Chapter 5, Problem 5.142P

(a)

Interpretation Introduction

Interpretation:

The cylinder that best represents the change in the volume of the ideal gas in cylinder A if the pressure is doubled at constant temperature and $n$ is to be identified.

Concept introduction:

The ideal gas law equation is as follows:

  $PV=nRT$        (1)

Here,

$P$ is the pressure of the gas.

$V$ is the volume occupied by gas.

$n$ is the number of moles of gas.

$T$ is the temperature in Kelvin.

$R$ is the universal gas constant.

For two different values of pressure, temperature, volume and number of moles equation (1) can be modified as follows:

  $\frac{P_1{V}_1}{n_1{T}_1}=\frac{P_2{V}_2}{n_2{T}_2}$        (2)

Here,

$P_1$ is the initial pressure of the gas.

$P_2$ is the final pressure of the gas.

$V_1$ is the initial volume of gas.

$V_2$ is the final volume of gas.

$n_1$ is the initial number of moles of gas.

$n_2$ is the final number of moles of gas.

Answer to Problem 5.142P

Cylinder B correctly represents the change in the volume of the ideal gas.

Explanation of Solution

The temperature and number of moles of gas is fixed and the pressure is doubled.

At constant temperature and number of moles equation (2) can be written as:

  $P_1{V}_1=P_2{V}_2$        (3)

Consider the initial pressure be $x$ so, the final pressure becomes $2x$.

Rearrange the equation (3) for $V_2$.

  $V_2=\frac{P_1{V}_1}{P_2}$        (4)

Substitute the value $x$ for $P_1$ and $2x\text{ atm}$ in equation (4).

  $\begin{array}{c}{V}_2=\frac{\left(x_1\right)\left(V_1\right)}{2x}\\ =\frac{1}{2}{V}_1.\end{array}$

As the pressure is doubled, the volume is decreased to half of the original volume. Therefore cylinder B best represents the change.

Conclusion

As pressure is doubled, the volume is decreased to half of the original volume. The cylinder that indicates the volume half of the volume in cylinder A is cylinder B. Therefore, cylinder B correctly represents the change in the volume of the ideal gas when the pressure of the gas is doubled.

(b)

Interpretation Introduction

Interpretation:

The cylinder that best represents the change in the volume of the ideal gas in cylinder A if the temperature is decreased from $400\text{ K}$ to $200\text{ K}$ is at constant pressure and $n$ is to be identified.

Concept introduction:

The ideal gas law equation is as follows:

  $PV=nRT$        (1)

Here,

$P$ is the pressure of the gas.

$V$ is the volume occupied by gas.

$n$ is the number of moles of gas.

$T$ is the temperature in Kelvin.

$R$ is the universal gas constant.

For two different values of pressure, temperature, volume and number of moles equation (1) can be modified as follows:

  $\frac{P_1{V}_1}{n_1{T}_1}=\frac{P_2{V}_2}{n_2{T}_2}$        (2)

Here,

$P_1$ is the initial pressure of the gas.

$P_2$ is the final pressure of the gas.

$V_1$ is the initial volume of gas.

$V_2$ is the final volume of gas.

$n_1$ is the initial number of moles of gas.

$n_2$ is the final number of moles of gas.

Answer to Problem 5.142P

Cylinder B correctly represents the change in the volume of the ideal gas.

Explanation of Solution

The pressure and number of moles of gas is fixed and the temperature is halved.

At constant pressure and number of moles equation (2) can be written as:

  $\frac{V_1}{T_1}=\frac{V_2}{T_2}$        (5)

Rearrange equation (5) for $V_2$.

  $V_2=\left(\frac{V_1}{T_1}\right)\left(T_2\right)$        (6)

Substitute the value $200\text{ K}$ for $T_2$ and $400\text{ K}$ for $T_1$ in the equation (6).

  $\begin{array}{c}{V}_2=\left(\frac{\left(V_1\right)\left(200\text{ K}\right)}{\left(400\text{ K}\right)}\right)\\ =\frac{1}{2}{V}_1.\end{array}$

As the temperature is decreased by a factor of 2, the volume is decreased to half of the original volume. Therefore cylinder B best represents the change

Conclusion

As the temperature is decreased by a factor of 2, the volume is decreased to half of the original volume. The cylinder that indicates the volume half of the volume in cylinder A is cylinder B. Therefore, cylinder B correctly represents the change in the volume of the ideal gas when the pressure of the gas is doubled.

(c)

Interpretation Introduction

Interpretation:

The cylinder that best represents the change in the volume of the ideal gas in cylinder A if the temperature is increased from $100°\text{C}$ to $200°\text{C}$ is at constant pressure and $n$ is to be identified.

Concept introduction:

The ideal gas law equation is as follows:

  $PV=nRT$        (1)

Here,

$P$ is the pressure of the gas.

$V$ is the volume occupied by gas.

$n$ is the number of moles of gas.

$T$ is the temperature in Kelvin.

$R$ is the universal gas constant.

For two different values of pressure, temperature, volume and number of moles equation (1) can be modified as follows:

  $\frac{P_1{V}_1}{n_1{T}_1}=\frac{P_2{V}_2}{n_2{T}_2}$        (2)

Here,

$P_1$ is the initial pressure of the gas.

$P_2$ is the final pressure of the gas.

$V_1$ is the initial volume of gas.

$V_2$ is the final volume of gas.

$n_1$ is the initial number of moles of gas.

$n_2$ is the final number of moles of gas.

Answer to Problem 5.142P

None of the cylinders show volume increases of $1.27$.

Explanation of Solution

Moles and pressure are constant.

The formula to convert $°\text{C}$ to Kelvin is:

  $T\left(\text{K}\right)=T\left(°\text{C}\right)+273.15$        (7)

Substitute $\text{100 }°\text{C}$ for $T\left(°\text{C}\right)$ in equation (7).

  $\begin{array}{c}T\left(\text{K}\right)=100°\text{C}+273.15\\ =373.15\text{ K}\end{array}$

Substitute $\text{200 }°\text{C}$ for $T\left(°\text{C}\right)$ in equation (7).

  $\begin{array}{c}T\left(\text{K}\right)=200°\text{C}+273.15\\ =473.15\text{ K}\end{array}$

Substitute the value $\text{373}\text{.15 K}$ for $T_2$ and $473.15\text{ K}$ for $T_1$ in the equation (6)

  $\begin{array}{c}{V}_2=\left(\frac{\left(V_1\right)\left(\text{473}\text{.15 K}\right)}{\left(\text{273}\text{.15 K}\right)}\right)\\ =1.27V_1.\end{array}$

As the temperature is increased, the volume is increased by a factor of $1.27$ of the original volume. Therefore none of the cylinders show volume increase of $1.27$.

Conclusion

As the temperature is increased by a factor of 2, the volume is increased of $1.27$ of the original volume. None of the cylinders show volume increase of $1.27$.

(d)

Interpretation Introduction

Interpretation:

The cylinder that best represents the change in the volume of the ideal gas in cylinder A if $0.1\text{ mol}$ of the gas is added at constant temperature and pressure is to be identified.

Concept introduction:

The ideal gas law equation is as follows:

  $PV=nRT$        (1)

Here,

$P$ is the pressure of the gas.

$V$ is the volume occupied by gas.

$n$ is the number of moles of gas.

$T$ is the temperature in Kelvin.

$R$ is the universal gas constant.

For two different values of pressure, temperature, volume and number of moles equation (1) can be modified as follows:

  $\frac{P_1{V}_1}{n_1{T}_1}=\frac{P_2{V}_2}{n_2{T}_2}$        (2)

Here,

$P_1$ is the initial pressure of the gas.

$P_2$ is the final pressure of the gas.

$V_1$ is the initial volume of gas.

$V_2$ is the final volume of gas.

$n_1$ is the initial number of moles of gas.

$n_2$ is the final number of moles of gas.

Answer to Problem 5.142P

Cylinder C correctly represents the change in the volume of the ideal gas.

Explanation of Solution

The temperature and pressure are constant.

Consider the initial moles be $0.1$ so the final moles become $0.2$.

At constant pressure and temperature equation (2) can be written as:

  $\frac{V_1}{n_1}=\frac{V_2}{n_2}$        (8)

Rearrange equation (8) for $V_2$.

  $V_2=\left(\frac{V_1}{n_1}\right)\left(n_2\right)$        (9)

Substitute the value $0.1$ for $n_1$ and $0.2$ for $n_2$ in the equation (9).

  $\begin{array}{c}{V}_2=\left(\frac{\left(V_1\right)}{\left(0.1\right)}\right)\left(0.2\right)\\ =2V_1.\end{array}$

As moles are increased, the volume is increased by a factor of 2 of the original volume. Therefore cylinder C best represents the change.

Conclusion

As moles are increased, the volume is increased by a factor of 2 of the original volume. The cylinder that indicates the volume double of the volume in cylinder A is cylinder C. Therefore cylinder C best represents the change.

(e)

Interpretation Introduction

Interpretation:

The cylinder that best represents the change in the volume of the ideal gas in cylinder A if $0.1\text{ mol}$ of the gas is added and pressure is doubled at constant temperature is to be identified.

Concept introduction:

The ideal gas law equation is as follows:

  $PV=nRT$        (1)

Here,

$P$ is the pressure of the gas.

$V$ is the volume occupied by gas.

$n$ is the number of moles of gas.

$T$ is the temperature in Kelvin.

$R$ is the universal gas constant.

For two different values of pressure, temperature, volume and number of moles equation (1) can be modified as follows:

  $\frac{P_1{V}_1}{n_1{T}_1}=\frac{P_2{V}_2}{n_2{T}_2}$        (2)

Here,

$P_1$ is the initial pressure of the gas.

$P_2$ is the final pressure of the gas.

$V_1$ is the initial volume of gas.

$V_2$ is the final volume of gas.

$n_1$ is the initial number of moles of gas.

$n_2$ is the final number of moles of gas.

Answer to Problem 5.142P

Cylinder D correctly represents the change in the volume of the ideal gas.

Explanation of Solution

Consider the initial moles will be $0.1$ so, the final moles become $0.2$.

Consider the initial pressure will be $P$ so, the final pressure becomes $2P$.

The temperature is constant.

At constant temperature equation (2) can be written as:

  $\frac{P_1{V}_1}{n_1}=\frac{P_2{V}_2}{n_2}$        (10)

Rearrange equation (10) for $V_2$.

  $V_2=\left(\frac{P_1{V}_1{n}_2}{P_2{n}_1}\right)$        (11)

Substitute the value $P$ for $P_1$, $2P$ for $P_2$, $0.1$ for $n_1$ and $0.2$ for $n_2$ in the equation (11).

  $\begin{array}{c}{V}_2=\left(\frac{\left(P\right)\left(V_1\right)\left(0.2\right)}{\left(2P\right)\left(0.1\right)}\right)\\ =V_1.\end{array}$

As moles and pressure are increased, the final volume has the same same as of initial volume. Therefore cylinder D best represents the change.

Conclusion

As moles and pressure are increased, the final volume has the same as of initial volume. The cylinder that indicates the volume same of the volume in cylinder A is cylinder D. Therefore cylinder D best represents the change.