Chapter 10, Problem 10.41P

Interpretation Introduction

(a)

Interpretation:

Degree of freedom for $\text{Tl-20 wt\% Pb}$ is to be determined at ${325}^{\circ }\text{C and }{400}^{\circ }\text{C}$.

Concept Introduction:

The Gibbs phase rule provides the relation between number of chemically independent component and the number of independent phases for a given system. When temperature and pressure of the system can be changed, it is defined as:

  $2+C=F+P$

Here, $C$ is used for the number of chemically independent variables. This means that number of different elements or compounds which are required to specify the given system.

  $F$ is the degree of freedom which are number of variables which are allowed to change in the given system without altering its number of phases. In this case, temperature and pressure are the two variables which are assumed to change which is represented in the equation by the constant "2". If pressure is constant for any system, then instead of " $2$ ", use " $1$ " is used in the above formula.

  $P$ is the number of phases present in the given system which are in equilibrium which each other.

Answer to Problem 10.41P

For temperature ${325}^{\circ }\text{C}$, $F=1$.

For temperature ${400}^{\circ }\text{C}$, $F=2$.

Explanation of Solution

The phase diagram given for the Tl-Pb solution is given as:

For $\text{Tl-2}0\text{ wt\% Pb}$, the straight line from $\text{20 wt\%}$ is drawn until temperature ${325}^{\circ }\text{C}$ and ${400}^{\circ }\text{C}$ as shown. At the point 'a' representing temperature ${325}^{\circ }\text{C}$, two phases are present which are solid solution and liquid. At point 'b', representing temperature ${400}^{\circ }\text{C}$, only one phase is present which is liquid.

For the given system, the pressure is assumed to be constant. Thus, the formula used to calculate degree of freedom is:

  $1+C=F+P$ ......... (1)

For temperature ${325}^{\circ }\text{C}$ ,

  $\begin{array}{c}C=2\\ P=2\end{array}$

Apply equation (1) and calculate degree of freedom as:

  $\begin{array}{c}1+C=F+P\\ 1+2=F+2\\ F=1\end{array}$

For temperature ${400}^{\circ }\text{C}$ ,

  $\begin{array}{c}C=2\\ P=1\end{array}$

Apply equation (1) and calculate degree of freedom as:

  $\begin{array}{c}1+C=F+P\\ 1+2=F+1\\ F=2\end{array}$

Interpretation Introduction

(b)

Interpretation:

Degree of freedom for $\text{Tl-40 wt\% Pb}$ is to be determined at ${325}^{\circ }\text{C and }{400}^{\circ }\text{C}$.

Concept Introduction:

The Gibbs phase rule provides the relation between number of chemically independent component and the number of independent phases for a given system. When temperature and pressure of the system can be changed, it is defined as:

  $2+C=F+P$

Here, $C$ is used for the number of chemically independent variables. This means that number of different elements or compounds which are required to specify the given system.

  $F$ is the degree of freedom which are number of variables which are allowed to change in the given system without altering its number of phases. In this case, temperature and pressure are the two variables which are assumed to change which is represented in the equation by the constant "2". If pressure is constant for any system, then instead of " $2$ ", use " $1$ " is used in the above formula.

  $P$ is the number of phases present in the given system which are in equilibrium which each other.

Answer to Problem 10.41P

For temperature ${325}^{\circ }\text{C}$, $F=2$.

For temperature ${400}^{\circ }\text{C}$, $F=2$.

Explanation of Solution

The phase diagram given for the Tl-Pb solution is given as:

For $\text{Tl-4}0\text{ wt\% Pb}$, the straight line from $\text{40 wt\%}$ is drawn until temperature ${325}^{\circ }\text{C}$ and ${400}^{\circ }\text{C}$ as shown. At the point 'a' representing temperature ${325}^{\circ }\text{C}$, only one phase is present which is solid. At point 'b', representing temperature ${400}^{\circ }\text{C}$, only one phase is present which is liquid.

For the given system, the pressure is assumed to be constant. Thus, the formula used to calculate degree of freedom is:

  $1+C=F+P$ ......... (1)

For temperature ${325}^{\circ }\text{C}$ ,

  $\begin{array}{c}C=2\\ P=1\end{array}$

Apply equation (1) and calculate degree of freedom as:

  $\begin{array}{c}1+C=F+P\\ 1+2=F+1\\ F=2\end{array}$

For temperature ${400}^{\circ }\text{C}$ ,

  $\begin{array}{c}C=2\\ P=1\end{array}$

Apply equation (1) and calculate degree of freedom as:

  $\begin{array}{c}1+C=F+P\\ 1+2=F+1\\ F=2\end{array}$

Interpretation Introduction

(c)

Interpretation:

Degree of freedom for $\text{Tl-90 wt\% Pb}$ is to be determined at ${325}^{\circ }\text{C and }{400}^{\circ }\text{C}$.

Concept Introduction:

The Gibbs phase rule provides the relation between number of chemically independent component and the number of independent phases for a given system. When temperature and pressure of the system can be changed, it is defined as:

  $2+C=F+P$

Here, $C$ is used for the number of chemically independent variables. This means that number of different elements or compounds which are required to specify the given system.

  $F$ is the degree of freedom which are number of variables which are allowed to change in the given system without altering its number of phases. In this case, temperature and pressure are the two variables which are assumed to change which is represented in the equation by the constant "2". If pressure is constant for any system, then instead of " $2$ ", use " $1$ " is used in the above formula.

  $P$ is the number of phases present in the given system which are in equilibrium which each other.

Answer to Problem 10.41P

For temperature ${325}^{\circ }\text{C}$, $F=1$.

For temperature ${400}^{\circ }\text{C}$, $F=2$.

Explanation of Solution

The phase diagram given for the Tl-Pb solution is given as:

For $\text{Tl-90 wt\% Pb}$, the straight line from $\text{90 wt\%}$ is drawn until temperature ${325}^{\circ }\text{C}$ and ${400}^{\circ }\text{C}$ as shown. At the point 'a' representing temperature ${325}^{\circ }\text{C}$, two phases are present which are solid solution and liquid. At point 'b', representing temperature ${400}^{\circ }\text{C}$, only one phase is present which is liquid.

For the given system, the pressure is assumed to be constant. Thus, the formula used to calculate degree of freedom is:

  $1+C=F+P$ ......... (1)

For temperature ${325}^{\circ }\text{C}$ ,

  $\begin{array}{c}C=2\\ P=2\end{array}$

Apply equation (1) and calculate degree of freedom as:

  $\begin{array}{c}1+C=F+P\\ 1+2=F+2\\ F=1\end{array}$

For temperature ${400}^{\circ }\text{C}$ ,

  $\begin{array}{c}C=2\\ P=1\end{array}$

Apply equation (1) and calculate degree of freedom as:

  $\begin{array}{c}1+C=F+P\\ 1+2=F+1\\ F=2\end{array}$