Chapter 1, Problem 1.37P

(a)

Interpretation Introduction

Interpretation:

SI unit conversion has to be performed for $\text{1}\text{.00 liters}$ to quartz and expressed in correct number of significant figures.

Concept Introduction:

Conversion factors:

Conversion of one unit into another unit is done by the use of unit conversion factor.  This involves multiplying or dividing by the power of 10.  For example, if centimeter has to be converted into meter, then the given value in centimeter has to be multiplied by the conversion factor as shown below.

    $1=\frac{1\text{m}}{100\text{cm}}$

Significant figures:

Precision of the quantities that are measured use the significant figures to express the result.  There are six rules for determining the number of significant figures.  They are given as follows.

• Nonzero digits and the zeros that are present between nonzero digits are significant.
• Zeros that are used merely to position the decimal are not significant.
• If zeros are present in the right end of the decimal point, then the zeros are significant.
• If zeros are present in the result are not in right of the decimal point, then the zeros may or may not be significant.
• If the zeros disappear when scientific notation is used means then those zeros are not significant.
• Exact numbers have no limit to their precision and they do not obey the rules given for significant figures.

Answer for a calculation should be always expressed in least significant figures in case of multiplication and division.

In case of addition and subtraction, the answer should have least significant figures after the decimal point.

Explanation of Solution

Given value is $\text{1}\text{.00 liters}$.  This has to be converted into quartz using the conversion factor.  Conversion factor for liters and quartz conversion is given below.

    $\frac{1\text{L}}{1.05669\text{quarts}}or\frac{1.05669\text{quarts}}{1\text{L}}$

As the given value is in liters, the second conversion factor is considered.  The SI unit conversion is performed as shown below.

    $1\text{L}\times \frac{1.05669\text{quarts}}{1\text{L}}=1.05669\text{quarts}$

Number $\text{1}\text{.00}\text{liters}$ has three significant figures and $\text{1}\text{.05669}\text{quarts}$ has six significant figures.  Therefore, the answer should contain only three significant figures.

Therefore,

  $1.05669\text{quarts}\approx \text{1}\text{.06}\text{quarts}$

(b)

Interpretation Introduction

Interpretation:

SI unit conversion has to be performed for $\text{186000}\text{miles per second}$ to meter per second and expressed in correct number of significant figures.

Concept Introduction:

Refer part (a).

Explanation of Solution

Given value is $\text{186000}\text{miles per second}$.  This has to be converted into meter per second using the conversion factor.  Conversion factor for miles per second and meter per second conversion is given below.

    $\frac{1\text{mile}{\text{s}}^{\text{-1}}}{1609.344{\text{ms}}^{-1}}or\frac{1609.344{\text{ms}}^{-1}}{1\text{mile}{\text{s}}^{\text{-1}}}$

As the given value is in miles per second, the second conversion factor is considered.  The SI unit conversion is performed as shown below.

    $186000\text{mile}{\text{s}}^{\text{-1}}\times \frac{1609.344{\text{ms}}^{-1}}{1\text{mile}{\text{s}}^{\text{-1}}}=\text{299337984}{\text{ms}}^{-1}$

Number $\text{186000}$ has three significant figures and $\text{299337984}$ has eight significant figures.  Therefore, the answer should contain only three significant figures.

Therefore,

  $\begin{array}{c}\text{299337984}{\text{ms}}^{-1}\approx \text{2}\text{.99337894}\times {\text{10}}^{\text{8}}{\text{ms}}^{-1}\\ \approx 3.00\times {10}^8{\text{ms}}^{-1}\end{array}$

(c)

Interpretation Introduction

Interpretation:

SI unit conversion has to be performed for $\text{8}\text{.3145}\text{J}\cdot {\text{K}}^{\text{-1}}\cdot {\text{mol}}^{\text{-1}}$ to $\text{cal}\cdot {\text{K}}^{\text{-1}}\cdot {\text{mol}}^{\text{-1}}$ and expressed in correct number of significant figures.

Concept Introduction:

Refer part (a).

Explanation of Solution

Given value is $\text{8}\text{.3145}\text{J}\cdot {\text{K}}^{\text{-1}}\cdot {\text{mol}}^{\text{-1}}$.  Conversion factors for the given conversion is shown below.

    $\frac{1\text{cal}}{4.18400\text{J}}or\frac{4.18400\text{J}}{1\text{cal}}$

As the given value is in joules, the first conversion factor is considered.  The SI unit conversion is performed as shown below.

    $\text{8}\text{.3145}\text{J}\cdot {\text{K}}^{\text{-1}}\cdot {\text{mol}}^{\text{-1}}\times \frac{1\text{cal}}{4.18400\text{J}}=\text{1}\text{.987213193116635}\text{cal}\cdot {\text{K}}^{\text{-1}}\cdot {\text{mol}}^{\text{-1}}$

Number $\text{8}\text{.3145}$ has five significant figures and $\text{4}\text{.18400}$ has six significant figures.  Therefore, the answer should contain only five significant figures.

Therefore,

  $\text{1}\text{.987213193116635}\text{cal}\cdot {\text{K}}^{\text{-1}}\cdot {\text{mol}}^{\text{-1}}\approx \text{1}\text{.9872}\text{cal}\cdot {\text{K}}^{\text{-1}}\cdot {\text{mol}}^{\text{-1}}$