Thermodynamics problem relating to Chemical Potentials

• JohnJ
In summary, the answer to part (a) is that the chemical potentials are the same at equilibrium. The answer to part (b) is that the same equation is used to calculate the chemical potentials relative to CO2 at 1 atm.
JohnJ
Homework Statement
A 1.05 L bottle of a carbonated soft drink contains 1.0 L of drink as is pressurized at 5 atm with
CO2 gas. At this pressure, the CO2 concentration in the drink is given by Henry’s law,
C = 0.031 P_g , where C is the concentration in moles/L and P_g is the partial pressure of the gas
(5 atm).
(a) What is the chemical potential of the CO2 in this system relative to the gas at atmospheric
pressure?
(b) The cap is briefly loosened, so that the gas comes fully into equilibrium with the atmosphere,
but the dissolved CO2 remains in solution. The bottle is then sealed.
What are the chemical potentials of the CO2 in the drink and the gas above?
(c) Finally, the bottle is shaken, such that the contents come into equilibrium. Describe what
happens, and calculate what is the pressure inside the bottle now?
Approximately how many times can this process be repeated before the drink goes flat?
Relevant Equations
Raoult's Law, Dalton's Law, Equations relating to chemical potentials
For part (a), I used this formula

where where the i's represent the substance being used and mu_i^0 represents some reference potential. However, to my knowledge this simply calculates the change in chemical potential from one state to another which is not of much help in finding the relative chemical potential of the gas and the liquid. So I think my answer to (a) is wrong.
For (b) I'm completely lost. I'm not sure if now we take it that the gas is in equilibrium with the atmosphere but is now not in equilibrium with the drink. So then the liquid in the drink and gas in the drink are different temperatures. I'm not sure what new information I can extract from the fact that the gas has come into thermal equilibrium with the atmosphere.
Thanks,

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JohnJ said:
So I think my answer to (a) is wrong.
As you haven't told us your answer to (a), how can we tell?
JohnJ said:
However, to my knowledge this simply calculates the change in chemical potential from one state to another which is not of much help in finding the relative chemical potential of the gas and the liquid.
If the gas and the liquid are in equilibrium , what does that tell you about the chemical potentials in the gas and the liquid?

In part (b), you can certainly take it that the gas is no longer in equilibrium with the liquid, and a new equilibrium has to be established. What I'm not sure of is whether it means simply that the gas pressure is reduced to 1 atm, or whether it implies full mixing with the atmosphere, so that the partial pressure of CO2 is the same as that in the atmosphere (which you are not told, but would need to answer part (b)). The latter is how I would understand "fully in equilibrium", but seems inconsistent with a brief loosening of the cap and the liquid not re-equilibrating. It would take time.

JohnJ
mjc123 said:
As you haven't told us your answer to (a), how can we tell?

If the gas and the liquid are in equilibrium , what does that tell you about the chemical potentials in the gas and the liquid?

In part (b), you can certainly take it that the gas is no longer in equilibrium with the liquid, and a new equilibrium has to be established. What I'm not sure of is whether it means simply that the gas pressure is reduced to 1 atm, or whether it implies full mixing with the atmosphere, so that the partial pressure of CO2 is the same as that in the atmosphere (which you are not told, but would need to answer part (b)). The latter is how I would understand "fully in equilibrium", but seems inconsistent with a brief loosening of the cap and the liquid not re-equilibrating. It would take time.
For part (a) are you suggesting that the answer is then zero, as the chemical potentials are the same due to equilibrium? I can't say that this is incorrect but it's not something that I've heard before.
For (b) I still don't know how I would handle given the information. If we say that the partial pressure of the CO2 is now 1 atm what formula would then be useable to calculate the chemical potential?

(a) Yes. At equilibrium the chemical potentials of CO2 in the gas and solution are the same. Now you can answer the question. (You had the right equation, just do the calculation.)
(b) The same formula. (I don't think you can, or are expected to, calculate absolute chemical potentials - calculate them, as in (a), relative to CO2(g) at 1 atm.)

1. What is the definition of chemical potential in thermodynamics?

The chemical potential is a thermodynamic quantity that represents the energy required to add one mole of a substance to a system at constant temperature and pressure. It is a measure of the tendency of a substance to move from one phase to another, and is related to the concentration of the substance in a system.

2. How is chemical potential related to Gibbs free energy?

The chemical potential is directly related to the Gibbs free energy through the equation μ = ∂G/∂n, where μ is the chemical potential, G is the Gibbs free energy, and n is the number of moles of the substance. This relationship allows for the determination of the chemical potential from known values of Gibbs free energy.

3. What is the significance of chemical potential in phase equilibria?

In phase equilibria, the chemical potential plays a crucial role in determining the stability and composition of a system. When two phases are in equilibrium, their chemical potentials must be equal. This allows for the calculation of equilibrium constants and the prediction of phase changes.

4. How does temperature affect chemical potential?

The chemical potential is directly proportional to temperature, meaning that as temperature increases, so does the chemical potential. This is because temperature affects the energy of a substance, and the chemical potential represents the energy required to add a substance to a system.

5. What is the relationship between chemical potential and entropy?

Chemical potential and entropy are inversely related, meaning that as one increases, the other decreases. This is due to the fact that chemical potential represents the energy required to add a substance to a system, while entropy represents the disorder or randomness of a system. As disorder increases, the energy required to add a substance decreases, leading to a decrease in chemical potential.

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