Chapter 2, Problem 2.100EP

A two-gram sample of a red-colored liquid is found to have a volume of four cubic centimeters. Calculate the liquid’s density using the following uncertainty specifications, and express your answers in scientific notation.

1. a. 2.00 g and 4.000 cm³
2. b. 2.000 g and 4.0 cm³
3. c. 2.0000 g and 4.000 cm³
4. d. 2.000 g and 4.0000 cm³

Interpretation Introduction

Interpretation:

If $2.00g$ of powdery white solid has a volume of $\text{4}\text{.000}{\text{cm}}^{\text{3}}$, the density should be calculated with specified uncertainties and the answer should be expressed in scientific notation.

Concept Introduction:

Density is a physical property and specific for each substance under fixed conditions.  It is defined as the ratio of mass of a substance to the volume of the substance.  If the answer obtained using the above density formula has to be expressed as a scientific notation, the operational rule can be applied here, which states that the number of significant figures in the answer after solving a multiplication or division problem is same as the number of fewest significant figures present in the measurement.

Answer to Problem 2.100EP

The density is $\text{5}\text{.0}\text{0}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}$, with three significant figures.

Explanation of Solution

Density can be calculated using the below formula.

  $\text{Density}\text{=}\left(\frac{\text{mass}}{\text{volume}}\right)$

$2.00g$ and $\text{4}\text{.000}{\text{cm}}^{\text{3}}$ are the given values for mass and volume respectively.  Calculating the density using the above formula.

  $\begin{array}{l}\text{three}\text{significant}\text{figures}\text{four}\text{significant}\text{figures}\\ \downarrow \downarrow \\ \text{2}\text{.00}\text{g}\text{÷}4.\text{000}{\text{cm}}^{\text{3}}\text{=}\text{5}\text{.00}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}\\ \uparrow \\ \text{three}\text{significant}\text{figures}\end{array}$

Since the significant figures for mass and volume are three and four respectively, the fewest significant figures here is three, hence the answer obtained too has a significant figure of three.

Conclusion

The density is $\text{5}\text{.0}\text{0}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}$, with a significant figure of three, since three is the fewest significant figures of the measurements made.

Interpretation Introduction

Interpretation:

If $2.000g$ of powdery white solid has a volume of $\text{4}\text{.0}{\text{cm}}^{\text{3}}$, the density should be calculated with specified uncertainties and the answer should be expressed in scientific notation.

Concept Introduction:

Density is a physical property and specific for each substance under fixed conditions.  It is defined as the ratio of mass of a substance to the volume of the substance.  If the answer obtained using the above density formula has to be expressed as a scientific notation, the operational rule can be applied here, which states that the number of significant figures in the answer after solving a multiplication or division problem is same as the number of fewest significant figures present in the measurement.

Answer to Problem 2.100EP

The density is $\text{5}\text{.0}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}$, with three significant figures.

Explanation of Solution

Density can be calculated using the below formula.

  $\text{Density}\text{=}\left(\frac{\text{mass}}{\text{volume}}\right)$

$2.000g$ and $\text{4}\text{.0}{\text{cm}}^{\text{3}}$ are the given values for mass and volume respectively.  Calculating the density using the above formula.

  $\begin{array}{l}\text{four}\text{significant}\text{figures}\text{two}\text{significant}\text{figures}\\ \downarrow \downarrow \\ \text{2}\text{.000}\text{g}\text{÷}4.{\text{0 cm}}^{\text{3}}\text{=}\text{5}\text{.0}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}\\ \uparrow \\ \text{two}\text{significant}\text{figures}\end{array}$

Since the significant figures for mass and volume are four and two respectively, the fewest significant figures here is two, hence the answer obtained too has a significant figure of two.

Conclusion

The density is $\text{5}\text{.0}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}$ with a significant figure of two, since two is the fewest significant figures of the measurements made.

Interpretation Introduction

Interpretation:

If $2.0000g$ of powdery white solid has a volume of $\text{4}\text{.000}{\text{cm}}^{\text{3}}$, the density should be calculated with specified uncertainties and the answer should be expressed in scientific notation.

Concept Introduction:

Density is a physical property and specific for each substance under fixed conditions.  It is defined as the ratio of mass of a substance to the volume of the substance.  If the answer obtained using the above density formula has to be expressed as a scientific notation, the operational rule can be applied here, which states that the number of significant figures in the answer after solving a multiplication or division problem is same as the number of fewest significant figures present in the measurement.

Answer to Problem 2.100EP

The density is $\text{5}\text{.0}\text{00}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}$, with four significant figures.

Explanation of Solution

Density can be calculated using the below formula.

  $\text{Density}\text{=}\left(\frac{\text{mass}}{\text{volume}}\right)$

$2.0000g$ and $\text{4}\text{.000}{\text{cm}}^{\text{3}}$ are the given values for mass and volume respectively.  Calculating the density using the above formula.

  $\begin{array}{l}\text{five}\text{significant}\text{figures}\text{four}\text{significant}\text{figures}\\ \downarrow \downarrow \\ \text{2}\text{.0000}\text{g}\text{÷}4.{\text{000 cm}}^{\text{3}}\text{=}\text{5}\text{.000}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}\\ \uparrow \\ \text{four}\text{significant}\text{figures}\end{array}$

Since the significant figures for mass and volume are five and four respectively, the fewest significant figures here is four, hence the answer obtained too has a significant figure of four.

Conclusion

The density is $\text{5}\text{.0}\text{00}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}$ with a significant figure of four, since four is the fewest significant figures of the measurements made.

Interpretation Introduction

Interpretation:

If $2.000g$ of powdery white solid has a volume of $\text{4}\text{.0000}{\text{cm}}^{\text{3}}$, the density should be calculated with specified uncertainties and the answer should be expressed in scientific notation.

Concept Introduction:

Density is a physical property and specific for each substance under fixed conditions.  It is defined as the ratio of mass of a substance to the volume of the substance.  If the answer obtained using the above density formula has to be expressed as a scientific notation, the operational rule can be applied here, which states that the number of significant figures in the answer after solving a multiplication or division problem is same as the number of fewest significant figures present in the measurement.

Answer to Problem 2.100EP

The density is $\text{5}\text{.0}\text{00}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}$, with two significant figures.

Explanation of Solution

Density can be calculated using the below formula.

  $\text{Density}\text{=}\left(\frac{\text{mass}}{\text{volume}}\right)$

$2.000g$ and $\text{4}\text{.0000}{\text{cm}}^{\text{3}}$ are the given values for mass and volume respectively.  Calculating the density using the above formula.

  $\begin{array}{l}\text{four}\text{significant}\text{figures}\text{five}\text{significant}\text{figures}\\ \downarrow \downarrow \\ \text{2}\text{.000}\text{g}\text{÷}4.{\text{0000 cm}}^{\text{3}}\text{=}\text{5}\text{.000}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}\\ \uparrow \\ \text{four}\text{significant}\text{figures}\end{array}$

Since the significant figures for mass and volume are four and five respectively, the fewest significant figures here is four, hence the answer obtained too has a significant figure of four.

Conclusion

The density is $\text{5}\text{.0}\text{00}\text{×}{\text{10}}^{\text{-1}}{\text{g/cm}}^{\text{3}}$ with a significant figure of four, since four is the fewest significant figures of the measurements made.