Chapter 1, Problem 56QP

Carry out the following conversions: (a) 22.6 m to decimeters, (b) 25.4 mg to kilograms, (c) 556 mL to liters, (d) $\text{10}{\text{.6 kg/m}}^{\text{3}}$ to ${\text{g/cm}}^{\text{3}}$.

Interpretation Introduction

Interpretation: The given conversions are to be carried out by dimensional analysis.

Concept introduction:

Dimensional analysis is a way to convert units of measurement. In order to convert from one unit to another, we need to know the relationship between those units. These relationships are called conversion factors. Dimensional analysis is used to set up and solve a unit conversion problem using conversion factors.

The conversion factor is a fraction obtained from a relationship between the units. It is written as a ratio and can be inverted to give two conversion factors for every relationship.

$\text{1 m = 10 dm}$, to convert meter to decimeters, the conversion factor is

$\frac{\text{10 dm}}{\text{1 m}}$

${\text{1 mg = 1×10}}^{-\text{3}}\text{ g}$, to convert milligram to gram, the conversion factor is

$\frac{{\text{1×10}}^{-\text{3}}\text{ g}}{\text{1 mg}}$

${\text{1 g = 1×10}}^{\text{-3}}\text{ kg}$, to convert gram to kilogram, the conversion factor is

$\frac{{\text{1×10}}^{\text{-3}}\text{ kg}}{\text{1 g}}$

$\text{1 kg }=\text{ 1000 g}$, to convert kilogram to gram, the conversion factor is

$\frac{\text{1000 g}}{\text{1 kg}}$

$\text{1 mL }={\text{ 1×10}}^{\text{-3}}\text{ L}$, to convert milliliters to liters, the conversion factor is

$\frac{{\text{1×10}}^{\text{-3}}\text{ L}}{\text{1 mL}}$

$\text{1 m = 100 cm}$, to convert m to centimeter, the conversion factor is

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

Answer to Problem 56QP

Solution:

a)

$226\text{ dm}$

b)

$\text{2}{\text{.54×10}}^{-5}\text{ kg}$

c)

$0.556\text{ L}$

d)

$\text{0}{\text{.0106 g/cm}}^{\text{3}}$

Explanation of Solution

a) $22.6\text{ m}$ to decimeters.

Convert the quantity $22.6\text{ m}$ to decimeters.

The conversion factors for the relationship $\text{1 m = 10 dm}$ are as follows:

$\frac{\text{1 m}}{\text{10 dm}}$ or $\frac{\text{10 dm}}{\text{1 m}}$

Now, set up the unit conversion problem using the appropriate conversion factor.

$\text{22}\text{.6 }\cancel{\text{m}}\times \frac{\text{10 dm}}{\text{1 }\cancel{\text{m}}}=\text{ 226 dm}$

The units in the conversion factor must cancel to give the correct unit for the answer.

Hence, $\text{22}\text{.6 m }=\text{ 226 dm}$.

b) $\text{25}\text{.4 mg}$ to kilograms.

Convert the quantity $\text{25}\text{.4 mg}$ to kilograms.

The conversion factors for the relationship ${\text{1 mg = 1×10}}^{-\text{3}}\text{ g}$ are as follows:

$\frac{\text{1 mg}}{{\text{1×10}}^{-\text{3}}\text{ g}}$ or $\frac{{\text{1×10}}^{-\text{3}}\text{ g}}{\text{1 mg}}$

The conversion factors for the relationship ${\text{1 g = 1×10}}^{\text{-3}}\text{ kg}$ are as follows:

$\frac{\text{1 g}}{{\text{1×10}}^{\text{-3}}\text{ kg}}$ or $\frac{{\text{1×10}}^{\text{-3}}\text{ kg}}{\text{1 g}}$

Now, set up the unit conversion problem using the appropriate conversion factors.

$\left(\text{25}\text{.4 }\cancel{\text{mg}}\text{ × }\frac{{\text{1×10}}^{-3}\cancel{\text{g}}}{\text{1 }\cancel{\text{mg}}}\text{ × }\frac{{\text{1×10}}^{-3}\text{ kg}}{\text{1 }\cancel{\text{g}}}\right)\text{ = 2}{\text{.54×10}}^{-5}\text{ kg}$

The units in the conversion factor must cancel to give the correct unit for the answer.

Hence,

$\text{25}\text{.4 mg }=\text{ 2}{\text{.54×10}}^{-5}\text{ kg}$.

c) $\text{556 mL}$ to liters.

Convert the quantity $\text{556 mL}$ to liters.

The conversion factors for the relationship $\text{1 mL }={\text{ 1×10}}^{\text{-3}}\text{ L}$ are as follows:

$\frac{\text{1 mL}}{{\text{1×10}}^{\text{-3}}\text{ L}}$ or $\frac{{\text{1×10}}^{\text{-3}}\text{ L}}{\text{1 mL}}$

Now, set up the unit conversion problem using the appropriate conversion factor.

$\left(\text{556 }\cancel{\text{mL}}\text{ × }\frac{{\text{1×10}}^{\text{-3}}\text{ L}}{\text{1 }\cancel{\text{mL}}}\right)=\text{ 0}\text{.556 L}$

The units in the conversion factor must cancel to give the correct unit for the answer.

Hence,

$\text{556 mL}=\text{0}\text{.556 L}$

d) $\text{10}{\text{.6 kg/m}}^{\text{3}}$ to g/cm³.

Convert the quantity $\text{10}{\text{.6 kg/m}}^{\text{3}}$ to g/cm³.

The conversion factors for the relationship $\text{1 kg }=\text{ 1000 g}$ are as follows:

$\frac{\text{1 kg}}{\text{1000 g}}$ or $\frac{\text{1000 g}}{\text{1 kg}}$

The conversion factors for the relationship $\text{1 m = 100 cm}$ are as follows:

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

Now, set up the unit conversion problem using the appropriate conversion factors.

$\left(\text{10}\text{.6 }\frac{\cancel{\text{kg}}}{{\cancel{\text{m}}}^{\text{3}}}\text{×}\frac{\text{1000 g}}{\text{1 }\cancel{\text{kg}}}\text{×}{\left(\frac{\text{1 }\cancel{\text{m}}}{\text{100 cm}}\right)}^{\text{3}}\right)=\text{0}{\text{.0106 g/cm}}^3$

The units in the conversion factor must cancel to give the correct unit for the answer.

Hence,

$\text{10}{\text{.6 kg/m}}^{\text{3}}=\text{ 0}{\text{.0106 g/cm}}^{\text{3}}$.