Cambridge Physics Problems: Avogadro's Constant

[johnconnor]

Sodium chloride crystallieses in a cubic lattice with ions of sodium and chlorine arranged alternately at the corners of a cube, so that any ion has six nearest neighbours of the other element. X-ray measurements show that the distance d, between the centres of adjacent sodium and chloride ions is 2.82E-10 m. The relative atomic massess of sodium and chlorine are 23 and 35.5 respectively, and the density of sodium chloride is 2.17E3 kg/m^3.

(i) Use this information to deduce a value for Avogadro constant.

(ii) Draw up a table showing the numbers and distances from P of the sodium ions in positions similar to ion A, the chloride ions in positions similar to ion B and the sodium ions in positions similar to ion C. Hence find an expression in terms of the inter-ionic distance d for the electrostatic potential energy Ves of the chloride ion P due to all other ions in this part of the crystal.

(iii) The expression obtained for Ves approximates to the electrostatic potential energy of the ion P in the whole crystal. Given that the distance d in sodium chloride is 2.8E-10m, estimate the binding energy of one mole of sodium chloride.

Attempt:
(i) Okay I really don't know what to do over here. I can't compare the ratio of the cubic volume of NaCl with that of ¹²C, and neither can I compare the number of atoms of NaCl in a volume of 24dm³ or 22.4. Comparing the mass doesn't help either. Can anyone please give me some pointers on this? Thank you!

 

[Infinitum]

For (1), density is given by

$$\rho = \frac{mass}{volume}$$

What is the mass of the cubic lattice in terms of the Avogadro's number and molar masses? What is the volume of the cube in terms of cube edge length? You have all the other values here except Avogadro's number, which you can calculate for.

 

[johnconnor]

Infinitum, have you any pointers or advice for the remaining parts, too? Thank you in advance.

 

[Infinitum]

I'm not really sure what ionic positions P, A, B, C signify. Was there a diagram to go along with this??

Also, how did you attempt these parts?

 

[johnconnor]

Oh dear. I'm sorry. I forgot to include them. here you go.

 

[Infinitum]

Okay, and your attempt, please?

 

[johnconnor]

Okay, and your attempt, please?

Okay. I was lying. I don't know how to do the second and third parts. In fact I'm still attempting the first part. =S

Please don't be mad at me?

 

[Infinitum]

Okay. I was lying. I don't know how to do the second and third parts. In fact I'm still attempting the first part. =S

Please don't be mad at me?

Okay, deal.

Please show how you attempted the first, after my hint??

 

[johnconnor]

Okay, deal.

Please show how you attempted the first, after my hint??

And then I'm stuck. I really don't know what "trick" am I supposed to apply here. Please help? Perhaps with a partial solution? Thank you very much!

 

[Infinitum]

Okay, so you have the mass of the cubic cell. This mass you got is the mass of 4 Na atoms and 4 Cl atoms. And one mole of NaCl has Avogadro number of Na and Cl atoms each. And you are given the masses of sodium and chlorine. Can you find the mass of one atom each of Na and Cl? So then you will have mass of 1 NaCl molecule in terms of Avogadro number, and the actual mass of 4NaCl molecules...Do you see a way to find N_A now?

 

[johnconnor]

Okay, so you have the mass of the cubic cell. This mass you got is the mass of 4 Na atoms and 4 Cl atoms. And one mole of NaCl has Avogadro number of Na and Cl atoms each. And you are given the masses of sodium and chlorine. Can you find the mass of one atom each of Na and Cl? So then you will have mass of 1 NaCl molecule in terms of Avogadro number, and the actual mass of 4NaCl molecules...Do you see a way to find N_A now?

No... I still can't. =( Could you please PM me the solution? I promise I'll learn from them. I have plenty of questions to go and I can't afford to get stuck on this forever. Thank you!

 

[Infinitum]

No... I still can't. =( Could you please PM me the solution? I promise I'll learn from them. I have plenty of questions to go and I can't afford to get stuck on this forever. Thank you!

Sorry, but no

Why do you want to miss the fun and excitement of this problem(Its a clever one!) by getting a ready made solution?? You are doing this for fun-spare time-learning, and I believe you shouldn't be looking at the quantity of sums, instead the fun and quality

I'll take the hint a step further...

The density of a mono atomic crystal structure is given by,

$$\rho = \frac{nM}{VN_A}$$

Where,
n is number of atoms in the unit cell
M is the mass of the atom in amu
V is the volume of the unit cellCan you see what the density of a diatomic crystal structure would be based on the above?