Thermal Physics Kittel chapter 6 -- Entropy of mixing problem

  • #1
bluepilotg-2_07
1
0
Homework Statement
Suppose that a system of N atoms of type A is placed
in diffusive contact with a system of N atoms of type B at the same temperature
and volume. Show that after diffusive equilibrium is reached the total entropy
is increased by 2N log 2. The entropy increase 2N log 2 is known as the entropy
of mixing. If the atoms are identical (A = B), show that there is no increase in
entropy when diffusive contact is established. The difference in the results has
been called the Gibbs paradox.
Relevant Equations
sigma = log(g), mu = tau*log(N/V*n_Q), sigma = N[log(V*n_Q/N)+5/2]
I've been working on this problem for the past 3 days. I have other papers with different ways of tackling the problem. However, I just cannot get to the answer (change in entropy = 2Nlog(2)).
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  • #2
Welcome to PF.

Your attached picture of your work is very light and hard to read. Can you upload a better image please?

Also, I will send you a message with tips for posting math equations at PF using LaTeX. That is a much better way to show your work in the future here. :smile:
 

FAQ: Thermal Physics Kittel chapter 6 -- Entropy of mixing problem

What is the entropy of mixing in the context of thermal physics?

The entropy of mixing refers to the increase in entropy that occurs when two or more different substances are mixed together. In thermal physics, it quantifies the degree of disorder or randomness that results from the combination of distinct components, such as gases or liquids, and is a key concept in understanding thermodynamic processes.

How is the entropy of mixing calculated for ideal gases?

For ideal gases, the entropy of mixing can be calculated using the formula: ΔS_mixing = -nR Σ (x_i ln x_i), where n is the total number of moles, R is the ideal gas constant, and x_i is the mole fraction of each component in the mixture. This equation reflects the contribution of each component to the overall increase in entropy upon mixing.

What assumptions are made in the entropy of mixing problem?

The main assumptions in the entropy of mixing problem include that the components are ideal gases, the mixing process is isothermal (occurring at constant temperature), and that there are no interactions between the molecules of different species. These assumptions simplify the calculations and allow for the use of the ideal gas law.

Why is the entropy of mixing important in thermodynamics?

The entropy of mixing is important in thermodynamics because it helps to explain spontaneous processes and the direction of chemical reactions. It provides insight into how systems evolve towards equilibrium and the conditions under which mixing occurs, reflecting the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease.

Can the entropy of mixing be negative, and under what conditions?

The entropy of mixing cannot be negative for an ideal mixture of different components. Mixing always results in an increase in entropy due to the increased number of accessible microstates. However, if there are strong interactions or if the components are not ideal, the effective entropy change could be less than expected, but it would not actually be negative in a typical mixing scenario.

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