# 4.5: "Component Within:" Equality Pattern and Conversions

- Page ID
- 213286

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

Upon establishing that a "component within" molar relationship is required to solve a problem, a corresponding "component within" equality must be developed. Then, using dimensional analysis, the resultant equality can be applied as a conversion factor, in order to bring about a desired unit transformation.

## "Component Within" Equality Pattern

As stated in Section 4.1, an equality pattern contains one number and * two* units on both sides of an equal sign. One of these units, "mol," is defined on both sides of a "component within" equality pattern, as shown below. Neither secondary unit is specified in this equality pattern. These positions, which are indicated as "blanks" in the equality pattern shown below, should be occupied by units that are relevant to the identities of the specific chemicals that are referenced in a given problem. As indicated below, the secondary unit on the left side of a "component within" equality should be the

*chemical formula*of the

*compound*that is referenced in the given problem, and the final unit position on the right side of the equality should be occupied by the

*chemical formula*of the specific

*element*that is being considered. Therefore, unlike in an Avogadro's number equality, the chemical formulas written on both sides of a "component within" equality

*do not match*. As stated in Section 4.3, a chemical name should

*not*be used in this, or any, equality. Finally, the relative order of the two units on the right side of a "component within" equality should not be interchanged.

As was the case in an Avogadro's number equality, a numerical value of "1" is specified on the left side of a "component within" equality. Recall that Avogadro's number, 6.02 × 10^{23}, occupies the numerical position on the right side of an Avogadro's number equality. However, a "blank" is present in the corresponding position in the "component within" equality pattern that is shown below. A "component within" equality establishes the relative ratios of the elements that are present within a compound, as indicated by the subscripts found on the elemental symbols within that compound's chemical formula. Therefore, the *subscript *for the particular element being considered should occupy the numerical position on the right side of a "component within" equality.

For example, consider a problem that requires a calculation of how many moles of Ag are present in 11.5 moles of Ag_{2}CO_{3}.

Because both a complete molecule, Ag_{2}CO_{3},_{ }silver carbonate, and a component element found within this molecule, Ag, silver, are referenced in the given problem, a "component within" equality should be developed. Since all chemical information was provided in the form of elemental symbols and/or chemical formulas, the symbols "Ag_{2}CO_{3}" and "Ag" are directly incorporated into the secondary unit positions on the left and right sides, respectively, of the equality that is being developed. Finally, since the subscript for Ag, the particular element that is being considered in this problem, is a "2" in the chemical formula for the compound, Ag_{2}CO_{3}, a 2 is inserted into the numerical position on the right side of this "component within" equality, as shown below.

1 mol Ag_{2}CO_{3} = 2 mol Ag

## Applying "Component Within" Equalities as Conversion Factors

Once an appropriate "component within" equality has been developed, the information that it contains can be re-written in the form of a conversion factor, which can then be applied to bring about a desired unit transformation. As stated previously, the quantity containing the unit being canceled must be written in the *denominator *of a conversion factor. This will cause the given unit, which appears in a numerator, to be divided by itself, since the same unit appears in the denominator of the conversion factor. Since any quantity that is divided by itself "cancels," orienting the conversation factor in this way results in the elimination of the undesirable unit. However, remember that both components of the equalities that are developed within this chapter contain *two *units. Therefore, in order to achieve complete unit cancelation, a conversion factor that results in the *simultaneous *elimination of *both units* must be applied.

For example, use a conversion factor based on the equality developed above for Ag to calculate how many moles of Ag are present in 11.5 moles of Ag_{2}CO_{3}.

As stated above, because both a complete molecule, Ag_{2}CO_{3},_{ }and a component element found within this molecule, Ag, are referenced in the given problem, a "component within" equality should be developed and applied to solve this problem. The equality that was generated based on the subscript information that corresponds to Ag in Ag_{2}CO_{3 }is replicated below.

1 mol Ag_{2}CO_{3} = 2 mol Ag

To create a conversion factor from this equality, the quantity on the left side of the equal sign is written in the numerator of a fraction, and the other quantity is written in the denominator. A second conversion factor can be developed by interchanging where each quantity is written, relative to the fraction bar. Both of the resultant conversion factors are shown below.

\( \dfrac{1 \text{ mol } \ce{Ag_2CO_3}}{2 {\text{ mol }} \text {Ag }} \) and \( \dfrac{2 {\text{ mol }} \text {Ag }}{1 \text{ mol } \ce{Ag_2CO_3}} \)

However, only one of these conversion factors will allow for the complete cancelation of the given unit, "moles of Ag_{2}CO_{3}," since *both *of the units that are being canceled must be written in the denominator of the conversion factor that should be applied to solve the given problem. Since the intent of this problem is to eliminate the unit "moles of Ag_{2}CO_{3}," the conversion factor on the right must be used. Therefore,

\( {11.5 \; \cancel{\rm{mol} \; \rm{Ag_2CO_3}}} \times\) \( \dfrac{2 \; \rm{mol} \; \rm{Ag}}{1 \; \cancel{\rm{mol} \; \rm{Ag_2CO_3}}}\)

The solution is calculated by multiplying the given number by the value in each numerator, and then dividing by the quantity in each denominator. Recall that, when using a calculator, each conversion factor should be entered in parentheses, * or* the "=" key should be used after

*division. In this case,*

*each*11.5 × (2 mol Ag ÷ 1) = 23 mol Ag ≈ 23.0 mol Ag

Finally, remember that the correct number of significant figures should be applied to any calculated quantity. Since the math involved in dimensional analysis is multiplication and division, the number of significant figures in each number being multiplied or divided must be counted, and the answer must be limited to the lesser count of significant figures. The numerical quantities within a "component within" equality are exact values, meaning that they are considered to have infinitely-many significant figures and will never limit the number of significant figures in a calculated answer. However, the given number, 11.5, is not exact, and its significant figures must be considered. As this value contains three significant figures, the final answer should be rounded to three significant digits, as shown above.