# Heat capcity

1. Nov 2, 2011

### ibysaiyan

1. The problem statement, all variables and given/known data

Atoms of the inert gas Krypton are adsorbed onto a smooth solid
surface at 90K. They can move freely over the surface but they cannot leave it. For a
sample with total surface area 2.5 m^2, and a surface density 3 nm-2, what is the heat
capacity of the krypton?

2. Relevant equations

$\Delta E$/ $\Delta T$ = Cv ( where Cv is specific heat @ constant volume)

$\Delta E$ = $\Delta W$ + $\DeltaQ$
U = 3/2nRT , U = 3/2NkbT

3. The attempt at a solution

Data given:

Krypton molecular weight : 84.8 /1000 = 0.0848 kg/mol.

I believe I have an idea on how to solve this. The fact that the gases can't escape implies work done = 0, however I am a little confused by the spatial dimensions provided.
Would I multiply 2.5 with 3 ? however that makes no sense to get mass. I was wondering.. could I use : Mass = Molecular weight / Avogadro's number and then plug in: Moles = mass/mr to give me 'n' which I could plug into U = 3/2nRT

Is my approach sensible ?

Last edited: Nov 2, 2011
2. Nov 2, 2011

### Staff: Mentor

Hmm. Surface area and surface density give you the number of moles of gas. The fact that the gas is trapped on the surface (but moves freely thereon) implies that you have a 2d gas! That is, one less degree of freedom for kinetic energy distribution for internal energy than a 3d volume of gas.

Perhaps you need to investigate equipartition of energy for an ideal gas with a given temperature, and how the specific heat constants are based upon the available degrees of freedom for distributing the energy. Hot search terms would be "specific heat", "internal energy", "equipartition", "kinetic temperature".

3. Nov 4, 2011

### ibysaiyan

Thanks for your reply. I am presuming that surface density times surface area would give number of moles, right ?
Also I am not quite sure if I have come across the term '2d' gas... nor it's consequence. I haven't done any reading yet, I ought to. I will do that tomorrow morning.

4. Nov 5, 2011

### ibysaiyan

I am so lost as to what to do with surface density and area... I tried to manipulate them to get something meaningful but to no avail. I mean if i divide or multiple the units.. they dont get anywhere sensible.. like for instance density has N/m^2.. which's kg/s^2 m...now if i multiply this with area... i get kgm/s^2... just a m factor away from getting energy.. any tips ?

5. Nov 5, 2011

### Staff: Mentor

The surface density is specified as 3 atoms of Krypton per square nanometer.

6. Nov 5, 2011

### ibysaiyan

Thanks for your reply.. the term 'surface density' is vague to me... so how does that relate to finding out mole of substance.

Last edited: Nov 5, 2011
7. Nov 5, 2011

### Staff: Mentor

It tells you how many atoms there are per square unit of surface area. What's the surface area? How many atoms does that make? How many atoms in a mole of atoms?

8. Nov 5, 2011

### ibysaiyan

edit: surface area is 2.5 m^2 , 3/ 2.5 = 1.2 so would n = 1.2 ?

Last edited: Nov 5, 2011
9. Nov 5, 2011

### Staff: Mentor

Your surface area was given as 2.5 m2. How many square nanometers is that?

10. Nov 5, 2011

### ibysaiyan

2.5 *10^(18)

11. Nov 5, 2011

### Staff: Mentor

So if there are 3 atoms of Krypton for ever square nm, how many atoms is that?

12. Nov 5, 2011

### ibysaiyan

2.5 *10^18 / 1*10^-9 =2.5 *10^27... so I will divide with this by Avogadro's number, right ?

13. Nov 5, 2011

### Staff: Mentor

Why did you divide by 10^-9?

14. Nov 5, 2011

### ibysaiyan

as you had said earlier.. there are three atoms per one nanometer so wouldn't i divide it by that ?

edit: it's about 4 am right now which may explain my incoherent thinking..

15. Nov 5, 2011

### Staff: Mentor

If you have 2 shoes per shoebox and you have a million shoeboxes, how many shoes do you have? Surely you don't divide a million by two!

Multiply the number of square nm by the number of atoms per nm to yield the total number of atoms. Once you have that, yes, you divide by Avogadro's number to find the number of moles.

Last edited: Nov 5, 2011
16. Nov 5, 2011

### ibysaiyan

Oh silly me....
so 2.5 *10^18 x 3 = 7.5*10^18 .. I hope i did this right...

17. Nov 5, 2011

### Staff: Mentor

Okay, so there are 7.5*10^18 atoms. Divide by NA to yield moles.

18. Nov 6, 2011

### ibysaiyan

I get approximately 1.25*10^-5 mol.
Now that gives me the 'n' value however I have to find specific heat @ 90k for krypton.
I know total energy of a system is given by : F/2 NkbT
Where f is the degrees of freedom but how do i figure out D.O.freedom for this gas, N , number of particles.
So if I find out the energy value and then divide that by T, then i should get my specific heat value, am I right ?
For number of particles I will multiply NA with moles .

edit: is the degrees of freedom in this case 2 , instead of three ? if so why.. is it because of the three independent squared terms i.e x,y,z one freezes [translational]?

19. Nov 6, 2011

### ehild

Yes, you have it correct: the atoms can not move away from the surface, so one from the usual 3 terms in the energy is zero.

ehild

20. Nov 6, 2011

### ibysaiyan

I get a value of 1.038 *10^-4, is that a reasonable value ? Since a quick google search shows it to be 0.247 J/g/C..