Using the Gibbs-Dalton Law to find Specific Heat Ratio

In summary, the problem involves finding the molecular weight and specific heat ratio of a mixture of gases at a given pressure and temperature. To do so, one must first calculate the number of moles and mole fractions of each constituent in the mixture. The specific heat of the mixture can then be determined by finding the specific heats of each constituent and combining them using their respective weights. This can be done using the equations \gamma\,=\,\frac{c_p}{c_v} and c_p\,=\,c_v\,+\,R, rearranged to solve for c_p in terms of \gamma and R.
  • #1
ColdFusion85
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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.
 
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  • #2
Anyone?
 
  • #3
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.
 
  • #4
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.
 
  • #5
One can try

[tex]\gamma\,=\,\frac{c_p}{c_v}[/tex] and [tex]c_p\,=\,c_v\,+\,R[/tex]

and rearrange terms to get cp[/sup] in terms of [itex]\gamma[/itex] and R.
 

1. How does the Gibbs-Dalton Law relate to finding specific heat ratio?

The Gibbs-Dalton Law is a thermodynamic principle that states the total energy of a system is equal to the sum of its internal energy, pressure-volume work, and surface energy. This law is useful in finding the specific heat ratio, which is the ratio of specific heats at constant pressure and constant volume.

2. What is the formula for calculating specific heat ratio using the Gibbs-Dalton Law?

The formula for specific heat ratio (γ) using the Gibbs-Dalton Law is: γ = Cp/Cv, where Cp is the specific heat at constant pressure and Cv is the specific heat at constant volume.

3. Can the Gibbs-Dalton Law be used for all types of substances?

Yes, the Gibbs-Dalton Law can be applied to all types of substances, including gases, liquids, and solids. However, the specific heat ratio may vary depending on the type of substance and its molecular structure.

4. How is the Gibbs-Dalton Law experimentally determined?

The Gibbs-Dalton Law is usually determined experimentally through calorimetry, which involves measuring the heat transfer between a substance and its surroundings. This can be done by measuring the change in temperature of a substance as it undergoes a physical or chemical change.

5. What are some practical applications of using the Gibbs-Dalton Law to find specific heat ratio?

The Gibbs-Dalton Law is commonly used in engineering and thermodynamics to calculate the specific heat ratio of different substances. It is also used in the design and optimization of heat engines, such as gas turbines and internal combustion engines. Additionally, the specific heat ratio is an important parameter in the study of thermodynamic processes and heat transfer.

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