Re: how do I solve for this
- To: mathgroup at smc.vnet.net
- Subject: [mg117629] Re: how do I solve for this
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Tue, 29 Mar 2011 06:49:53 -0500 (EST)
Sure, no problem, other than there being 3 rather complicated branches of
v:
D[v /. Solve[p == n*kb*t/(v - n*b) - a*n^2/v^2, v], t]
{(kb n)/(3 p) + (2^(
1/3) (3 a n^2 p - (-b n p - kb n t)^2) (-9 a kb n^3 p +
6 b^2 kb n^3 p^2 + 12 b kb^2 n^3 p t +
6 kb^3 n^3 t^2 + (2 (-9 a kb n^3 p + 6 b^2 kb n^3 p^2 +
12 b kb^2 n^3 p t + 6 kb^3 n^3 t^2) (18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3) +
24 kb n (-b n p -
kb n t) (3 a n^2 p - (-b n p -
kb n t)^2)^2)/(2 \[Sqrt]((18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p -
kb n t)^2)^3))))/(9 p (18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 +
2 kb^3 n^3 t^3 + \[Sqrt]((18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))^(
4/3)) + (-9 a kb n^3 p + 6 b^2 kb n^3 p^2 + 12 b kb^2 n^3 p t +
6 kb^3 n^3 t^2 + (2 (-9 a kb n^3 p + 6 b^2 kb n^3 p^2 +
12 b kb^2 n^3 p t + 6 kb^3 n^3 t^2) (18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3) +
24 kb n (-b n p -
kb n t) (3 a n^2 p - (-b n p -
kb n t)^2)^2)/(2 \[Sqrt]((18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3)))/(9 2^(1/3)
p (18 a b n^3 p^2 + 2 b^3 n^3 p^3 - 9 a kb n^3 p t +
6 b^2 kb n^3 p^2 t + 6 b kb^2 n^3 p t^2 +
2 kb^3 n^3 t^3 + \[Sqrt]((18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))^(2/3)) - (2 2^(1/3)
kb n (-b n p - kb n t))/(3 p (18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t + 6 b kb^2 n^3 p t^2 +
2 kb^3 n^3 t^3 + \[Sqrt]((18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))^(1/3)), (kb n)/(
3 p) - ((1 +
I Sqrt[3]) (3 a n^2 p - (-b n p - kb n t)^2) (-9 a kb n^3 p +
6 b^2 kb n^3 p^2 + 12 b kb^2 n^3 p t +
6 kb^3 n^3 t^2 + (2 (-9 a kb n^3 p + 6 b^2 kb n^3 p^2 +
12 b kb^2 n^3 p t + 6 kb^3 n^3 t^2) (18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3) +
24 kb n (-b n p -
kb n t) (3 a n^2 p - (-b n p -
kb n t)^2)^2)/(2 \[Sqrt]((18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))))/(9 2^(2/3)
p (18 a b n^3 p^2 + 2 b^3 n^3 p^3 - 9 a kb n^3 p t +
6 b^2 kb n^3 p^2 t + 6 b kb^2 n^3 p t^2 +
2 kb^3 n^3 t^3 + \[Sqrt]((18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))^(
4/3)) - ((1 - I Sqrt[3]) (-9 a kb n^3 p + 6 b^2 kb n^3 p^2 +
12 b kb^2 n^3 p t +
6 kb^3 n^3 t^2 + (2 (-9 a kb n^3 p + 6 b^2 kb n^3 p^2 +
12 b kb^2 n^3 p t + 6 kb^3 n^3 t^2) (18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3) +
24 kb n (-b n p -
kb n t) (3 a n^2 p - (-b n p -
kb n t)^2)^2)/(2 \[Sqrt]((18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))))/(18 2^(1/3)
p (18 a b n^3 p^2 + 2 b^3 n^3 p^3 - 9 a kb n^3 p t +
6 b^2 kb n^3 p^2 t + 6 b kb^2 n^3 p t^2 +
2 kb^3 n^3 t^3 + \[Sqrt]((18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))^(2/3)) + (2^(
1/3) (1 + I Sqrt[3]) kb n (-b n p -
kb n t))/(3 p (18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t + 6 b kb^2 n^3 p t^2 +
2 kb^3 n^3 t^3 + \[Sqrt]((18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))^(1/3)), (kb n)/(
3 p) - ((1 -
I Sqrt[3]) (3 a n^2 p - (-b n p - kb n t)^2) (-9 a kb n^3 p +
6 b^2 kb n^3 p^2 + 12 b kb^2 n^3 p t +
6 kb^3 n^3 t^2 + (2 (-9 a kb n^3 p + 6 b^2 kb n^3 p^2 +
12 b kb^2 n^3 p t + 6 kb^3 n^3 t^2) (18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3) +
24 kb n (-b n p -
kb n t) (3 a n^2 p - (-b n p -
kb n t)^2)^2)/(2 \[Sqrt]((18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))))/(9 2^(2/3)
p (18 a b n^3 p^2 + 2 b^3 n^3 p^3 - 9 a kb n^3 p t +
6 b^2 kb n^3 p^2 t + 6 b kb^2 n^3 p t^2 +
2 kb^3 n^3 t^3 + \[Sqrt]((18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))^(
4/3)) - ((1 + I Sqrt[3]) (-9 a kb n^3 p + 6 b^2 kb n^3 p^2 +
12 b kb^2 n^3 p t +
6 kb^3 n^3 t^2 + (2 (-9 a kb n^3 p + 6 b^2 kb n^3 p^2 +
12 b kb^2 n^3 p t + 6 kb^3 n^3 t^2) (18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3) +
24 kb n (-b n p -
kb n t) (3 a n^2 p - (-b n p -
kb n t)^2)^2)/(2 \[Sqrt]((18 a b n^3 p^2 +
2 b^3 n^3 p^3 - 9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))))/(18 2^(1/3)
p (18 a b n^3 p^2 + 2 b^3 n^3 p^3 - 9 a kb n^3 p t +
6 b^2 kb n^3 p^2 t + 6 b kb^2 n^3 p t^2 +
2 kb^3 n^3 t^3 + \[Sqrt]((18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))^(2/3)) + (2^(
1/3) (1 - I Sqrt[3]) kb n (-b n p -
kb n t))/(3 p (18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t + 6 b kb^2 n^3 p t^2 +
2 kb^3 n^3 t^3 + \[Sqrt]((18 a b n^3 p^2 + 2 b^3 n^3 p^3 -
9 a kb n^3 p t + 6 b^2 kb n^3 p^2 t +
6 b kb^2 n^3 p t^2 + 2 kb^3 n^3 t^3)^2 +
4 (3 a n^2 p - (-b n p - kb n t)^2)^3))^(1/3))}
You might choose a nicer or more specific form using the following (if you
have ranges for the constants):
Reduce[p == n*kb*t/(v - n*b) - a*n^2/v^2, v]
(p == 0 && n == 0 &&
t v != 0) || (p !=
0 && (v ==
Root[-a b n^3 + a n^2 #1 + (-b n p - kb n t) #1^2 + p #1^3 &,
1] || v ==
Root[-a b n^3 + a n^2 #1 + (-b n p - kb n t) #1^2 + p #1^3 &,
2] || v ==
Root[-a b n^3 + a n^2 #1 + (-b n p - kb n t) #1^2 + p #1^3 &,
3]) && b n v - v^2 != 0) || (p == 0 &&
kb n t !=
0 && (v == (a n - Sqrt[a] n Sqrt[a - 4 b kb t])/(2 kb t) ||
v == (a n + Sqrt[a] n Sqrt[a - 4 b kb t])/(2 kb t)) &&
a b n - a v + b kb t v != 0) || (t == 0 && p == 0 && n == 0 &&
v != 0) || (p == 0 && n t != 0 && kb == 0 && a == 0 &&
b n v - v^2 != 0) || (t == 0 && p == 0 && n != 0 && a == 0 &&
b n v - v^2 != 0)
Bobby
On Thu, 24 Mar 2011 06:31:55 -0500, thinktank1985 <dibya at umich.edu> wrote:
> Say I have
>
> p=N*kb*T/(V-N*b)-a*N^2/V^2
>
> I want to evaluate the derivative of V with respect to T, keeping
> p,N,kb,b,a constant.
>
> I understand that this can be done by hand. I just want to know whether
> this can be done by using mathematica. I couldnt understand how to use
> Solve to do this. maybe there is something else I am not aware of
>
--
DrMajorBob at yahoo.com