Chapter 11, Problem 11.27P

Interpretation Introduction

Interpretation:

The percentage of a and β phase in the phase diagram needs to be determined.

Concept introduction:

Alloy is a combination of two or more metals with another element or metal to form a new metal. The formed metal by combining two metals has different properties from pure metal like increased hardness, toughness, surface roughness;etc.an alloy may be a solid solution or combination of metallic phase. Alloys are having different metallic bonding character.

Answer to Problem 11.27P

At, ${\text{620}}^{\text{o}}\text{C}$, percent obtained are 66.7% weight L and 33.3% weight a

At, ${\text{600}}^{\text{o}}\text{C}$, percent obtained are 43.75% weight L and 56.25% weight a

At, ${578}^{\text{o}}\text{C}$, percent obtained are 35.29% weight L and 64.71% weight a

At, ${576}^{\text{o}}\text{C}$, percent obtained are 2.806% weight β and 97.197% weight a

At, ${\text{500}}^{\text{o}}\text{C}$, percent obtained are 3.158% weight β and 96.842% weight a

Explanation of Solution

Given:

5 percent weight of Si alloy at temperature ${\text{620}}^{\text{o}}\text{C,}{\text{600}}^{\text{o}}\text{C},{578}^{\text{o}}\text{C},{576}^{\text{o}}\text{C},50{\text{0}}^{\text{o}}\text{C}$ are given.

Calculation:

The given phase diagram is as follows:

The given diagram is a graph of plotted against temperature and weight percent of silicon.

So, to calculate percent of a-phase and liquidphase first,

Draw a vertical linefrom alloy composition Al-5wt%Si

At temperature ${\text{620}}^{\text{o}}\text{C}$ ,draw a horizontal line and mark the intersection point.

Now for liquid phase,

From the given diagram is a graph plotted verses temperature weight percentage of silicon.

For temperature ${\text{620}}^{\text{o}}\text{C}$

The percentage of liquid phase is calculated as,

  $wt\%Liquid=\frac{C_2-C_1}{C_3-C_1}$

Where,

  $C_3$ = percent of silicon in the alloy

  $C_2$ =percent of silicon in the liquid phase

  $C_1$ =percent of silicon in the solid phase

Put,

  $C_3$ = 5

  $C_2$ =7

  $C_1$ =1

  $\begin{array}{l}wt\%Liquid=\frac{5-1}{7-1}\\ =0.667\times 100\\ =66.7wt\%Liquid\end{array}$

Now for a-phase,

So, percentage of phase at temperature ${\text{620}}^{\text{o}}\text{C}$

Percent of phase at calculate percent of as,

  $wt\%Liquid=\frac{C_3-C_2}{C_3-C_1}$

Where,

  $C_3$ = percent of silicon in the alloy

  $C_2$ =percent of silicon in the liquid phase

  $C_1$ =percent of silicon in the solid phase

Put,

  $C_3$ = 7

  $C_2$ =5

  $C_1$ =1

Therefore, weight percentage is,

  $\begin{array}{l}wt\%Liquid=\frac{7-5}{7-1}\\ =0.333\times 100\\ =33.3wt\%Liquid\end{array}$

So, the percentage of phase at ${\text{620}}^{\text{o}}\text{C}$ is 66.7wt% and 33.3wt%a.

Now for liquid phase,

The diagram is a graph plotted against temperature and weight percent of silicon.

Calculate the percent of liquid phase by the formula,

  $wt\%Liquid=\frac{C_2-C_1}{C_3-C_1}$

Where,

  $C_3$ = percent of silicon in the alloy

  $C_2$ =percent of silicon in the liquid phase

  $C_1$ =percent of silicon in the solid phase

Put,

  $C_3$ = 9.5

  $C_2$ =5

  $C_1$ =1.5

Therefore, weight percentage is,

  $\begin{array}{l}wt\%Liquid=\frac{5-1.5}{9.5-1.5}\\ =0.4375\times 100\\ =43.75wt\%Liquid\end{array}$

Now, calculate the percent for a-phase,

  $wt\%Liquid=\frac{C_3-C_2}{C_3-C_1}$

Where,

  $C_3$ = percent of silicon in the alloy

  $C_2$ =percent of silicon in the liquid phase

  $C_1$ =percent of silicon in the solid phase

Put,

  $C_3$ = 9.5

  $C_2$ =5

  $C_1$ =1.5

  $\begin{array}{l}wt\%Liquid=\frac{9.5-5}{9.5-1.5}\\ =0.5625\times 100\\ =56.25wt\%Liquid\end{array}$

Therefore, at ${\text{600}}^{\text{o}}\text{C}$ the percentage phase is 43.75wt% liquid and 56.25wt%a

Now for temperature at ${578}^{\text{o}}\text{C}$ ,

The given diagram is a graph plotted against temperature and weight percent of silicon.

Calculate the percentage of liquid phase as,

  $wt\%Liquid=\frac{C_2-C_1}{C_3-C_1}$

Where,

  $C_3$ = percent of silicon in the alloy

  $C_2$ =percent of silicon in the liquid phase

  $C_1$ =percent of silicon in the solid phase

Put,

  $C_3$ = 10.5

  $C_2$ =5

  $C_1$ =2

  $\begin{array}{l}wt\%Liquid=\frac{5-2}{10.5-2}\\ =0.3529\times 100\\ =35.29wt\%Liquid\end{array}$

Now, calculate the percentage of a-phase.

  $wt\%Liquid=\frac{C_3-C_2}{C_3-C_1}$

Where,

  $C_3$ = percent of silicon in the alloy

  $C_2$ =percent of silicon in the liquid phase

  $C_1$ =percent of silicon in the solid phase

Put,

  $C_3$ = 10.5

  $C_2$ =5

  $C_1$ =2

  $\begin{array}{l}wt\%Liquid=\frac{10.5-5}{10.5-2}\\ =0.6471\times 100\\ =64.71wt\%Liquid\end{array}$

Therefore, at ${578}^{\text{o}}\text{C}$ the percent of phase is 35.29wt% liquid and 64.71wt%a

Now for temperature ${576}^{\text{o}}\text{C}$

The given diagram is a graph plotted against temperature and weight percent of silicon.

Now calculate the percentage of phase

Therefore, the percentage of β-phase,

  $wt\%Liquid=\frac{C_2-C_1}{C_3-C_1}$

Where,

  $C_3$ = percent of silicon in the alloy

  $C_2$ =percent of silicon in the liquid phase

  $C_1$ =percent of silicon in the solid phase

Put,

  $C_3$ = 98.5

  $C_2$ =5

  $C_1$ =2.3

  $\begin{array}{l}wt\%\beta =\frac{5-2.3}{98.5-2.3}\\ =0.02806\times 100\\ =2.806wt\%\beta \end{array}$

Calculate the percentage of a- phase.

  $wt\%Liquid=\frac{C_2-C_1}{C_3-C_1}$

Where,

  $C_3$ = percent of silicon in the alloy

  $C_2$ =percent of silicon in the liquid phase

  $C_1$ =percent of silicon in the solid phase

Put,

  $C_3$ = 98.5

  $C_2$ =5

  $C_1$ =2.3

  $\begin{array}{l}wt\%\alpha =\frac{98.5-5}{98.5-2.3}\\ =0.97194\times 100\\ =97.194wt\%\alpha \end{array}$

Therefore, at ${576}^{\text{o}}\text{C}$ the percent of phase is 2.806% weight β and 97.194% weight a

Now for temperature ${\text{500}}^{\text{o}}\text{C}$ ,

The given diagram is a graph plotted against temperature and weight percent of silicon.

Now calculate the percentage of phase

Therefore, the percentage of β-phase,

  $wt\%Liquid=\frac{C_2-C_1}{C_3-C_1}$

Where,

  $C_3$ = percent of silicon in the alloy

  $C_2$ =percent of silicon in the liquid phase

  $C_1$ =percent of silicon in the solid phase

Put,

  $C_3$ = 97

  $C_2$ =5

  $C_1$ =2

  $\begin{array}{l}wt\%\beta =\frac{5-2}{97-2}\\ =0.03158\times 100\\ =3.158wt\%\beta \end{array}$

Calculate the percentage of a- phase.

  $wt\%\alpha =\frac{C_3-C_2}{C_3-C_1}$

Where,

  $C_3$ = percent of silicon in the alloy

  $C_2$ =percent of silicon in the liquid phase

  $C_1$ =percent of silicon in the solid phase

Put,

  $C_3$ = 97

  $C_2$ =5

  $C_1$ =2

  $\begin{array}{l}wt\%\alpha =\frac{97-5}{97-2}\\ =0.96842\times 100\\ =96.842wt\%\alpha \end{array}$

Therefore, at ${\text{500}}^{\text{o}}\text{C}$ the percent of phase is 3.158% weight β and 96.842% weight a

Conclusion

At, ${\text{620}}^{\text{o}}\text{C}$, percent obtained are 66.7% weight L and 33.3% weight a

At, ${\text{600}}^{\text{o}}\text{C}$, percent obtained are 43.75% weight L and 56.25% weight a

At, ${578}^{\text{o}}\text{C}$, percent obtained are 35.29% weight L and 64.71% weight a

At, ${576}^{\text{o}}\text{C}$, percent obtained are 2.806% weight β and 97.197% weight a

At, ${\text{500}}^{\text{o}}\text{C}$, percent obtained are 3.158% weight β and 96.842% weight a