Thermal physics - ratio of multiplicites

[solidus_E]

Homework Statement

A penny, made of crystalline copper, is heated at 1,083 degrees celsius and melts. The density of copper is 8.5 g/cm^3. The energy needed to melt copper is 200 J/g. You will need to estimate the volume of a penny.
By what factor does the multiplicity of the copper change when the penny is melted? Express your answer as 10^(some power).

Homework Equations

roe = m/v
T (in kelvins) = T (in celsius) + 273.15
q = Em
ln (initial multiplicity / final multiplicity) = q/kT
v = .360 cm^3 (googled, haha)
k = 1.381 x 10^-23 J/K

The Attempt at a Solution

m = (roe)(v) = (8.5 g/cm^3)(0.360cm^3) = 3.06g

q = (200 J/g)(3.06g) = 200 J

ln (ratio of multiplicities) = (200 J) / [(1.4 x 10^-23 J/K)(1356.15 K)] = 3.22 x 10^22

e^ln (ratio of multiplicities) = e^3.22 x 10^22

(from book) e = 10^log(base10)e = 10^0.434

initial multiplicity / final multiplicity = (10^0.434)^(3.22 x 10^22) = 10^(1.4 x 10^22)

So my question is, did I do everything right? Is this the correct answer?

I followed an example used in the book and arrived at this conclusion from it.

Thanks for any help guys.

 

[Jhero]

Your calculation seems correct, but there are a few things you could clarify and double-check for accuracy.

First, make sure to convert the temperature to kelvins before using it in the equation for multiplicity (you did this correctly in your calculation for q, but not in the equation for multiplicity).

Also, it might be helpful to include units in your final answer - the multiplicity should be unitless, but you could specify that it is 10^(some power).

Lastly, it's always good to double-check your final answer and make sure it makes sense. In this case, the multiplicity should be a large number, since the number of possible arrangements of atoms increases significantly when the penny is melted. So a number like 10^(1.4 x 10^22) seems reasonable.