help substituting one formula into another

First of all apologies if this is in the wrong section, I really wasn't sure where to put it. I'm a physics student in my 2nd year at university and came across the following in a text book:

Note that all derivatives are partial

(1) U=(1/C)*(P - T*(dP/dT))

(2) P*v = RT(1+B/v)

Substitute (2) into (1) gives:

U= (-1/C) * ((RT^2)/v^2) * (dB/dT)

The book just gives the result and misses out all the mathematical steps and when I try it myself I just can't get that result.

from (2) P= ((RT)/v) + ((RTB)/v^2)

dp/dt = R/v + (RB)/v^2

Substitute in gives:

U = (1/c) * [ { ((RT)/v) + ((RTB)/v^2) } - { T*(R/v + (RB)/v^2) } ]

Which just equals zero?

I'm sure I must be doing something obvious wrong, but can't see it.. any help would be appreciated! :)

Re: help substituting one formula into another

Quote:

Originally Posted by

**dexev** First of all apologies if this is in the wrong section, I really wasn't sure where to put it. I'm a physics student in my 2nd year at university and came across the following in a text book:

Note that all derivatives are partial

(1) U=(1/C)*(P - T*(dP/dT))

(2) P*v = RT(1+B/v)

Substitute (2) into (1) gives:

U= (-1/C) * ((RT^2)/v^2) * (dB/dT)

The book just gives the result and misses out all the mathematical steps and when I try it myself I just can't get that result.

from (2) P= ((RT)/v) + ((RTB)/v^2)

dp/dt = R/v + (RB)/v^2

Substitute in gives:

U = (1/c) * [ { ((RT)/v) + ((RTB)/v^2) } - { T*(R/v + (RB)/v^2) } ]

Which just equals zero?

I'm sure I must be doing something obvious wrong, but can't see it.. any help would be appreciated! :)

Here's a rundown for you.

Step by step.

$\displaystyle Pv = RT \left ( 1 + \frac{B}{v} \right ) $

$\displaystyle P = RT \left ( \frac{1}{v} + \frac{B}{v^2} \right ) $

Now, v is an independent variable here, so no T dependence.

$\displaystyle P_T = R \left ( \frac{1}{v} + \frac{B}{v^2} \right ) + \frac{RT}{v^2} \cdot B_T$

Putting this into the U equation:

$\displaystyle U = \frac{1}{C} \cdot \left [ RT \left ( \frac{1}{v} + \frac{B}{v^2} \right ) - T \left ( R \left ( \frac{1}{v} + \frac{B}{v^2} \right ) + \frac{RT}{v^2}B_T \right ) \right ] $

Upon simplifying (Hah!) you get the book's answer.

-Dan