Question

DIFFERENTIAL EQUATION: The ideal gas law relating pressure ,temperature and volume is given by:

P =(CT)/V where c is a constant.
Find TdP/dT * dV/dT

Answer 1

\[T \frac{ \delta P }{ \delta T } *\frac{ \delta V }{ \delta T }\]

Answer 2

A hint: once write P in terms of T and then differentiate, another time write V in term of T and then differentiate.

Answer 3

P in terms of T:\[P= \frac{ CT }{ V}\] and \[\frac{ \delta P }{ \delta T } = C/V ?\]

Answer 4

first differentiate P w.r.t T assuming V constant after that differentiate V w.r.t. T as P constant.
then put in the equation.

Answer 5

@MERTICH , I'm not sure, but you might be interested in using the \(\LaTeX\) symbol \(\partial\) (\partial). That's what I use, but maybe you meant to use \(\delta\)! I just know that \(\LaTeX\) isn't common knowledge.

Answer 6

@MERTICH Looks good so far!

Answer 7

Take care, all.

Answer 8

thanks, let me take care of the other one

Answer 9

Yes, you did OK. try to do the second one.

Answer 10

\[V = \frac{ CT }{ P }, \frac{ \delta V }{ \delta T } = \frac{ C }{ P }\]

Answer 11

Yep, you have done all OK.

Answer 12

thank you @Saeeddiscover , now, do i just need to multiply them?

Answer 13

there is a dot . between them

Answer 14

Yes, it is what you need to do.

Answer 15

thank you!

Answer 16

No problem!

Answer 17

so what happpens with the T ,\[T \frac{ \delta P }{ \delta T} *\frac{ \delta V }{ \delta T}\], WHEN I MULTIPLY, is that T at first included?

Answer 18

Hi! I notice that you haven't quite finished! You do multiply.

If you substitute in the partial derivatives that you found, you're left with
\(T\left(\dfrac C V\right)\dot\ \left(\dfrac C P \right)\)

This can be simplified, so you should.
\(\dfrac {T\ C} V\dot\ \dfrac C P=\dfrac {C\ T} V\dot\ \dfrac C P\)

That was a very slight algebraic move there.. I know you're not on but would like to think about this if you were here.
Does that
\(\dfrac{CT} V\)
look familiar?


\(\\\downarrow\\\downarrow\\\downarrow\)


It is \(P\)! Look at the pressure equation from the beginning.

So we have \(P\ \dot\ \dfrac C P\)

That is,
\(T \dfrac{ \delta P }{ \delta T} *\dfrac{ \delta V }{ \delta T}=P\ \dot\ \dfrac C P=C\)

And then \(that\) would be your simplified solution.