Thermo question on mean square of concentration

This will give us:- 2<x1><δ\hat{N1} \hat{N}> = - 2<x1><N1><N> + 2<x1><N><N>Using this, we can finally write the equation for the mean square deviation of concentration as:<(δ\hat{x1})^2> = <N^2{}(δ\hat{x1})^2> - 2<x1><N1><N> + <x1^2 (δ\hat{N})^2>To calculate the value of <(δ\hat{x1})^2>, we need to take the expectation value of each term
  • #1
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This problem is from Callen. In the 1st edition it is 15.4-3. I am stuck on the last part of this problem. It also appears, in a slightly different form, in the second edition of Callen as problem 19.3-5.

Homework Statement



15.4-3 Consider a small volume V within a two-component simple system. Let x1= N1/(N1+N2), in which N1 and N2 are the mole numbers within V. Show that

[itex]N^2{}[/itex](δ[itex]\hat{x1}[/itex])^2 = (δ[itex]\hat{N1}[/itex])^2 - 2x1(δ[itex]\hat{N1}[/itex] [itex]\hat{N}[/itex]) + x1^2 (δ[itex]\hat{N}[/itex])^2

and compute the mean square deviation of concentration <(δ[itex]\hat{x1}[/itex])^2>.

Homework Equations



Actually don't know. It is in a section of the book which is related to fluctuations and correlation moments. Nothing looks relevant to me.

The Attempt at a Solution



Done the "show that"..its just taking a derivative. Quite straightforward. But I don't know how to compute the mean square deviation of concentration based on this. Nobody in my peer group knows what to do. Took it to the TA and he was also unsure what to do with it. Just looking for how to get started.
 
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  • #2


Hello,

Thank you for bringing this problem to our attention. I am familiar with the problem you mentioned from Callen, and I can understand why you are stuck on the last part. The last part of this problem involves calculating the mean square deviation of concentration, which is a measure of the spread or variability of the concentration values within the small volume V.

To calculate this, we first need to understand what the terms in the equation represent. The term N^2(δ\hat{x1})^2 represents the square of the deviation of the concentration x1, while (δ\hat{N1})^2 and (δ\hat{N})^2 represent the square of the deviations of the mole numbers N1 and N, respectively. The term 2x1(δ\hat{N1} \hat{N}) represents the covariance between the deviations of N1 and N.

To calculate the mean square deviation of concentration, we need to take the expectation value of the equation given in the problem. This means that we need to take the average value of each term in the equation, which is denoted by < >. In this case, we need to take the average value of the square of the deviation of concentration, which is given by <(δ\hat{x1})^2>.

To calculate this, we can use the equation given in the problem and plug in the expectation values of the terms. This will give us:

<(δ\hat{x1})^2> = <N^2{}(δ\hat{x1})^2> - 2<x1(δ\hat{N1} \hat{N})> + <x1^2 (δ\hat{N})^2>

Now, we need to use the definition of covariance to write the second term on the right-hand side in terms of the variances of N1 and N. The covariance between two variables is given by:

Cov(X,Y) = <XY> - <X><Y>

Using this definition, we can rewrite the second term as:

- 2<x1(δ\hat{N1} \hat{N})> = - 2<x1><δ\hat{N1} \hat{N}> + 2<x1><δ\hat{N}><N>

Now, we can use the definition of the mean value < > to write the terms in the above equation in terms
 

1) What is the mean square of concentration in thermodynamics?

The mean square of concentration in thermodynamics refers to the average value of the concentration squared over a given system. It is a measure of the spread or variability of the concentration values within the system.

2) How is the mean square of concentration calculated?

The mean square of concentration can be calculated by taking the sum of the concentration squared values and dividing it by the total number of data points in the system.

3) What is the significance of the mean square of concentration in thermodynamics?

In thermodynamics, the mean square of concentration is used to determine the extent of mixing or homogeneity within a system. It can also provide information about the thermodynamic properties and behavior of the system.

4) Can the mean square of concentration change over time?

Yes, the mean square of concentration can change over time as the system undergoes changes in temperature, pressure, and composition. It is a dynamic measure that reflects the current state of the system.

5) How does the mean square of concentration relate to entropy?

The mean square of concentration is directly related to entropy, as it is a measure of the disorder or randomness within a system. A higher mean square of concentration indicates a higher entropy, and vice versa.

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