Foundations of Materials Science and Engineering
6th Edition
ISBN: 9781259696558
Author: SMITH
Publisher: MCG
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Chapter 4.8, Problem 5KCP
During solidification, how does the degree of undercooling affect the critical nucleus size? Assume homogeneous nucleation.
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Q1/ In a homogeneous solidification process, assume molten metal solidifies into a spherical
nucleus with a BCC structure. The given data are; lattice parameter (0.292 nm), the heat of
fusion energy (1.85×10-9 J/m³), latent surface free energy (0.204 J/m²), critical radius (1-35
nm), equilibrium melting temperature (1516 K), and room temperature (27 °C). Calculate the
following for this metal;
(a) supercooling value temperature (b) activation tree energy (c) number of atoms in a
nucleus of critical size.
(a) For the solidification of iron, calculate the critical radius r* and the activation free energy ΔG* if nucleation is homogeneous. Values for the latent heat of fusion and surface free energy are –1.85 × 109 J/m3 and 0.204 J/m2, respectively. Use the supercooling value ΔT = 286 K, and the melting point of iron is 1538°C.
(Critical radius, r* in nm and Activation Free Energy ΔG* in J)
(b) Now calculate the number of atoms found in a nucleus of critical size. Assume a lattice parameter of 0.292 nm for solid iron at its melting temperature.
(Number of Atoms for Critical Size in atoms/critical nuclues)
4.a) For the solidification of iron, calculate the critical radius r* and the activation free
energy AG* if nucleation is homogeneous. Values for the latent heat of fusion and
surface free energy are -1.85 x 10° J/m3 and 0.204 J/m?, respectively. Use the
supercooling value of 295°C for iron.
4.b) Calculate the number of atoms found in a nucleus of critical size. Assume a lattice
parameter of 0.292 nm for solid iron at its melting temperature..
Chapter 4 Solutions
Foundations of Materials Science and Engineering
Ch. 4.8 - Prob. 1KCPCh. 4.8 - Define the homogeneous nucleation process for the...Ch. 4.8 - In the solidification of a pure metal, what are...Ch. 4.8 - In the solidification of a metal, what is the...Ch. 4.8 - During solidification, how does the degree of...Ch. 4.8 - Distinguish between homogeneous and heterogeneous...Ch. 4.8 - Describe the grain structure of a metal ingot that...Ch. 4.8 - Distinguish between equiaxed and columnar grains...Ch. 4.8 - How can the grain size of a cast ingot be refined?...Ch. 4.8 - Prob. 10KCP
Ch. 4.8 - Prob. 11KCPCh. 4.8 - Prob. 12KCPCh. 4.8 - Distinguish between a substitutional solid...Ch. 4.8 - What are the conditions that are favorable for...Ch. 4.8 - Prob. 15KCPCh. 4.8 - Prob. 16KCPCh. 4.8 - Prob. 17KCPCh. 4.8 - Prob. 18KCPCh. 4.8 - Describe the structure of a grain boundary. Why...Ch. 4.8 - Describe and illustrate the following planar...Ch. 4.8 - Prob. 21KCPCh. 4.8 - Describe the optical metallography technique. What...Ch. 4.8 - Prob. 23KCPCh. 4.8 - Prob. 24KCPCh. 4.8 - Prob. 25KCPCh. 4.8 - Prob. 26KCPCh. 4.8 - Prob. 27KCPCh. 4.8 - Prob. 28KCPCh. 4.8 - Prob. 29KCPCh. 4.8 - Prob. 30KCPCh. 4.8 - Prob. 31KCPCh. 4.8 - Calculate the size (radius) of the critically...Ch. 4.8 - Prob. 33AAPCh. 4.8 - Prob. 34AAPCh. 4.8 - Calculate the number of atoms in a critically...Ch. 4.8 - Prob. 36AAPCh. 4.8 - Prob. 37AAPCh. 4.8 - Prob. 38AAPCh. 4.8 - Prob. 39AAPCh. 4.8 - Prob. 40AAPCh. 4.8 - Prob. 41AAPCh. 4.8 - Prob. 42AAPCh. 4.8 - Determine, by counting, the ASTM grain-size number...Ch. 4.8 - Prob. 44AAPCh. 4.8 - For the grain structure in Problem 4.43, estimate...Ch. 4.8 - Prob. 46AAPCh. 4.8 - Prob. 47SEPCh. 4.8 - Prob. 48SEPCh. 4.8 - Prob. 49SEPCh. 4.8 - Prob. 50SEPCh. 4.8 - In Chapter 3 (Example Problem 3.11), we calculated...Ch. 4.8 - Prob. 52SEPCh. 4.8 - Prob. 53SEPCh. 4.8 - Prob. 54SEPCh. 4.8 - Prob. 55SEPCh. 4.8 - Prob. 56SEPCh. 4.8 - Prob. 57SEPCh. 4.8 - Prob. 58SEPCh. 4.8 - Prob. 59SEPCh. 4.8 - Prob. 60SEPCh. 4.8 - Prob. 61SEPCh. 4.8 - Prob. 62SEPCh. 4.8 - Prob. 63SEPCh. 4.8 - Prob. 64SEPCh. 4.8 - Prob. 65SEPCh. 4.8 - Prob. 66SEP
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- For the solidification of nickel, calculate the critical radius r* and the activation energy ΔG* if nucleation is homogeneous. Values for the latent heat of fusion and surface free energy are -2.53×109 J/m3 and 0.255 J/m2, respectively. The supercooling value ΔT for the homogeneous nucleation of Ni is 319 oC. The melting point for Ni is 1455 oC.arrow_forwardDescribe the cooling of a peritectic alloy with the concentration CO and sketch the microstructure during solidification. T S+α 8 S S+B a+ß В сarrow_forwardQuestion-6. For solidification of a piece of FCC-metal at 860 °C. The melting point of the metal is 1260 °C. The latent heat of fusion and surface free energy are -2.16 x108 J/m³ and 0.126 J/m², respectively. If nucleation is homogeneous, answer the following questions: (a) Compute the critical radius r* in nm (b) Compute the activation free energy AG* in J (c) If the lattice parameter is 0.26 nm at the melting temperature, compute the number of atoms found in a nucleus of critical size (d) Compute the critical radius at the supercooling degree of 260 K.arrow_forward
- b) For the solidification of a metal, Tm =1000K with undercooling of 200K, calculate the rate of homogeneous nucleation in nuclei/m³/s. Neglect activation energy. Assume v=1o12/s and s'pd estimated as 1028/m³, AH=-1.26x10° J/m³, yLs= 0.16 J/m².arrow_forwardThe nucleation rate for a phase transformation depends on which factors? Density O Yield Strength O Undercooling (Temperature Energy) O Diffusivity of atoms O Holding time above Temperaturearrow_forwarda- Calculate the size of the critical nucleus of tin (Sn) after it has been supercooled by 20 °C. Assume that nucleation is uniform. For the solidification of tin (melting point is 232 °C), the enthalpy change is 0.42 GJ/m³. The interfacial energy between liquid and solid is 0.055 J/m². b- Answer the following questions? (Chose only two) 1- What are the advantages of the Hydrothermal growth method? 2- What are the parameters that the perfection of the final crystal is based on by using the growth from the Gel method? 3- What are the types of techniques for growing crystals from the vapor phase?arrow_forward
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- A 50%Ni-50%Cu alloy is slowly cooled from 1400°C to 1200°C. Determine (1) At what temperature does the first solid phase form andwhat is the solid’s composition?(2) At what temperature does all liquid solidify and what is thelast remaining liquid’s composition?(3) At 1300°C, estimate the compositions of the solid andliquid.(4) At 1300°C, estimate the weight fractions of the solid andliquid.arrow_forwardA cube shaped casting solidifies in 5 minutes. The solidification time in minutes for a cube of the same material, which is 8 times heavier than the original casting will be (A) 10 (B) 20 (C) 24 (D) 40arrow_forward1) From the figure of cooling curve find the following: (a) Pouring temperature (b) Cooling rate (c) Solidus temperature (d) Super heating (e) Super cooling (f) Solidification range 800 (g) Local solidification time (h) Total solidification time 600 400 8 10 Time (min) 2) From the figure of cooling curve find the following: (a) Pouring temperature (b) Cooling rate (c) Liquidus temperature (d) Solidus temperature (e) Super heating (f) Super cooling (g) Solidification range 300 (h) Local solidification time 200E (i) Total solidification time 100 4. 8 12 16 20 24 28 32 36 Times (sec) Temperature (°C) Temperature (°C)arrow_forward
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Explanation of Solidification of Metals & Alloys | Manufacturing Processes; Author: Magic Marks;https://www.youtube.com/watch?v=G5z9KknF_s8;License: Standard Youtube License