One mole of tungsten (6 1023 atoms) has a mass of 184 grams, and its density is

Question

One mole of tungsten (6 1023 atoms) has a mass of 184 grams, and its density is 19.3 grams per cubic centimeter, so the center-to-center distance between atoms is 2.51 10-10 m. You have a long thin bar of tungsten, 2.6 m long, with a square cross section, 0.12 cm on a side. You hang the rod vertically and attach a 265 kg mass to the bottom, and you observe that the bar becomes 1.3 cm longer. From these measurements, it is possible to determine the stiffness of one interatomic bond in tungsten. 1) What is the spring stiffness of the entire wire, considered as a single macroscopic (large scale), very stiff spring? ks = N/m 2) How many side-by-side atomic chains (long springs) are there in this wire? This is the same as the number of atoms on the bottom surface of the tungsten wire. Note that the cross-sectional area of one tungsten atom is (2.51 10-10)2 m2. Number of side-by-side long chains of atoms = 3) How many interatomic bonds are there in one atomic chain running the length of the wire? Number of bonds in total length = 4) What is the stiffness of a single interatomic "spring"? ks,i = N/m An interatomic bond in tungsten is stiffer than a slinky, but less stiff than a pogo stick. The stiffness of a single interatomic bond is very much smaller than the stiffness of the entire wire. Please show steps.

Explanation / Answer

a) Force applied on the bar (F) = mg = 119 * 9.8 = 1166.2 N

Extension in the wire (L) = 1.27 cm = 0.0127

so, stiffness of wire = F/L = 1166.2 / 0.0127

stiffness of wire = 91826.77 N/m

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b) number of atoms in one layer of cross section = (8e-4)2 / (2.51e-10)2

  number of atoms in one layer of cross section = 1.0158e13

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c) No. of bond along the length = 2.5 / 2.51e-10

No. of bond along the length = 9.96e9

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d) Bodn stiffness = force applied to bond / bond strain

force applied to bond = 1.152e-20 N

strain = (0.0127 / 2.5) * 2.51e-10 = 1.275e-12 m

so, stiffness = 1.152e-20 / 1.275e-12  

stiffness = 9.035e-9 N/m ( I am not sure about this part, please check )

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