Question
A hanging wire made out of titanium with diameter 0.080 cm is initially 2.3 m long. When a 78 kg mass is hung from it, the wire stretches an amount 2.89 cm. A mole of titanium has a mass of 47.9 grams, and its density is 4.51 g/cm3. Based on these experimental measurements, what is the Young's modulus for titanium?
Find the effective spring stiffness of one interatomic bond in titanium.
Find the effective spring stiffness of one interatomic bond in titanium.
Answers
Young's modulus = (tensile stress)/(strain)
In this case the strain is
(delta L)/L = 2.89cm/230 cm
= 1.257*10^-2
and the stress is
sigma = (78*9.8 N)/[(pi/4)*(8^10^-4m)^2] = 764.4 N/5.027*10^-7 m^2)
= 1.52*10^9 N/m^2
So E = 1.21*10^11 N/m^2
= 121 GPa
A spring stiffness for an individual Ti-Ti interatomic pair can be estimated by dividing
(Tensile force)/(area occupied by one molecule)
by (stretch per intermolecular molecule pair).
You will need a characteristic intermolecular distance or diameter for Ti atoms in the solid. Call it d. Get that from the number density of Ti atoms, n.
d = n^(-1/3)
n = [4.51 g/cm^3/(47.9g/mole)]*6.02*10^23 atom/mole = 5.56*10^22 atom/cm^3
n^-1/3 = d = 2.6*10^-8 cm
= 2.6*10^-10 m
spring stiffness = k
=(Tension/d^2)/(strain*d)
= E/d = 1.21*10^11/2.6*10^-10
= 4.6*10^20 N/m
In this case the strain is
(delta L)/L = 2.89cm/230 cm
= 1.257*10^-2
and the stress is
sigma = (78*9.8 N)/[(pi/4)*(8^10^-4m)^2] = 764.4 N/5.027*10^-7 m^2)
= 1.52*10^9 N/m^2
So E = 1.21*10^11 N/m^2
= 121 GPa
A spring stiffness for an individual Ti-Ti interatomic pair can be estimated by dividing
(Tensile force)/(area occupied by one molecule)
by (stretch per intermolecular molecule pair).
You will need a characteristic intermolecular distance or diameter for Ti atoms in the solid. Call it d. Get that from the number density of Ti atoms, n.
d = n^(-1/3)
n = [4.51 g/cm^3/(47.9g/mole)]*6.02*10^23 atom/mole = 5.56*10^22 atom/cm^3
n^-1/3 = d = 2.6*10^-8 cm
= 2.6*10^-10 m
spring stiffness = k
=(Tension/d^2)/(strain*d)
= E/d = 1.21*10^11/2.6*10^-10
= 4.6*10^20 N/m