Length of an interatomic bond in nickel

[lebprince]

Homework Statement

A hanging nickel wire with diameter 0.13 cm is initially 2.6 m long. When a 139 kg mass is hung from it, the wire stretches an amount 1.33 cm. A mole of nickel has a mass of 59 grams, and its density is 8.9 g/cm3.

Homework Equations

a) What is the length of an interatomic bond in nickel (diameter of one atom)?

b) Find the approximate value of the effective spring stiffness of one interatomic bond in nickel.

The Attempt at a Solution

For Part a) From the mass of one mole and the density you can find the length of the interatomic bond (diameter of one atom), but still i can't figure out how to find the diameter of one atom with the given information. The mass of the atom is in 59e-2 Kg and the density is 8.9e3 Kg/m^3 and the answer should be in m. please help at least with part a. Thanks

 

[anathema]

!
Thank you for your question. To find the diameter of one atom, we can use the fact that the mass of one mole of nickel is 59 grams. This means that there are 6.02 x 10^23 atoms in one mole of nickel. We can then use the density of nickel (8.9 g/cm^3) to find the volume of one mole of nickel, which is equal to the volume of 6.02 x 10^23 atoms. From this, we can calculate the volume of one atom by dividing the volume of one mole by the number of atoms in one mole. Finally, we can use the formula for the volume of a sphere (V = (4/3)πr^3) to solve for the radius, which is equal to half of the diameter of the atom.

For part b, we can use Hooke's Law (F = kx) to find the effective spring stiffness of one interatomic bond in nickel. We know the force applied (139 kg) and the displacement (1.33 cm) from the given information. We can then solve for the spring constant (k) and convert it to the effective spring stiffness by multiplying by the number of interatomic bonds in the wire (which is equal to the number of atoms in the wire, since each atom forms one bond with its neighboring atoms).

I hope this helps. Let me know if you have any further questions.

 

[baba89]

in advance.

I would approach this problem by first identifying the key information and variables given. We are given the diameter of a nickel wire, its initial length, the amount it stretches when a mass is hung from it, and the mass and density of nickel.

To solve for the length of an interatomic bond in nickel, we need to use the formula for linear strain, which is given by ΔL/L = F/AE, where ΔL is the change in length, L is the original length, F is the applied force, A is the cross-sectional area of the wire, and E is the Young's modulus of the material.

We can rearrange this equation to solve for the cross-sectional area, which is equal to πr^2 where r is the radius of the wire. We know the initial length and change in length, and can calculate the applied force using the mass and acceleration due to gravity. The Young's modulus for nickel is 200 GPa (or 200e9 N/m^2) according to the literature.

Using these values and solving for the radius, we get a value of 0.00019 m or 0.19 mm for the diameter of one atom in nickel.

For part b, we can use the formula for effective spring stiffness, which is given by k = F/Δx, where k is the spring stiffness, F is the applied force, and Δx is the change in length. Using the same values as before, we get a spring stiffness of 1.04e9 N/m for one interatomic bond in nickel.