Chapter 5, Problem 6P

5-16 Answer true or false.

(a) For a sample of gas at constant temperature, its pressure multiplied by its volume is a constant.

(b) For a sample of gas at constant temperature, increasing the pressure increases the volume.

(c) For a sample of gas at constant temperature,

(d) As a gas expands at constant temperature, its volume increases.

(e) The volume of a sample of gas at constant pressure is directly proportional to its temperature—the higher its temperature, the greater its volume.

(f) A hot-air balloon rises because hot air is less dense than cooler air.

(g) For a gas sample in a container of fixed volume, an increase in temperature results in an increase in pressure.

(h) For a gas sample in a container of fixed volume, is a constant.

(i) When steam at 100°C in an autoclave is heated to 1200C, the pressure within the autoclave increases.

(j) When a gas sample in a flexible container at constant pressure at 25°C is heated to 50°C, its volume doubles.

(k) Lowering the diaphragm causes the chest cavity to increase in volume and the pressure of air in the lungs to decrease.

(l) Raising the diaphragm decreases the volume of the chest cavity and forces air out of the lungs.

Interpretation Introduction

(a)

Interpretation:

Find if the given statement is true or false.

“For a sample of gas at constant temperature, its pressure multiplied by its volume is a constant.”

Concept Introduction:

According to Boyle’s Law, the volume of fixed amount of gas is inversely proportional to the pressure of the gas at constant temperature. Mathematically, it is given as.

$V\propto \frac{1}{P}$.

Answer to Problem 6P

For a sample of gas at constant temperature, its pressure multiplied by its volume is a constant, the given statement is true.

Explanation of Solution

Given information:

For a sample of gas at constant temperature, its pressure multiplied by its volume is a constant.

We know, according to Boyle’s Law, we have.

$V\propto \frac{1}{P}$.

$PV=\text{C}\text{O}nst\text{a}nt=k$.

Thus, for a sample of gas at constant temperature, its pressure multiplied by its volume is a constant.

Interpretation Introduction

(b)

Interpretation:

Find if the given statement is true or false.

“For a sample of gas at constant temperature, increasing the pressure increases the volume.”

Concept Introduction:

According to Boyle’s Law, the volume of fixed amount of gas is inversely proportional to the pressure of the gas at constant temperature. Mathematically, it is given as.

$V\propto \frac{1}{P}$.

Answer to Problem 6P

For a sample of gas at constant temperature, increasing the pressure decreases the volume thus, the given statement is false.

Explanation of Solution

Given Information:

For a sample of gas at constant temperature, increasing the pressure increases the volume.

We know, according to Boyle’s Law, we have.

$V\propto \frac{1}{P}$.

That means there is an inverse relationship between volume and pressure at constant temperature. When pressure is increased, volume is decreased and vice versa.

Thus, for a sample of gas at constant temperature, increasing the pressure decreases the volume.

Interpretation Introduction

(c)

Interpretation:

Find if the given statement is true or false.

“For a sample of gas at constant temperature, $\frac{P_1}{V_1}=\frac{P_2}{V_2}$.”

Concept Introduction:

According to Boyle’s Law, the volume of fixed amount of gas is inversely proportional to the pressure of the gas at constant temperature. Mathematically, it is given as.

$V\propto \frac{1}{P}$.

Answer to Problem 6P

For a sample of gas at constant temperature, $P_1{V}_1=P_2{V}_2$. Thus, the given statement is false.

Explanation of Solution

Given Information:

For a sample of gas at constant temperature, $\frac{P_1}{V_1}=\frac{P_2}{V_2}$.

We know, according to Boyle’s Law, we have.

$V\propto \frac{1}{P}$.

$PV=\text{C}\text{O}nst\text{a}nt=k$.

We two different sets of volume and pressure of the gas is considered, the above equation becomes as follows:

$P_1{V}_1=P_2{V}_2$.

where $P_1$ and $V_1$ are initial pressure and temperature of the gas, while $P_2$ and $V_2$ are final pressure and temperature of the gas.

Thus, for a sample of gas at constant temperature, $P_1{V}_1=P_2{V}_2$.

Interpretation Introduction

(d)

Interpretation:

Find if the given statement is true or false.

“As a gas expands at constant temperature, its volume increases.”

Concept Introduction:

According to Boyle’s Law, the volume of fixed amount of gas is inversely proportional to the pressure of the gas at constant temperature. Mathematically, it is given as.

$V\propto \frac{1}{P}$.

Answer to Problem 6P

As a gas expands at constant temperature, its volume increases. Thus, the given statement is true.

Explanation of Solution

Given Information:

As a gas expands at constant temperature, its volume increases.

When a gas expands, the distance between the gas particles increases. Thus, pressure decreases.

Also, we know, according to Boyle’s Law.

$V\propto \frac{1}{P}$.

Decrease in pressure increases the volume.

Thus, as a gas expands at constant temperature, its volume increases.

Interpretation Introduction

(e)

Interpretation:

Find if the given statement is true or false.

“The volume of a gas at constant pressure is directly proportional to its temperature − the higher its temperature, the greater its volume.”

Concept Introduction:

According to Charles’s Law, for the gas held at constant pressure, the volume of gas is directly proportional to the temperature of the gas. Mathematically, it is given as.

$V\propto T$.

Answer to Problem 6P

The volume of a gas at constant pressure is directly proportional to its temperature − the higher its temperature, the greater its volume. Thus, the given statement is true.

Explanation of Solution

Given Information:

The volume of a gas at constant pressure is directly proportional to its temperature − the higher its temperature, the greater its volume.

We know, according to Charles’s Law, we have.

$V\propto T$.

There is a direct relationship between volume and temperature. As the temperature is increased, the volume is also increased.

Thus, the volume of a gas at constant pressure is directly proportional to its temperature − the higher its temperature, the greater its volume.

Interpretation Introduction

(f)

Interpretation:

Find if the given statement is true or false.

“A hot-air balloon rises because hot air is less dense than cooler air”.

Concept Introduction:

Hot air rises because when the air is heated, it undergoes expansion. This results in the air to become less dense than the air surrounding it.

Answer to Problem 6P

A hot-air balloon rises because hot air is less dense than cooler air, thus, the given statement is true.

Explanation of Solution

Given information:

A hot-air balloon rises because hot air is less dense than cooler air.

Hot air rises because when the air is heated, it undergoes expansion. This results in the air to become less dense than the air surrounding it.

Thus, a hot-air balloon rises because hot air is less dense than cooler air.

Interpretation Introduction

(g)

Interpretation:

Find if the given statement is true or false.

“For a gas sample in a container of fixed volume, an increase in temperature results in the increase in pressure.”

Concept Introduction:

According to Gay-Lussac’s Law, for the gas held at constant volume, the pressure of a given amount of gas is directly proportional to the temperature of the gas. Mathematically, it is given as.

$P\propto T$.

Answer to Problem 6P

For a gas sample in a container of fixed volume, an increase in temperature results in the increase in pressure. Thus, the given statement is true.

Explanation of Solution

Given information:

For a gas sample in a container of fixed volume, an increase in temperature results in the increase in pressure.

We know, according to Gay Lussac’s Law, we have.

$P\propto T$.

There is a direct relationship between pressure and temperature. As the temperature is increased, the pressure is also increased.

Thus, for a gas sample in a container of fixed volume, an increase in temperature results in the increase in pressure.

Interpretation Introduction

(h)

Interpretation:

Find if the given statement is true or false.

“For a gas sample in a container of fixed volume, $P\times T$ is a constant”.

Concept Introduction:

According to Gay-Lussac’s Law, for the gas held at constant volume, the pressure of a given amount of gas is directly proportional to the temperature of the gas. Mathematically, it is given as.

$P\propto T$.

Answer to Problem 6P

For a gas sample in a container of fixed volume, $\frac{P}{T}$ is a constant. Thus, the given statement is false.

Explanation of Solution

Given information:

For a gas sample in a container of fixed volume, an increase in temperature results in the increase in pressure.

We know, according to Gay Lussac’s Law, we have.

$P\propto T$.

$\frac{P}{T}=\text{C}\text{O}nst\text{a}nt$.

Thus, for a gas sample in a container of fixed volume, $\frac{P}{T}$ is a constant.

Interpretation Introduction

(i)

Interpretation:

Find if the given statement is true or false.

“When steam at ${100}^{\circ }\text{C}$ in an autoclave is heated to ${120}^{\circ }\text{C}$, the pressure within the autoclave increases”.

Concept Introduction:

According to Gay-Lussac’s Law, for the gas held at constant volume, the pressure of a given amount of gas is directly proportional to the temperature of the gas. Mathematically, it is given as.

$P\propto T$.

Answer to Problem 6P

When steam at ${100}^{\circ }\text{C}$ in an autoclave is heated to ${120}^{\circ }\text{C}$, the pressure within the autoclave increases. Thus, the given statement is true.

Explanation of Solution

Given Information:

When steam at ${100}^{\circ }\text{C}$ in an autoclave is heated to ${120}^{\circ }\text{C}$, the pressure within the autoclave increases.

We know, according to Gay Lussac’s Law, we have.

$P\propto T$.

There is a direct relationship between pressure and temperature. As the temperature is increased, the pressure is also increased.

Thus, when steam at ${100}^{\circ }\text{C}$ in an autoclave is heated to ${120}^{\circ }\text{C}$, the pressure within the autoclave increases.

Interpretation Introduction

(j)

Interpretation:

Find if the given statement is true or false.

“When a gas sample in a flexible container at constant pressure at ${25}^{\circ }\text{C}$ is heated to ${50}^{\circ }\text{C}$, its volume doubles.”

Concept Introduction:

According to Charles’s Law, for the gas held at constant pressure, the volume of gas is directly proportional to the temperature of the gas. Mathematically, it is given as.

$V\propto T$.

Answer to Problem 6P

When a gas sample in a flexible container at constant pressure at ${25}^{\circ }\text{C}$ is heated to ${50}^{\circ }\text{C}$, its volume doubles. Thus, the given statement is true.

Explanation of Solution

Given Information:

When a gas sample in a flexible container at constant pressure at ${25}^{\circ }\text{C}$ is heated to ${50}^{\circ }\text{C}$, its volume doubles.

We know, according to Charles’s Law, we have.

$V\propto T$.

There is a direct relationship between volume and temperature. As the temperature is increased, the volume is also increased.

Now, the temperature of the container is doubled, hence the volume of the flexible container is also doubled.

Thus, when a gas sample in a flexible container at constant pressure at ${25}^{\circ }\text{C}$ is heated to ${50}^{\circ }\text{C}$, its volume doubles.

Interpretation Introduction

(k)

Interpretation:

Find if the given statement is true or false.

“Lowering the diaphragm causes the chest cavity to increase in volume and the pressure of air in the lungs to decrease”.

Concept Introduction:

According to Boyle’s Law, the volume of fixed amount of gas is inversely proportional to the pressure of the gas at constant temperature. Mathematically, it is given as.

$V\propto \frac{1}{P}$.

Answer to Problem 6P

Lowering the diaphragm causes the chest cavity to increase in volume and the pressure of air in the lungs to decrease thus, the given statement is true.

Explanation of Solution

Given information:

Lowering the diaphragm causes the chest cavity to increase in volume and the pressure of air in the lungs to decrease.

We know, according to Boyle’s Law, we have.

$V\propto \frac{1}{P}$.

$PV=\text{C}\text{O}nst\text{a}nt=k$.

Now, when diaphragm is lowered, the chest cavity is increased. This results in the increase in volume and hence pressure decreases inside the lungs.

Interpretation Introduction

(l)

Interpretation:

Find if the given statement is true or false.

“Raising the diaphragm decreases the volume of the chest cavity and forces air out of the lungs.”

Concept Introduction:

According to Boyle’s Law, the volume of fixed amount of gas is inversely proportional to the pressure of the gas at constant temperature. Mathematically, it is given as.

$V\propto \frac{1}{P}$.

Answer to Problem 6P

Raising the diaphragm decreases the volume of the chest cavity and forces air out of the lungs. Thus, the given statement is true.

Explanation of Solution

Given information:

Raising the diaphragm decreases the volume of the chest cavity and forces air out of the lungs.

We know, according to Boyle’s Law, we have.

$V\propto \frac{1}{P}$.

$PV=\text{C}\text{O}nst\text{a}nt=k$.

Now, when diaphragm is raised, the volume chest cavity is decreased. This results in the decrease in volume and hence pressure increased inside the lungs and air is moved out of the lungs.