating the Atom Count

Substitute the given values into the ratio derived in Step 2:

$\text{Total mass} = 52 \, u$

$\text{Mass of one atom} = 4 \, u$

$\text{Number of atoms} = \dfrac{52 \, u}{4 \, u} = 13$

Visual Representation of Atomic Mass Quantification

Step 4: Analytical Verification

Since the problem provides the mass in $u$ (atomic mass units) rather than in grams ($g$), we do not require Avogadro's constant ($N_A = 6.022 \times 10^{23} \, \text{atoms/mol}$). Avogadro's constant is only utilized when converting from molar mass (grams/mol) to atomic count. [By Dimensional Analysis, the units of $u$ cancel out, leaving a dimensionless integer representing the discrete count of atoms].

Final Solution: The number of atoms in 52 u of He is 13 atoms.

Step 1: Identifying the Decimal Point and Target Position

In the decimal number $234,000$, the decimal point is implicitly located to the far right of the last digit: $234,000.0$. To represent this in scientific notation, we must express the number in the form $N \times 10^n$, where $N$ is a number between $1$ and $10$ (i.e., $1 \le N < 10$), and $n$ is an integer [Per the standard convention of Scientific Notation].

Step 2: Determining the Shift Factor

To obtain a value $N$ such that $1 \le N < 10$, we must shift the decimal point to the left so that it resides immediately after the first non-zero digit ($2$).

• Original position: After the final zero ($234000.$)
• Target position: Between the $2$ and the $3$ ($2.34000$)

The number of places the decimal point is moved to the left is calculated as follows:

[Justification: Shifting the decimal $n$ places to the left is equivalent to multiplying the original number by $10^n$. Since we are expressing the number as $N \times 10^n$, we compensate by using a positive exponent for the power of $10$].

Step 3: Constructing the Scientific Notation

By moving the decimal point $5$ places to the left, we transform $234,000$ into $2.34$. To maintain the equality of the expression, we multiply this by $10^5$.

Calculation: $234,000 = 2.34 \times 10^5$

Visual Representation of Decimal Shift

Step 4: Verification of Significance

In the original integer $234,000$, the number of significant figures is typically considered to be three ($2, 3, 4$) unless specific measurements imply the zeros are significant. In scientific notation, $2.34 \times 10^5$ clearly maintains these three significant figures.

Final Solution: $2.34 \times 10^5$

Step 1: Balanced Chemical Equation and Stoichiometry

The balanced chemical equation for the formation of ammonia is:

$N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$

[Per the Law of Conservation of Mass, the equation must be balanced to reflect the conservation of atoms.]

Step 2: Calculation of Molar Masses

Using standard atomic masses ($N = 14.01 \, \text{g/mol}$, $H = 1.008 \, \text{g/mol}$):

• Molar mass of $N_2 = 2 \times 14.01 = 28.02 \, \text{g/mol}$
• Molar mass of $H_2 = 2 \times 1.008 = 2.016 \, \text{g/mol}$
• Molar mass of $NH_3 = 14.01 + (3 \times 1.008) = 17.034 \, \text{g/mol}$

Step 3: Determination of Moles of Reactants

Given initial masses:

• $n(N_2) = \frac{2.00 \times 10^3 \, \text{g}}{28.02 \, \text{g/mol}} \approx 71.38 \, \text{mol}$
• $n(H_2) = \frac{1.00 \times 10^3 \, \text{g}}{2.016 \, \text{g/mol}} \approx 496.03 \, \text{mol}$

Step 4: Identification of the Limiting Reagent

According to the stoichiometry, $1 \, \text{mol}$ of $N_2$ requires $3 \, \text{mol}$ of $H_2$:

Required $H_2 = 71.38 \, \text{mol} \times 3 = 214.14 \, \text{mol}$

Since we have $496.03 \, \text{mol}$ of $H_2$, which is significantly greater than $214.14 \, \text{mol}$, $N_2$ is the limiting reagent. [The limiting reagent determines the maximum product yield.]

Step 5: Calculation of Theoretical Yield of Ammonia

From the balanced equation, $1 \, \text{mol}$ of $N_2$ produces $2 \, \text{mol}$ of $NH_3$:

$n(NH_3) = 2 \times n(N_2) = 2 \times 71.38 = 142.76 \, \text{mol}$

Mass of $NH_3 = 142.76 \, \text{mol} \times 17.034 \, \text{g/mol} \approx 2431.77 \, \text{g}$

Final Solution: The mass of ammonia produced is approximately 2432 g (or 2.43 × 10³ g).

Step 1: Identification of Input Values and Significant Figure Rules

To determine the number of significant figures in the product of a calculation, we must analyze the precision of each operand involved. The given expression is $5 \times 5.364$.

• Operand 1 ($5$): This represents an exact count or a defined integer constant. According to the rules of measurement and scientific notation, exact numbers are considered to have an infinite number of significant figures [Per the definition of exact numbers in measurement theory].
• Operand 2 ($5.364$): This is a measured quantity. Following the rules for significant figures:
  • All non-zero digits are significant.
  • The digits $5$, $3$, $6$, and $4$ are all significant.
  • Total significant figures = $4$.

Step 2: Theoretical Principle for Multiplication

According to the standard rules of significant figures in multiplication and division, the result of a calculation must be reported to the same number of significant figures as the measurement with the least number of significant figures present in the calculation [Rule derived from propagation of uncertainty].

Step 3: Execution of the Calculation

Performing the arithmetic operation:

$5 \times 5.364 = 26.82$

Step 4: Application of Constraints

We evaluate the significant figures of the operands:

Since the exact number has infinite precision, the limiting factor is the measured value, $5.364$, which contains $4$ significant figures. Therefore, the result must be expressed with $4$ significant figures.

Final Solution: The result should be reported with 4 significant figures.

Definition of Significant Figures

Significant figures are the meaningful digits in a measured or calculated quantity. They represent the precision of a measurement, including all digits that are known with certainty plus the first digit that is uncertain. [Per the conventions of analytical chemistry and physical measurements].

Rules for Determining Significant Figures

The determination of significant figures in a measured value follows these specific criteria:

• Non-zero digits: All non-zero digits are significant. (e.g., $285$ has 3 significant figures).
• Zeros between non-zero digits: Zeros preceding a non-zero digit are not significant, but zeros located between two non-zero digits are significant. (e.g., $2.005$ has 4 significant figures).
• Leading zeros: Zeros to the left of the first non-zero digit are not significant; they merely indicate the position of the decimal point. (e.g., $0.005$ has 1 significant figure).
• Trailing zeros:
  • In numbers with a decimal point, trailing zeros are significant. (e.g., $0.200$ has 3 significant figures).
  • In numbers without a decimal point, trailing zeros may or may not be significant depending on the context of the measurement precision.
• Exact numbers: Numbers derived from counting objects or defined constants have an infinite number of significant figures. (e.g., in $2$ balls or $1$ inch = $2.54$ cm, the $2$ and $2.54$ are considered to have infinite significance).

Visual Representation of Measurement Precision

Analytical Significance

Significant figures are critical in chemical calculations because they prevent the reporting of results that imply a higher level of accuracy than the equipment used for measurement actually possesses. When performing arithmetic operations, the final result is constrained by the least precise measurement used in the calculation.

Final Solution: Significant figures are the total number of digits in a measurement consisting of all certain digits plus one uncertain digit, reflecting the precision of the observational tool used.

Step 1: Identification of Given Quantities and Constants

To determine the number of atoms in a given mass of an element, we identify the following variables:

• Mass of Helium ($m$) = $52\text{ g}$
• Molar mass of Helium ($M$) = $4.0026\text{ g/mol}$ (approximated to $4.00\text{ g/mol}$ for standard stoichiometric calculations) [Per the definition of Molar Mass]
• Avogadro’s constant ($N_A$) = $6.022 \times 10^{23}\text{ atoms/mol}$ [Per the Avogadro constant definition]

Step 2: Calculation of the Amount of Substance (Moles)

The amount of substance ($n$) in moles is calculated using the ratio of the given mass to the molar mass of the element.

$n = \frac{m}{M}$

$n = \frac{52\text{ g}}{4.00\text{ g/mol}}$

$n = 13\text{ mol}$

[Justification: By the definition of the mole, $n = \frac{\text{mass}}{\text{molar mass}}$]

Step 3: Determination of the Total Number of Atoms

The total number of atoms ($N$) is derived by multiplying the number of moles ($n$) by Avogadro’s constant ($N_A$).

$N = n \times N_A$

$N = 13\text{ mol} \times (6.022 \times 10^{23}\text{ atoms/mol})$

$N = 78.286 \times 10^{23}\text{ atoms}$

[Justification: $N = n \cdot N_A$ represents the conversion from macroscopic moles to microscopic particle counts]

Step 4: Standard Scientific Notation Conversion

To adhere to standard scientific notation (where the coefficient is between 1 and 10), we adjust the exponent:

$N = 7.8286 \times 10^{24}\text{ atoms}$

Visual Representation of Atomic Scale

Final Solution: The number of atoms in 52 g of Helium is $7.83 \times 10^{24}$ atoms.