Derive the lever rule for the amount in weight percent of each phase in two-phase regions of a binary phase diagram. Use a phase diagram in which two elements are completely soluble in each other.

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Answer 1

Textbook Question

Chapter 8.15, Problem 50SEP

Derive the lever rule for the amount in weight percent of each phase in two-phase regions of a binary phase diagram. Use a phase diagram in which two elements are completely soluble in each other.

Expert Solution & Answer

To determine

Obtain the weight percentages of the liquid and solid phases in any two phase region of binary equilibrium phase diagram by the help of lever rule.

Explanation of Solution

Let the phase diagram of two elements A and B that are totally soluble in each other. Let the alloy composition of interest be x and its weight fraction of B in A as w0.

The lever equations can be obtained by the help of weight balances. The first equation of lever rule is derived by taking the sum of the weight fraction of liquid and solid phase as 1.

Write the sum of the phases as,

Xl+Xs=1

Xl=1−Xs

Xs=1−Xl

Here, weight fraction of liquid and solid phase is Xl and Xs respectively.

Derive the second equation of the lever rule by a using a weight balance of B in the alloy as a whole and the sum of B in the two different phases.

Write the weight balance by considering 1 g of alloy.

[Grams of B intwo phase mix]=[Grams of B inliquid phase]+[Grams of B insolid phase](1 g)(1)(%w0/100)=[(1 g)(Xl)(%wl/100)]+[(1 g)(Xs)(%ws/100)]w0=Xlwl+Xsws (I)

Substitute 1−Xs for Xl in Equation (I).

w0=(1−Xs)wl+Xsws=wl−Xswl+Xsws

Xsws−Xswl=w0−wl (II)

Rewrite Equation (II) to get weight fraction of solid phase.

Xs=w0−wlws−wl (III)

Substitute 1−Xl for Xs in Equation (III).

1−Xl=w0−wlws−wl

Xl=1−w0−wlws−wl=ws−wl−w0+wlws−wl=ws−w0ws−wl (IV)

Equations (III) and (IV) are called as lever rule equations for solid and liquid phase respectively.

Hence, the lever rule equations are Xs=w0−wlws−wl and Xl=ws−w0ws−wl.

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Answer 2

Textbook Question

Chapter 8.15, Problem 50SEP

Derive the lever rule for the amount in weight percent of each phase in two-phase regions of a binary phase diagram. Use a phase diagram in which two elements are completely soluble in each other.

Expert Solution & Answer

To determine

Obtain the weight percentages of the liquid and solid phases in any two phase region of binary equilibrium phase diagram by the help of lever rule.

Explanation of Solution

Let the phase diagram of two elements A and B that are totally soluble in each other. Let the alloy composition of interest be x and its weight fraction of B in A as w0.

The lever equations can be obtained by the help of weight balances. The first equation of lever rule is derived by taking the sum of the weight fraction of liquid and solid phase as 1.

Write the sum of the phases as,

Xl+Xs=1

Xl=1−Xs

Xs=1−Xl

Here, weight fraction of liquid and solid phase is Xl and Xs respectively.

Derive the second equation of the lever rule by a using a weight balance of B in the alloy as a whole and the sum of B in the two different phases.

Write the weight balance by considering 1 g of alloy.

[Grams of B intwo phase mix]=[Grams of B inliquid phase]+[Grams of B insolid phase](1 g)(1)(%w0/100)=[(1 g)(Xl)(%wl/100)]+[(1 g)(Xs)(%ws/100)]w0=Xlwl+Xsws (I)

Substitute 1−Xs for Xl in Equation (I).

w0=(1−Xs)wl+Xsws=wl−Xswl+Xsws

Xsws−Xswl=w0−wl (II)

Rewrite Equation (II) to get weight fraction of solid phase.

Xs=w0−wlws−wl (III)

Substitute 1−Xl for Xs in Equation (III).

1−Xl=w0−wlws−wl

Xl=1−w0−wlws−wl=ws−wl−w0+wlws−wl=ws−w0ws−wl (IV)

Equations (III) and (IV) are called as lever rule equations for solid and liquid phase respectively.

Hence, the lever rule equations are Xs=w0−wlws−wl and Xl=ws−w0ws−wl.

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Answer 3

Textbook Question

Chapter 8.15, Problem 50SEP

Derive the lever rule for the amount in weight percent of each phase in two-phase regions of a binary phase diagram. Use a phase diagram in which two elements are completely soluble in each other.

Expert Solution & Answer

To determine

Obtain the weight percentages of the liquid and solid phases in any two phase region of binary equilibrium phase diagram by the help of lever rule.

Explanation of Solution

Let the phase diagram of two elements A and B that are totally soluble in each other. Let the alloy composition of interest be x and its weight fraction of B in A as w0.

The lever equations can be obtained by the help of weight balances. The first equation of lever rule is derived by taking the sum of the weight fraction of liquid and solid phase as 1.

Write the sum of the phases as,

Xl+Xs=1

Xl=1−Xs

Xs=1−Xl

Here, weight fraction of liquid and solid phase is Xl and Xs respectively.

Derive the second equation of the lever rule by a using a weight balance of B in the alloy as a whole and the sum of B in the two different phases.

Write the weight balance by considering 1 g of alloy.

[Grams of B intwo phase mix]=[Grams of B inliquid phase]+[Grams of B insolid phase](1 g)(1)(%w0/100)=[(1 g)(Xl)(%wl/100)]+[(1 g)(Xs)(%ws/100)]w0=Xlwl+Xsws (I)

Substitute 1−Xs for Xl in Equation (I).

w0=(1−Xs)wl+Xsws=wl−Xswl+Xsws

Xsws−Xswl=w0−wl (II)

Rewrite Equation (II) to get weight fraction of solid phase.

Xs=w0−wlws−wl (III)

Substitute 1−Xl for Xs in Equation (III).

1−Xl=w0−wlws−wl

Xl=1−w0−wlws−wl=ws−wl−w0+wlws−wl=ws−w0ws−wl (IV)

Equations (III) and (IV) are called as lever rule equations for solid and liquid phase respectively.

Hence, the lever rule equations are Xs=w0−wlws−wl and Xl=ws−w0ws−wl.

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Answer 4

Textbook Question

Chapter 8.15, Problem 50SEP

Derive the lever rule for the amount in weight percent of each phase in two-phase regions of a binary phase diagram. Use a phase diagram in which two elements are completely soluble in each other.

Expert Solution & Answer

To determine

Obtain the weight percentages of the liquid and solid phases in any two phase region of binary equilibrium phase diagram by the help of lever rule.

Explanation of Solution

Let the phase diagram of two elements A and B that are totally soluble in each other. Let the alloy composition of interest be x and its weight fraction of B in A as w0.

The lever equations can be obtained by the help of weight balances. The first equation of lever rule is derived by taking the sum of the weight fraction of liquid and solid phase as 1.

Write the sum of the phases as,

Xl+Xs=1

Xl=1−Xs

Xs=1−Xl

Here, weight fraction of liquid and solid phase is Xl and Xs respectively.

Derive the second equation of the lever rule by a using a weight balance of B in the alloy as a whole and the sum of B in the two different phases.

Write the weight balance by considering 1 g of alloy.

[Grams of B intwo phase mix]=[Grams of B inliquid phase]+[Grams of B insolid phase](1 g)(1)(%w0/100)=[(1 g)(Xl)(%wl/100)]+[(1 g)(Xs)(%ws/100)]w0=Xlwl+Xsws (I)

Substitute 1−Xs for Xl in Equation (I).

w0=(1−Xs)wl+Xsws=wl−Xswl+Xsws

Xsws−Xswl=w0−wl (II)

Rewrite Equation (II) to get weight fraction of solid phase.

Xs=w0−wlws−wl (III)

Substitute 1−Xl for Xs in Equation (III).

1−Xl=w0−wlws−wl

Xl=1−w0−wlws−wl=ws−wl−w0+wlws−wl=ws−w0ws−wl (IV)

Equations (III) and (IV) are called as lever rule equations for solid and liquid phase respectively.

Hence, the lever rule equations are Xs=w0−wlws−wl and Xl=ws−w0ws−wl.

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution & Answer

To determine

Obtain the weight percentages of the liquid and solid phases in any two phase region of binary equilibrium phase diagram by the help of lever rule.

Explanation of Solution

Let the phase diagram of two elements A and B that are totally soluble in each other. Let the alloy composition of interest be x and its weight fraction of B in A as w0.

The lever equations can be obtained by the help of weight balances. The first equation of lever rule is derived by taking the sum of the weight fraction of liquid and solid phase as 1.

Write the sum of the phases as,

Xl+Xs=1

Xl=1−Xs

Xs=1−Xl

Here, weight fraction of liquid and solid phase is Xl and Xs respectively.

Derive the second equation of the lever rule by a using a weight balance of B in the alloy as a whole and the sum of B in the two different phases.

Write the weight balance by considering 1 g of alloy.

[Grams of B intwo phase mix]=[Grams of B inliquid phase]+[Grams of B insolid phase](1 g)(1)(%w0/100)=[(1 g)(Xl)(%wl/100)]+[(1 g)(Xs)(%ws/100)]w0=Xlwl+Xsws (I)

Substitute 1−Xs for Xl in Equation (I).

w0=(1−Xs)wl+Xsws=wl−Xswl+Xsws

Xsws−Xswl=w0−wl (II)

Rewrite Equation (II) to get weight fraction of solid phase.

Xs=w0−wlws−wl (III)

Substitute 1−Xl for Xs in Equation (III).

1−Xl=w0−wlws−wl

Xl=1−w0−wlws−wl=ws−wl−w0+wlws−wl=ws−w0ws−wl (IV)

Equations (III) and (IV) are called as lever rule equations for solid and liquid phase respectively.

Hence, the lever rule equations are Xs=w0−wlws−wl and Xl=ws−w0ws−wl.

Answer 8

Expert Solution & Answer

To determine

Obtain the weight percentages of the liquid and solid phases in any two phase region of binary equilibrium phase diagram by the help of lever rule.

Explanation of Solution

Let the phase diagram of two elements A and B that are totally soluble in each other. Let the alloy composition of interest be x and its weight fraction of B in A as w0.

The lever equations can be obtained by the help of weight balances. The first equation of lever rule is derived by taking the sum of the weight fraction of liquid and solid phase as 1.

Write the sum of the phases as,

Xl+Xs=1

Xl=1−Xs

Xs=1−Xl

Here, weight fraction of liquid and solid phase is Xl and Xs respectively.

Derive the second equation of the lever rule by a using a weight balance of B in the alloy as a whole and the sum of B in the two different phases.

Write the weight balance by considering 1 g of alloy.

[Grams of B intwo phase mix]=[Grams of B inliquid phase]+[Grams of B insolid phase](1 g)(1)(%w0/100)=[(1 g)(Xl)(%wl/100)]+[(1 g)(Xs)(%ws/100)]w0=Xlwl+Xsws (I)

Substitute 1−Xs for Xl in Equation (I).

w0=(1−Xs)wl+Xsws=wl−Xswl+Xsws

Xsws−Xswl=w0−wl (II)

Rewrite Equation (II) to get weight fraction of solid phase.

Xs=w0−wlws−wl (III)

Substitute 1−Xl for Xs in Equation (III).

1−Xl=w0−wlws−wl

Xl=1−w0−wlws−wl=ws−wl−w0+wlws−wl=ws−w0ws−wl (IV)

Equations (III) and (IV) are called as lever rule equations for solid and liquid phase respectively.

Hence, the lever rule equations are Xs=w0−wlws−wl and Xl=ws−w0ws−wl.

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

To determine

Obtain the weight percentages of the liquid and solid phases in any two phase region of binary equilibrium phase diagram by the help of lever rule.

Answer 12

Explanation of Solution

Let the phase diagram of two elements A and B that are totally soluble in each other. Let the alloy composition of interest be x and its weight fraction of B in A as w0.

The lever equations can be obtained by the help of weight balances. The first equation of lever rule is derived by taking the sum of the weight fraction of liquid and solid phase as 1.

Write the sum of the phases as,

Xl+Xs=1

Xl=1−Xs

Xs=1−Xl

Here, weight fraction of liquid and solid phase is Xl and Xs respectively.

Derive the second equation of the lever rule by a using a weight balance of B in the alloy as a whole and the sum of B in the two different phases.

Write the weight balance by considering 1 g of alloy.

[Grams of B intwo phase mix]=[Grams of B inliquid phase]+[Grams of B insolid phase](1 g)(1)(%w0/100)=[(1 g)(Xl)(%wl/100)]+[(1 g)(Xs)(%ws/100)]w0=Xlwl+Xsws (I)