Re-arranging equation to get P as fcn of mu_p, gamma_half, M fcn of gamma only

[irfantai]

Hi,

So my maths skills aren't the strongest so I was hoping someone could help me with a problem that seems fairly trivial but I can't seem to solve.
So I have the following equation:

. . .\(\displaystyle \large{ Z\, =\, \dfrac{1\, +\, (P\, -\, 1)\, M}{P} }\)

. . . . .\(\displaystyle \large{ =\, \dfrac{\mu_w}{\mu_p}\,+\, \dfrac{1\, -\, \dfrac{\mu_w}{\mu_p}}{1\, +\, \left[\dfrac{\gamma}{\gamma_{\frac{1}{2}}}\right]^{P_{\alpha} - 1}} }\)

. . . . . . .\(\displaystyle \large{ =\, \dfrac{\gamma_{\frac{1}{2}}^{P_{\alpha} - 1}\, +\, \dfrac{\mu_w}{\mu_p}\, \gamma^{P_{\alpha} - 1}}{\gamma_{\frac{1}{2}}^{P_{\alpha} - 1}\, +\, \gamma^{P_{\alpha} - 1}} }\)


I need to find P and M such that P is a function of mu_p and gamma_half, and M is only a function of gamma.

. . .\(\displaystyle \large{ P\, =\, P(\mu_p,\, \gamma_{\frac{1}{2}}) }\)

. . .\(\displaystyle \large{ M\, =\, M(\gamma) }\)


mu_w and P_alpha are constants so they can go anywhere. Any help would be greatly appreciated!!

Thanks in advance!

 

[JeffM]

So I have the following equation:

. . .\(\displaystyle \large{ Z\, =\, \dfrac{1\, +\, (P\, -\, 1)\, M}{P} }\)

. . . . .\(\displaystyle \large{ =\, \dfrac{\mu_w}{\mu_p}\,+\, \dfrac{1\, -\, \dfrac{\mu_w}{\mu_p}}{1\, +\, \left[\dfrac{\gamma}{\gamma_{\frac{1}{2}}}\right]^{P_{\alpha} - 1}} }\)

. . . . . . .\(\displaystyle \large{ =\, \dfrac{\gamma_{\frac{1}{2}}^{P_{\alpha} - 1}\, +\, \dfrac{\mu_w}{\mu_p}\, \gamma^{P_{\alpha} - 1}}{\gamma_{\frac{1}{2}}^{P_{\alpha} - 1}\, +\, \gamma^{P_{\alpha} - 1}} }\)


I need to find P and M such that P is a function of mu_p and gamma_half, and M is only a function of gamma.

. . .\(\displaystyle \large{ P\, =\, P(\mu_p,\, \gamma_{\frac{1}{2}}) }\)

. . .\(\displaystyle \large{ M\, =\, M(\gamma) }\)


mu_w and P_alpha are constants so they can go anywhere. Any help would be greatly appreciated!!

What you have given is not a problem, but rather a bunch of statements. I am not at all sure that you have provided sufficient information to find functions for P and M. Given values for all the variables and constants on the right hand sides of your equations, you get a single value that equals an expression involving both P and M. There are infinite possibilities for P and M. What is the exact and complete wording of the question?

 

[irfantai]

What you have given is not a problem, but rather a bunch of statements. I am not at all sure that you have provided sufficient information to find functions for P and M. Given values for all the variables and constants on the right hand sides of your equations, you get a single value that equals an expression involving both P and M. There are infinite possibilities for P and M. What is the exact and complete wording of the question?

Sorry, I was struggling to word the question! But essentially I want to rearrange the equation on the right to resemble the equation on the left with M and P's.
So mu_p and gamma_half are dependent on concentration so I want to couple the two so that I have one function which is concentration Vs P. I also have gamma Vs Z but M needs to be a function of gamma such that a plot of M vs gamma would be the same for any given concentration. I'm not sure if that helps or just makes it worse :/

 

[irfantai]

What you have given is not a problem, but rather a bunch of statements. I am not at all sure that you have provided sufficient information to find functions for P and M. Given values for all the variables and constants on the right hand sides of your equations, you get a single value that equals an expression involving both P and M. There are infinite possibilities for P and M. What is the exact and complete wording of the question?

Also, I don't know if this really helps but, if I allow M to be a function of gamma and gamma_half, then M = 1/(1+(gamma/gamma_half)^(Pa-1)) and P = mu_p/mu_w, but I need M to only depend on gamma, not gamma_half?

 

[Denis]

So I have the following equation:

. . .\(\displaystyle \large{ Z\, =\, \dfrac{1\, +\, (P\, -\, 1)\, M}{P} }\)

Let's go slow...to see where you're at:

can you solve that equation for P?

 

[JeffM]

Sorry, I was struggling to word the question! But essentially I want to rearrange the equation on the right to resemble the equation on the left with M and P's.
So mu_p and gamma_half are dependent on concentration so I want to couple the two so that I have one function which is concentration Vs P. I also have gamma Vs Z but M needs to be a function of gamma such that a plot of M vs gamma would be the same for any given concentration. I'm not sure if that helps or just makes it worse :/

I am gathering that this is not a homework problem.

You don't even have a proper equation.

Z = expression in M and P = first expression in other variables = second expression in the same variables as the previous expression is THREE equations so when you say the right side of the equation, which equation are you talking about?

Is Z a dependent variable or an independent variable? What are V and s?

Moreover the notation is hard to work with, and I won't even try to do so. First I am going to use minuscule m, p, and z instead of majuscule M, P, and Z. I am also going to use

\(\displaystyle x = 1 - P_{\alpha}, q = \mu_p,\ r = \mu_w,\ c = \gamma_{1/2},\ \text { and } d = \gamma.\)

\(\displaystyle z = \dfrac{1 + (1 - p)m}{p} \implies p \ne 0,\ m = \dfrac{1 - pz}{p - 1} \text { if } p \ne 1, \text { and}\)

\(\displaystyle z = 1 \text { if } p = 1 \text { and then } m \text { is indeterminate.}\)

\(\displaystyle z = \dfrac{r}{q} + \dfrac{1 - \dfrac{r}{q}}{1 + \dfrac{d^x}{c^x}} =\)

\(\displaystyle \dfrac{r}{q} + \dfrac{\dfrac{q - r}{q}}{\dfrac{c^x + d^x}{c^x}} = \dfrac{r}{q} + \dfrac{qc^c - rc^x}{q(c^x + d^x)} = \)

\(\displaystyle \dfrac{rc^x + rd_x + qc^x - rc^x}{q(c^x + d_x)} = \dfrac{qc^x + rd^x}{q(c^x + d^x)}.\)

That at least is a little easier to work with. And

\(\displaystyle z = {c^x + \dfrac{r}{q} * d^x}{c^x + d^x} * \dfrac{q}{q} = \dfrac{qc^x + rd_x}{q(c^x + d_x)}.\)

As far as I can see so far, there is no way to get functions for m and p out of that mathematically. Maybe I am missing something, but why do you believe that such functions exist?

 

[irfantai]

I am gathering that this is not a homework problem.

You don't even have a proper equation.

Z = expression in M and P = first expression in other variables = second expression in the same variables as the previous expression is THREE equations so when you say the right side of the equation, which equation are you talking about?

[When I say the right hand side, I mean either of the equations containing \(\displaystyle \gamma.\), the first expression containing \(\displaystyle \gamma.\) is how the equation was given to me, and the second is rearranged, I thought it might be easier to understand in that form.

Is Z a dependent variable or an independent variable? What are V and s?

[Z is a dependent variable]

Moreover the notation is hard to work with, and I won't even try to do so. First I am going to use minuscule m, p, and z instead of majuscule M, P, and Z. I am also going to use

\(\displaystyle x = 1 - P_{\alpha}, q = \mu_p,\ r = \mu_w,\ c = \gamma_{1/2},\ \text { and } d = \gamma.\)

\(\displaystyle z = \dfrac{1 + (1 - p)m}{p} \implies p \ne 0,\ m = \dfrac{1 - pz}{p - 1} \text { if } p \ne 1, \text { and}\)

\(\displaystyle z = 1 \text { if } p = 1 \text { and then } m \text { is indeterminate.}\)

\(\displaystyle z = \dfrac{r}{q} + \dfrac{1 - \dfrac{r}{q}}{1 + \dfrac{d^x}{c^x}} =\)

\(\displaystyle \dfrac{r}{q} + \dfrac{\dfrac{q - r}{q}}{\dfrac{c^x + d^x}{c^x}} = \dfrac{r}{q} + \dfrac{qc^c - rc^x}{q(c^x + d^x)} = \)

\(\displaystyle \dfrac{rc^x + rd_x + qc^x - rc^x}{q(c^x + d_x)} = \dfrac{qc^x + rd^x}{q(c^x + d^x)}.\)

That at least is a little easier to work with. And

\(\displaystyle z = {c^x + \dfrac{r}{q} * d^x}{c^x + d^x} * \dfrac{q}{q} = \dfrac{qc^x + rd_x}{q(c^x + d_x)}.\)

As far as I can see so far, there is no way to get functions for m and p out of that mathematically. Maybe I am missing something, but why do you believe that such functions exist?

Thank you

Indeed it isn't a homework problem, it's for a piece of research and I need to input data for Z in the form of the equation with the M and P, but I just realised, Z is dimensionless and M and P also need to be dimensionless and the only way to make M dimensionless is if \(\displaystyle \gamma.\) is divided by \(\displaystyle \gamma_{1/2}\), which leads me to thing that the software requires tweaking.

But thank you all for your time, it's much appreciated