Help calculating equilibrium number of vacancies

[digeratus]

Homework Statement

Calculate the fraction of atom sites that are vacant for lead at its melting temperature of 327°C (600 K). Assume an energy for vacancy formation of 0.55 eV/atom.

Homework Equations

N = (Na)(p)/A

Nv = N exp (-Q/kT)

Na = avogadro's number
p = density
A = atomic weight
Nv = number of vacancies
N = number of atomic sites / number of atoms
Q = the energy for vacancy formation
k = Boltzmann constant (can be 8.62E-5 eV/K or 1.38E-23 J/(atom)(K) )
T = temperature in Kelvin

The Attempt at a Solution

I have a sample problem done for me in the textbook. Doing the exact equation that's in the book, with all of the variables given to me, I get an incorrect answer. This happens no matter what calculator I use. Intermittent rounding gives a very wrong answer. Plugging it all in at once gives a consistent answer of 0.

Using lead A = 207.2, the eV/K boltzmann's constant, and all the other information, I get either 0 or 7.46E24 vacancies. The options are: 1.52E-11, 5.84E-8, 4.03E-6, and 16.66 .

 

[Borek]

I have a sample problem done for me in the textbook. Doing the exact equation that's in the book, with all of the variables given to me, I get an incorrect answer. This happens no matter what calculator I use. Intermittent rounding gives a very wrong answer. Plugging it all in at once gives a consistent answer of 0.

Not seeing the book nor your work it is hard to comment on. You need to show more details.

What is N again? And what are its units?

If you need just a fraction, a second equation should be enough, first is not necessary.

 

[JohnRC]

It looks to me as though your second answer might be a correct calculation of the number of vacant sites per cubic metre of lead. (I got 8E24 with a rough calculation in my head). But that is not what you were asked for. Read the actual question carefully! And also read carefully the definitions and units for each of the quantities in your equations. I am fairly sure that you have stated at least one of them wrongly. But I cannot get a match with any of the alternative answers given either.

 

[digeratus]

Because it says "fraction of vacant sites" does that mean I need to put the vacant sites over the total number of atoms? They don't do that in the book, but that may be because the one in the book is for "equilibrium number of vacancies?" In any case, I can't calculate that one correctly either.

I didn't show work because it's all one equation and everything is given. I'm sorry though:

I do Nv = [(6.022E23 atoms/mol)(10.66 g/cm3)(10E6 cm3/m3)]/[207.2 g/mol] all multiplied by (e^(-((.55eV)/[(8.62E-5 eV/K)(600 K)]))

If I calculate the two aspects separately and then multiply them, I get the one number. If i do everything at once, I get 0. The sample problem in the book is exactly the same except for the element and temperature (copper & 1273 K). That one is:

Nv = [(6.022E23 atoms/mol)(8.4 g/cm3)(10E6 cm3/m3)]/[63.5 g/mol] all multiplied by (e^(-((.9eV)/[(8.62E-5 eV/K)(1273 K)]))

But I have the same problem with that, even though the values are all literally given to me. The answer is 2.2E25 vacancies/m3, but I either get 0 or a wrong answer.

 

[Borek]

The answer is 2.2E25 vacancies/m3, but I either get 0 or a wrong answer.

That's the number I am getting by simply plugging the expression as given into my calculator (I assumed 10E6 is intended to mean 10⁶, not 10x10⁶). Again, hard to tell what you are doing wrong while keying numbers into your calculator, but apparently that's the first thing you should work on. I suppose you are misunderstanding something about the way you should enter exponents and exponential function arguments.

 

[digeratus]

That's odd; I've had the same calculator for 4 years and haven't had problems before. Thank you though. I'll see what WolframAlpha gives me or something else.

 

[JohnRC]

Digeratus, the problem that most often comes up with a calculation like this is with the exponent.

When you are doing (e^(-((.9eV)/[(8.62E-5 eV/K)(1273 K)])) it is very easy to key into your calculator

" 0.9 / 8.62E-5 * 1273 " instead of " 0.9 / 8.62E-5 / 1273 "

If you do make this mistake it will give you a huge number for the exponent, and exp(large negative number) will be taken as zero by your calculator.