# Le Chatelier and Le Chatelier Braun

• Telemachus
In summary, the fluctuation on the entropy has an effect on temperature and pressure. The response attenuates the fluctuation, but it doesn't seem to oppose the change.
Telemachus
Hi there. I'm trying to solve this problem, which says:

A system is in equilibrium with its envornment at a common temperature and a common pressure. The entropy of the system is increased slightly (by a fluctation in which heat flows into the system, or by the purposeful injection of heat into the system). Explain the implication of both the Le Chatelier and the Le Chatelier-Braun principles to the ensuing processes, proving your assertions in detail.

I know form the book that if I have a spontaneous fluctuation $$dX_1^f$$ in my composite system then this fluctuation will be accompanied by a change in intensive parameter $$P_1$$ of the subsystem, and then I have this equation:

$$dP_1^f=\frac{\partial P_1}{\partial X_1}dX_1$$

And that the fluctuation $$dX_1^f$$ also alters intensive parameter $$P_2$$

$$dP_2^f=\frac{\partial P_2}{\partial X_1}dX_1$$

This is how the book explains it, then it makes a reasoning based on the responses. The thing is I don't know how to relate this with the specific problem I'm dealing with, I don't know how to use the fluctuation on the entropy, and I don't know which kind of responses that will produce.

Would you help me?

May I have in consideration that $$S(U,V,N)$$?

Ok. Some time have passed, anyway, I'm facing this problem again.

This is my approach now. At first, about the notation: (r) corresponds to response, (f) to fluctuation, and (res) is for reservoir.
I've considered that the fluctuation on the entropy has a primary effect on temperature

$$dT^f=\left ( \frac{\partial T}{\partial s} \right )_v ds^f=\frac{1}{c_v}dQ_1^f$$

And then that there is also consequently an effect over the pressure:

$$dP^f=\frac{\partial P}{\partial s}ds^f=\frac{\alpha}{c_v k_t}dQ^f$$

Then for the response:
$$d(U+U^{res})=(T-T^{res})ds^r+(P-P^{res})dv^r \leq 0$$
$$dT^fds^r+dP^fdv^r\leq 0$$

Because the two members at the right are independent this leads to:
$$dT^fds^r=\frac{\partial T}{\partial s}ds^fds^r \leq 0$$
Because of the convexity criterion this can be written as:

$$\frac{\partial T}{\partial s}ds^f \frac{\partial T}{\partial s} ds^r=dT^fdT^r \leq 0$$

And
$$dP^fdv^r=\left ( \frac{\partial P}{\partial s} \right )_v ds^f dv^r \leq 0$$

Now, I've used the Maxwell relations on this, and here is the problem I think.
$$\left ( \frac{\partial P}{\partial s}\right )_v=-\left ( \frac{\partial T}{\partial v}\right )_s$$
The minus sign there is what bothers me. Because the response should attenuate the fluctuation, and written like that I get to:
$$\left ( \frac{\partial T}{\partial s} \right )_v ds^f \left(-\left ( \frac{\partial T}{\partial v}\right )_s dv^r \right ) \leq 0$$
So, it doesn't seems to be opposing to the change.

Perhaps I'm missing something here, or perhaps there is something wrong about the whole thing. Any suggestion or help will be thanked.

PD: By the way, I think this should go on advanced Physics. Please, move it if you can.

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## 1. What is the Le Chatelier principle?

The Le Chatelier principle is a concept in chemistry that states that when a system at equilibrium is subjected to a change, it will adjust in order to counteract that change and reach a new equilibrium.

## 2. How does the Le Chatelier principle apply to chemical reactions?

The Le Chatelier principle applies to chemical reactions by predicting how changing the concentration, pressure, or temperature of the reactants or products will affect the equilibrium of the reaction. It helps scientists understand how a system will respond to changes in its environment.

## 3. What is the difference between Le Chatelier and Le Chatelier-Braun?

Le Chatelier and Le Chatelier-Braun are both names for the same principle. Le Chatelier was the initial discoverer of the principle, while Le Chatelier-Braun refers to the addition of the chemist Carl Braun's name, who expanded on the principle and applied it to electrochemical systems.

## 4. How can the Le Chatelier principle be used to control industrial processes?

The Le Chatelier principle can be used to control industrial processes by manipulating the conditions in which the reaction takes place. This can help to maximize the production of desired products and minimize the production of unwanted byproducts.

## 5. What are some limitations of the Le Chatelier principle?

One limitation of the Le Chatelier principle is that it only applies to reversible reactions. It also cannot predict the extent of the equilibrium shift, only the direction. Additionally, it does not take into account the kinetics of a reaction, only the thermodynamics.

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