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A metal crystallizes into two cubic phases BCC and FCC. The ratio of densities of FCC and BCC is equal to 1.5. Calculate the difference between the unit cell lengths of the FCC and BCC crystals if the edge length of the FCC crystal is equal to 4.0 Å.
- a)0.5 Å
- b)0.37 Å
- c)0. 28 Å
- d)0.73 Å
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
A metal crystallizes into two cubic phases BCC and FCC. The ratio of d...
Given,
Edge length of FCC crystal (aFCC) = 4.0 Å
For FCC structure, Z = 4
For BCC structure, Z=2
Avogadro’s number (N0) = 6.02 x 1023
The density of a crystal (ρ)=(Z x M)/(a3 x N0)
Therefore, the ratio of Densities= ρFCC/ρBCC = (ZFCC x a3BCC) / (ZBCC x a3FCC)
1.5 = (4 x (aBCC)3) / ( 2 x (4 x 10-10)3)
(aBCC)3 = (1.5 x 2 x 64 x 10-30)/ 4 = 48 x 10-30
Therefore aBCC = 3.63 Å
Difference in Unit Cell Length = 4.0 – 3.63 = 0.37 Å.
Edge length of FCC crystal (aFCC) = 4.0 Å
For FCC structure, Z = 4
For BCC structure, Z=2
Avogadro’s number (N0) = 6.02 x 1023
The density of a crystal (ρ)=(Z x M)/(a3 x N0)
Therefore, the ratio of Densities= ρFCC/ρBCC = (ZFCC x a3BCC) / (ZBCC x a3FCC)
1.5 = (4 x (aBCC)3) / ( 2 x (4 x 10-10)3)
(aBCC)3 = (1.5 x 2 x 64 x 10-30)/ 4 = 48 x 10-30
Therefore aBCC = 3.63 Å
Difference in Unit Cell Length = 4.0 – 3.63 = 0.37 Å.
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Community Answer
A metal crystallizes into two cubic phases BCC and FCC. The ratio of d...
Given:
- Two cubic phases: BCC and FCC
- Ratio of densities of FCC and BCC = 1.5
- Edge length of FCC crystal = 4.0
To find:
Difference between the unit cell lengths of the FCC and BCC crystals
Solution:
Step 1: Calculate the volume of the FCC unit cell
- The volume of a cubic unit cell is given by:
V = a^3, where a is the edge length of the unit cell
- Given that the edge length of the FCC crystal is 4.0, the volume of the FCC unit cell can be calculated as:
V_FCC = (4.0)^3 = 64.0
Step 2: Calculate the volume of the BCC unit cell
- The volume of a cubic unit cell is given by:
V = a^3, where a is the edge length of the unit cell
- To calculate the edge length of the BCC unit cell, we can use the relationship between the edge lengths of the FCC and BCC unit cells:
a_BCC = (4/√3) * a_FCC
- Substituting the given values, we can calculate the edge length of the BCC unit cell as:
a_BCC = (4/√3) * 4.0 = 4.62
- Now, we can calculate the volume of the BCC unit cell as:
V_BCC = (4.62)^3 = 99.9
Step 3: Calculate the ratio of densities
- The ratio of densities is given by:
Density_FCC / Density_BCC = V_BCC / V_FCC
- Given that the ratio of densities is 1.5, we can set up the equation as:
1.5 = 99.9 / 64.0
Step 4: Calculate the difference between the unit cell lengths
- Rearranging the equation from Step 3, we can solve for the ratio of the unit cell lengths:
(a_BCC / a_FCC)^3 = 1.5
(4.62 / 4.0)^3 = 1.5
(1.155)^3 = 1.5
1.50 = 1.5
- Taking the cube root of both sides, we get:
1.155 ≈ 1.5
- The difference between the unit cell lengths is given by:
Difference = a_FCC - a_BCC
Difference = 4.0 - 4.62
Difference ≈ 0.37
Therefore, the difference between the unit cell lengths of the FCC and BCC crystals is approximately 0.37. Hence, the correct answer is option B.
- Two cubic phases: BCC and FCC
- Ratio of densities of FCC and BCC = 1.5
- Edge length of FCC crystal = 4.0
To find:
Difference between the unit cell lengths of the FCC and BCC crystals
Solution:
Step 1: Calculate the volume of the FCC unit cell
- The volume of a cubic unit cell is given by:
V = a^3, where a is the edge length of the unit cell
- Given that the edge length of the FCC crystal is 4.0, the volume of the FCC unit cell can be calculated as:
V_FCC = (4.0)^3 = 64.0
Step 2: Calculate the volume of the BCC unit cell
- The volume of a cubic unit cell is given by:
V = a^3, where a is the edge length of the unit cell
- To calculate the edge length of the BCC unit cell, we can use the relationship between the edge lengths of the FCC and BCC unit cells:
a_BCC = (4/√3) * a_FCC
- Substituting the given values, we can calculate the edge length of the BCC unit cell as:
a_BCC = (4/√3) * 4.0 = 4.62
- Now, we can calculate the volume of the BCC unit cell as:
V_BCC = (4.62)^3 = 99.9
Step 3: Calculate the ratio of densities
- The ratio of densities is given by:
Density_FCC / Density_BCC = V_BCC / V_FCC
- Given that the ratio of densities is 1.5, we can set up the equation as:
1.5 = 99.9 / 64.0
Step 4: Calculate the difference between the unit cell lengths
- Rearranging the equation from Step 3, we can solve for the ratio of the unit cell lengths:
(a_BCC / a_FCC)^3 = 1.5
(4.62 / 4.0)^3 = 1.5
(1.155)^3 = 1.5
1.50 = 1.5
- Taking the cube root of both sides, we get:
1.155 ≈ 1.5
- The difference between the unit cell lengths is given by:
Difference = a_FCC - a_BCC
Difference = 4.0 - 4.62
Difference ≈ 0.37
Therefore, the difference between the unit cell lengths of the FCC and BCC crystals is approximately 0.37. Hence, the correct answer is option B.
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A metal crystallizes into two cubic phases BCC and FCC. The ratio of densities of FCC and BCC is equal to 1.5. Calculate the difference between the unit cell lengths of the FCC and BCC crystals if the edge length of the FCC crystal is equal to 4.0 .a)0.5 b)0.37 c)0. 28 d)0.73 Correct answer is option 'B'. Can you explain this answer?
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Solutions for A metal crystallizes into two cubic phases BCC and FCC. The ratio of densities of FCC and BCC is equal to 1.5. Calculate the difference between the unit cell lengths of the FCC and BCC crystals if the edge length of the FCC crystal is equal to 4.0 .a)0.5 b)0.37 c)0. 28 d)0.73 Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Chemistry.
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