Using the Gibbs-Dalton Law to find Specific Heat Ratio

[ColdFusion85]

Homework Statement

A mixture of gases containing 10 kg of nitrogen, 10 kg of hydrogen, and 15 kg of helium is contained at a pressure of 6.7 MPa and a temperature of 300 K. If the constituents are taken to be perfect gases and the Gibbs-Dalton law holds, what are the molecular weight and specific heat ratio of the mixture?

Nitrogen: Mole wt = 28, Specific Heat Ratio = 1.4
Hydrogen: Mole wt = 2 , Specific Heat Ratio = 1.4
Helium: Mole wt = 4, Specific Heat Ratio = 1.67


2. The Attempt at the Solution

Basically I first found the number of moles of each constituent by dividing the amount of each constituent in the mixture by its mole weight. Then I found the mole fractions by dividing the # of moles of each constituent by the total number of moles in the mixture. I then multiplied the mole fraction of each constituent by the mole weight for each constituent and divided each by the sum of these products to get the mole fraction of each constituent. This is where I get stuck. To get the specific heat ratio of the mixture I need to determine the specific heat at either constant pressure of volume (doesn't matter which one since you can get the other later on via other relations). How do I determine C_p, for example? I know it depends on temperature, but is there a formula I use to calculate C_p? Where does the pressure of the mixture come into play? Is the fact that the specific heat ratio of each constituent is given trying to hint at something? I guess I am confused as to where to go from here.

 

[ColdFusion85]

Anyone?

 

[Astronuc]

It sounds like you did the first part correctly. It would be helpful if one showed the formulae and calculations.

Let me think about the ratio, I may have misread the problem.

 

[ColdFusion85]

Mole weight was given. To get moles, for example I would take N_2 (mole weight 28), and since N_2 is 10kg in the problem, there are 10kg/28kg = .36 moles of N_2 in the mixture. You do the same for the other two gases then add them up to get the total number of moles in the mixture (9.11). Then the mole fraction for each gas is just, for example, .36/9.11 for nitrogen. Then I multiply that number by the mole weight to get the weight of the gas. Adding them up gives the molecular weight of the mixture. Then, the mass fraction is just the weight of the gas/weight of mixture. Now to get the specific heat of the mixture (at constant pressure for example) I need to find the specific heats for the constituents. Then i would take the specific heats of each part and multiply by the respective weight of that part, and add them up to get the C_p of the mixture. From there I can handle the rest of the problem. However, I don't know how to get C_p. In class he did it at standard temp and pressure, but he didn't say how to do it at a different temp and pressure, as in this problem. This is where I am stuck.

 

[Astronuc]

One can try

\gamma\,=\,\frac{c_p}{c_v} and c_p\,=\,c_v\,+\,R

and rearrange terms to get c_{p[/sup] in terms of \gamma and R.}