Chapter 6, Problem 6.46E

(a)

Interpretation Introduction

Interpretation:

The units of surface tension $\gamma$ is to be predicted.

Concept introduction:

The Gibbs free energy is the thermodynamic quantity that defines the available energy in the system. It is used to describe the spontaneity of the chemical reaction.

Answer to Problem 6.46E

The units of surface tension $\gamma$ is ${\text{J/m}}^2$.

Explanation of Solution

At constant pressure and temperature, the equation is represented as shown below.

$dG={\mu }_{\text{phase}}d{n}_{\text{phase}}+\gamma dA$

The unit of surface tension is calculated as follows:

$\begin{array}{rll}\gamma & =& \left(\frac{dG}{dA}\right)\\ & =& \frac{\text{J}}{{\text{m}}^2}\end{array}$

Hence, the units of surface tension is ${\text{J/m}}^2$.

Conclusion

The units of surface tension $\text{γ}$ is ${\text{J/m}}^2$.

(b)

Interpretation Introduction

Interpretation:

The equation 6.26 is to be verified by taking the derivative of $A$ and $V$.

Concept introduction:

The Gibbs free energy is the thermodynamic quantity that defines the available energy in the system. It is used to describe the spontaneity of the chemical reaction.

Answer to Problem 6.46E

Explanation of Solution

The volume of the spherical droplet is $4/3\pi r^3$. The partial derivative of volume with respect to radius of droplet $r$ is as follows:

$\begin{array}{rll}V& =& 4/3\pi r^3\\ dV& =& \frac{4}{3}\times 3\pi r^{2}dr\\ dV& =& 4\pi r^{2}dr\end{array}$

Thus, the value of $dV$ is $4\pi r^{2}dr$.

The area of the spherical droplet is $4\pi r^2$. The partial derivative of area with respect to radius of droplet $r$ is as follows.

$\begin{array}{rll}A& =& 4\pi r^2\\ dA& =& 4\times 2\pi rdr\\ dA& =& 8\pi rdr\end{array}$

Thus, the value of $dA$ is $8\pi rdr$.

The equation 6.16 is given below.

$dA=\frac{2dV}{r}$

Where,

• $dA$ is the change in surface area of the droplet.

• $dV$ is the change in volume of the droplet.

Substitute the value of $dV$ in the above equation.

$\begin{array}{rll}dA& =& \frac{2\times 4\pi r^{2}dr}{r}\\ dA& =& 8\pi rdr\end{array}$

Thus, the value of $dA$ is $8\pi rdr$ after substituting the value of $dV$ in the equation 6.16.

Similarly, substitute the value of $dA$ in the above equation.

$\begin{array}{rll}8\pi rdr& =& \frac{2dV}{r}\\ dV& =& 4\pi r^{2}dr\end{array}$

Thus, the value of $dV$ is $4\pi r^{2}dr$ after substituting the value of $dA$ in the equation 6.16.

Hence, the equation 6.26 is verified by taking the derivation of $A$ and $V$.

Conclusion

The equation 6.26 is verified by taking the derivation of $A$ and $V$.

(c)

Interpretation Introduction

Interpretation:

The new equation in terms of $dV$ using equation 6.26 is to be derived.

Concept introduction:

The Gibbs free energy is the thermodynamic quantity that defines the available energy in the system. It is used to describe the spontaneity of the chemical reaction.

Answer to Problem 6.46E

The new equation in terms of $dV$ using equation 6.26 is $dG={\mu }_{\text{phase}}d{n}_{\text{phase}}+\frac{2\gamma dV}{r}$.

Explanation of Solution

The equation 6.16 is given below.

$dA=\frac{2dV}{r}$

Where,

• $dA$ is the change in surface area of the droplet.

• $dV$ is the change in volume of the droplet.

At constant pressure and temperature, the equation is represented as shown below.

$dG={\mu }_{\text{phase}}d{n}_{\text{phase}}+\gamma dA$

Substitute the value of $dA$ in the above equation.

$\begin{array}{rll}dG& =& {\mu }_{\text{phase}}d{n}_{\text{phase}}+\gamma \left(\frac{2dV}{r}\right)\\ dG& =& {\mu }_{\text{phase}}d{n}_{\text{phase}}+\frac{2\gamma dV}{r}\end{array}$

Hence, the new equation in terms of $dV$ is $dG={\mu }_{\text{phase}}d{n}_{\text{phase}}+\frac{2\gamma dV}{r}$.

Conclusion

The new equation in terms of $dV$ is $dG={\mu }_{\text{phase}}d{n}_{\text{phase}}+\frac{2\gamma dV}{r}$.

(d)

Interpretation Introduction

Interpretation:

Whether a large droplet radius or a small droplet radius contributes to a large $dG$ value if a spontaneous change in phase were to be accompanied by a positive $dG$ value is to be stated.

Concept introduction:

The Gibbs free energy is the thermodynamic quantity that defines the available energy in the system. It is used to describe the spontaneity of the chemical reaction.

Answer to Problem 6.46E

The small droplet radius contributes to a large $dG$ value.

Explanation of Solution

The spontaneous change in phase is accompanied by a positive value of $dG$.The value of $dG$ is negative when the value of $dV$ will be maximum or positive. Therefore, the radius of small droplets will make the value $dV$ positive. Hence, the small droplet radius contributes to a large $dG$ value.

Conclusion

The small droplet radius contributes to a large $dG$ value.

(e)

Interpretation Introduction

Interpretation:

The droplet which evaporates faster is to be predicted.

Concept introduction:

The Gibbs free energy is the thermodynamic quantity that defines the available energy in the system. It is used to describe the spontaneity of the chemical reaction.

Answer to Problem 6.46E

The small droplet evaporates faster because at this stage the $dG$ becomes negative.

Explanation of Solution

Evaporation is a process in which the water molecule at the surface evaporates and converts into gas phase.

The size of the droplet decreases when it evaporates. Thus, the value of $dV$ becomes negative. In this case, the value of $dG$ becomes positive. Thus, the reaction becomes non-spontaneous. The value of $dG$ becomes negative when the value of $dV$ will be maximized or positive. Hence, the small droplet will evaporate faster.

Conclusion

The small droplet evaporates faster because at this stage the value of $dG$ becomes negative.

(f)

Interpretation Introduction

Interpretation:

The validation of the method of delivery of many perfumes and colognes via so-called atomizers is to be explained.

Concept introduction:

The Gibbs free energy is the thermodynamic quantity that defines the available energy in the system. It is used to describe the spontaneity of the chemical reaction.

Answer to Problem 6.46E

The small droplets have tendency to move faster and diffuses the perfumes and colognes via so-called atomizers over a large area.

Explanation of Solution

The atomizers are the devices that emit a liquid into a large area. The small droplets evaporate faster than the large droplets. Thus, the small droplets have the tendency to move faster. Hence, the small droplets produced by the perfume will diffuse over a large area.

Conclusion

The small droplets have tendency to move faster and diffuse the perfumes and colognes via so-called atomizers over a large area.