Chapter 1, Problem 1.78P

(a)

Interpretation Introduction

Interpretation:

Whether the ball float or not if it is evacuated is to be determined.

Concept introduction:

Density is defined as mass per unit volume. Mass and volume are physical quantities and the units of mass and volume are fundamental units. Density is the ratio of mass to the volume. The unit of volume is derived from the units of mass and volume. The SI unit of density is ${\text{kg/m}}^{\text{3}}$. The formula to calculate density is,

  $\text{Density}=\frac{\text{Mass}}{\text{Volume}}$        (1)

An object can float in the air if its density is less than the density of air.

Answer to Problem 1.78P

The ball will float if it is evacuated.

Explanation of Solution

The density of air is $1.189\text{g/L}$.

Substitute $560{\text{cm}}^3$ for volume and $0.12\text{g}$ for mass in the equation (1) to calculate the density of ball.

  $\begin{array}{c}\text{Density}=\left(\frac{0.12\text{g}}{560{\text{cm}}^3}\right)\left(\frac{\text{1}{\text{cm}}^3}{1\text{mL}}\right)\left(\frac{1000\text{mL}}{1\text{L}}\right)\\ =0.21\text{g/L}\text{.}\end{array}$

The ball float will float because the density of the ball is less than the air.

Conclusion

When the density of an object is less than the density of air, then that object floats in the air.

(b)

Interpretation Introduction

Interpretation:

Whether the ball filled with carbon dioxide will float or not is to be determined.

Concept introduction:

Density is defined as mass per unit volume. Mass and volume are physical quantities and the units of mass and volume are fundamental units. Density is the ratio of mass to the volume. The unit of volume is derived from the units of mass and volume. The SI unit of density is ${\text{kg/m}}^{\text{3}}$. The formula to calculate density is,

  $\text{Density}=\frac{\text{Mass}}{\text{Volume}}$        (1)

An object can float in the air if its density is less than the density of air.

Answer to Problem 1.78P

The ball filled with carbon dioxide will not float.

Explanation of Solution

The density of air is $1.189\text{g/L}$ and the density of carbon dioxide is $1.830\text{g/L}$.

The density of carbon dioxide is greater than the density of air and therefore the ball filled with carbon dioxide will not float.

Conclusion

When the density of an object is less than the density of air, then that object floats in the air.

(c)

Interpretation Introduction

Interpretation:

Whether the ball filled with hydrogen will float or not is to be determined.

Concept introduction:

Density is defined as mass per unit volume. Mass and volume are physical quantities and the units of mass and volume are fundamental units. Density is the ratio of mass to the volume. The unit of volume is derived from the units of mass and volume. The SI unit of density is ${\text{kg/m}}^{\text{3}}$. The formula to calculate density is,

  $\text{Density}=\frac{\text{Mass}}{\text{Volume}}$        (1)

An object can float in the air if its density is less than the density of air.

Answer to Problem 1.78P

The ball filled with hydrogen will float.

Explanation of Solution

The density of air is $1.189\text{g/L}$.

The density of the ball filled with hydrogen is calculated as follows:

  $\begin{array}{c}\text{Total Density}=\left(\text{density of ball}\right)+\left(\text{density of hydrogen}\right)\\ =0.21\text{g/L}+0.0899\text{g/L}\\ \approx \text{0}\text{.30}\text{g/L}\text{.}\end{array}$

The density of ball filled with hydrogen is less than the density of air and therefore the ball filled with hydrogen will float.

Conclusion

When the density of an object is less than the density of air, then that object floats in the air.

(d)

Interpretation Introduction

Interpretation:

Whether the ball filled with oxygen will float or not is to be determined.

Concept introduction:

Density is defined as mass per unit volume. Mass and volume are physical quantities and the units of mass and volume are fundamental units. Density is the ratio of mass to the volume. The unit of volume is derived from the units of mass and volume. The SI unit of density is ${\text{kg/m}}^{\text{3}}$. The formula to calculate density is,

  $\text{Density}=\frac{\text{Mass}}{\text{Volume}}$        (1)

An object can float in the air if its density is less than the density of air.

Answer to Problem 1.78P

The ball filled with oxygen will not float.

Explanation of Solution

The density of air is $1.189\text{g/L}$ and the density of oxygen is $1.330\text{g/L}$.

The density of oxygen is greater than the density of air and therefore the ball filled with oxygen will not float.

Conclusion

When the density of an object is less than the density of air, then that object floats in the air.

(e)

Interpretation Introduction

Interpretation:

Whether the ball filled with nitrogen will float or not is to be determined.

Concept introduction:

Density is defined as mass per unit volume. Mass and volume are physical quantities and the units of mass and volume are fundamental units. Density is the ratio of mass to the volume. The unit of volume is derived from the units of mass and volume. The SI unit of density is ${\text{kg/m}}^{\text{3}}$. The formula to calculate density is,

  $\text{Density}=\frac{\text{Mass}}{\text{Volume}}$        (1)

An object can float in the air if its density is less than the density of air.

Answer to Problem 1.78P

The ball filled with nitrogen will not float but sink.

Explanation of Solution

The density of air is $1.189\text{g/L}$.

The density of the ball filled with nitrogen is calculated as follows:

  $\begin{array}{c}\text{Total Density}=\left(\text{density of ball}\right)+\left(\text{density of nitrogen}\right)\\ =0.21\text{g/L}+1.165\text{g/L}\\ \approx 1.\text{38}\text{g/L}\text{.}\end{array}$

The density of ball filled with nitrogen is greater than the density of air and therefore the ball filled with nitrogen will not float.

Conclusion

When the density of an object is less than the density of air, then that object floats in the air.

(f)

Interpretation Introduction

Interpretation:

The weight that must be added to make it sink is to be calculated.

Concept introduction:

Density is defined as mass per unit volume. Mass and volume are physical quantities and the units of mass and volume are fundamental units. Density is the ratio of mass to the volume. The unit of volume is derived from the units of mass and volume. The SI unit of density is ${\text{kg/m}}^{\text{3}}$. The formula to calculate density is,

  $\text{Density}=\frac{\text{Mass}}{\text{Volume}}$        (1)

An object can float in the air if its density is less than the density of air.

Answer to Problem 1.78P

$0.55\text{g}$ of gas must be added to sink the ball. In the case of hydrogen, $0.50\text{g}$ of hydrogen must be added to sink the ball.

Explanation of Solution

Rearrange the equation (1) to calculate the mass required of ball and gas to sink.

  $\text{Mass required of ball and gas}=\left(\text{Density}\right)\left(\text{Volume}\right)$        (2)

Substitute $1.189\text{g/L}$ for density and $560{\text{cm}}^3$ for volume in the equation (2) to calculate the mass required of ball and gas to sink.

  $\begin{array}{c}\text{Mass required of ball and gas}=\left(1.189\text{g/L}\right)\left(560{\text{cm}}^3\right)\left(\frac{1\text{mL}}{\text{1}{\text{cm}}^3}\right)\left(\frac{1\text{L}}{1000\text{mL}}\right)\\ =0.66584 \text{g}\end{array}$

The mass required of ball and gas to sink is $0.66584 \text{g}$ and the mass of the ball is $0.12\text{g}$ Therefore the mass added for the evacuated ball to sink is calculated as follows:

  $\begin{array}{c}\text{Mass for evacuated ball}=\text{mass}\text{required of ball and gas}-\text{mass of ball}\\ = 0.66584 \text{g}-0.12\text{g}\\ =0.5458\text{g}\\ \approx 0.55\text{g}\end{array}$

Therefore $0.55\text{g}$ of gas must be added to sink the ball.

Rearrange the equation (1) to calculate the mass of hydrogen in the ball.

  $\text{Mass of hydrogen in the ball}=\left(\text{Density}\right)\left(\text{Volume}\right)$        (3)

Substitute $0.0899\text{g/L}$ for density and $560{\text{cm}}^3$ for volume in the equation (3) to calculate the mass of hydrogen in the ball.

  $\begin{array}{c}\text{Mass of hydrogen in the ball}=\left(0.0899\text{g/L}\right)\left(560{\text{cm}}^3\right)\left(\frac{1\text{mL}}{\text{1}{\text{cm}}^3}\right)\left(\frac{1\text{L}}{1000\text{mL}}\right)\\ =0.0503\text{g}\end{array}$

The mass of hydrogen is $0.0503\text{g}$ and the mass of the ball is $0.12\text{g}$. Therefore the mass of hydrogen and ball is calculated as follows:

  $\begin{array}{c}\text{Mass of hydrogen and ball}=\text{Mass of hydrogen}+\text{Mass of ball}\\ =0.0503\text{g}+0.12\text{g}\\ =0.17\text{g}\end{array}$

The mass required of ball and gas to sink is $0.66584 \text{g}$ and the mass of hydrogen and ball is $0.17\text{g}$. Therefore the mass added for ball filled with hydrogen to sink is calculated as follows:

  $\begin{array}{c}\text{Mass for ball filled with hydrogen}=\left(\begin{array}{c}\left(\text{mass}\text{required of ball and gas}\right)-\\ \left(\text{mass of hydrogen and ball}\right)\end{array}\right)\\ = 0.66584 \text{g}-0.17\text{g}\\ =0.490\text{g}\\ \approx 0.50\text{g}\text{.}\end{array}$

Therefore $0.50\text{g}$ of hydrogen must be added to sink the ball.

Conclusion

When the density of an object is less than the density of air, then that object floats in the air.