Chapter 1, Problem 1.42PAE

Perform these calculations and express the result with the proper number of significant figures.

(a) $\left(4.850\text{ g}-\text{2}\text{.34 g}\right)/1.3\text{ mL}$

(b) $V=\pi r^3$, where $r=4.112\text{ cm}$

(c) $\left(4.66\times {10}^{-3}\right)\times 4.666$

(d) 0.003400/65.2

Interpretation Introduction

(a)

To perform:

The calculations of $\frac{(4.850g-2.34g)}{1.3mL}$and express the result in proper number of significant figure.

Explanation of Solution

First, set the rules for “significant numbers”

• zeroes to the left are not significant.
• zeroes to the right are not significant if not specified as “exact” zeroes.
• all other digits other than zero are significant.
• zeroes between non-zero numbers are significant.

Second, define the addition/subtraction and multiplication/division rules for significant numbers.

• Addition/Subtraction ( Always keep the least significant number after the decimal point.
• Multiplication/Division ( Always keep the least significant number of the whole number.
• If addition/subtraction and Multiplication/division are included, separate the operation to simple operations and keep following the rules.

Now, for the given operation.$\frac{(4.850g-2.34g)}{1.3mL}$

Perform subtraction.$\frac{2.510g}{1.3mL}$

Since there are only two significant digits after the decimal, keep only 2 significant figures in the product.$\frac{2.51g}{1.3mL}$

Now, for multiplication, perform total operation.

$1.9307\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$mL$}\right.$

According to sig. fig. rules, we must keep only 2 sig. fig. since that is the least amount of sig. fig. for 1.3 mL. Round if required.

$1.93\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$mL$}\right.$

Rounding is required. Report:

$1.9\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$mL$}\right.$.

Interpretation Introduction

(b)

To perform:

The calculations of ${\text{V=πr}}^{\text{3}}\text{;}\text{r = 4}\text{.112 cm}$and express the result in proper number of significant figure.

Explanation of Solution

First, set the rules for “significant numbers”

• zeroes to the left are not significant.
• zeroes to the right are not significant if not specified as “exact” zeroes.
• all other digits other than zero are significant.
• zeroes between non-zero numbers are significant.

Second, define the addition/subtraction and multiplication/division rules for significant numbers.

• Addition/Subtraction ( Always keep the least significant number after the decimal point.
• Multiplication/Division ( Always keep the least significant number of the whole number.
• If addition/subtraction and Multiplication/division are included, separate the operation to simple operations and keep following the rules.

Now, for the given operation. Substitute$V=\pi r^3;r=4.112cm$

$V=\pi x{(4.112cm)}^3$

Assume that pi is actually an exact number, i.e. has infinite amount of significant figures.

$V=218.42839c{m}^3$

According to sig. fig. rules, we must keep only 4 sig. fig. since that is the least amount of significant figures for $4.112$. Round if required$V=218.4c{m}^3$

Interpretation Introduction

(c)

To perform:

The calculations of $\text{(4}{\text{.66x10}}^{\text{-3}}\text{)x4}\text{.666}$and express the result in proper number of significant figure.

Explanation of Solution

First, set the rules for “significant numbers”

• zeroes to the left are not significant.
• zeroes to the right are not significant if not specified as “exact” zeroes.
• all other digits other than zero are significant.
• zeroes between non-zero numbers are significant

Second, define the addition/subtraction and multiplication/division rules for significant numbers.

• Addition/Subtraction ( Always keep the least significant number after the decimal point.
• Multiplication/Division ( Always keep the least significant number of the whole number.
• If addition/subtraction and Multiplication/division are included, separate the operation to simple operations and keep following the rules.

Now, for the given operation. Perform calculation$\begin{array}{l}(4.66x{10}^{-3})x4.666\\ 21743.56\end{array}$

Total sig. fig. reported are 3, for 4.66.

Round if required$\begin{array}{l}21743.56\\ 21400\end{array}$

It can also be reported in scientific notation.$\text{2}{\text{.14x10}}^{\text{4}}$.

Interpretation Introduction

(d)

To perform:

The calculations of $\frac{0.003400}{65.2}$and express the result in proper number of significant figure.

Explanation of Solution

First, set the rules for “significant numbers”.

• zeroes to the left are not significant.
• zeroes to the right are not significant if not specified as “exact” zeroes.
• all other digits other than zero are significant.
• zeroes between non-zero numbers are significant.

Second, define the addition/subtraction and multiplication/division rules for significant numbers.

• Addition/Subtraction ( Always keep the least significant number after the decimal point.
• Multiplication/Division ( Always keep the least significant number of the whole number.
• If addition/subtraction and Multiplication/division are included, separate the operation to simple operations and keep following the rules.

Now, for the given operation. Perform calculation$\frac{0.003400}{65.2}=0.000052147$

Total sig. fig. are 4 for $0.003400$ and 3 for 65.2. Choose 3 significant figures in total.

Round if required $\begin{array}{l}0.00005214\\ 0.0000521\end{array}$

Conclusion

Always ensure to keep the correct significant numbers when performing addition, subtraction, multiplication and division. It is relevant to keep track of accuracy in the measurements.