Setting up integration, thermodynamics

[Markel]

I have a general problem with finding the right parts for integrations.

For example, for an Isothermal expansion, both pressure and volume are changing. For some reason we call V=dv and relate P to V by the ideal gas law and integrate to get work:

W= nRT ln (V_F/ V_i)

Now, could we do the same thing, and call work the integral of:

W= $$\int$$Vdp
Which would lead to

W= nRT ln (P_f/ P_i)
If both quantities are changing? How do we know which to set as the differential, and which to relate in terms of the other variable?

or why couldn't I do a double integral? first over the volume, then over the pressure?


Also: Why does the LAtex code show up all messed up when I hit 'preview post'...?thanks

 

[jbsteele]

The way to decide which to set as the differential and which to relate in terms of the other is by considering the nature of the process that you are modeling. For example, in the case of an isothermal expansion, the pressure and volume are changing in a specific way. The pressure is decreasing while the volume increases, so we use the ideal gas law to relate the pressure and volume at each point. This means that the volume is the independent variable, and the pressure is the dependent variable. Therefore, we can set the differential as dV and relate it to dP using the ideal gas law. In your example, it is not possible to do a double integral because a double integral requires two independent variables. In this case, the pressure and volume are not independent variables since they are related by the ideal gas law. As for the LaTeX code, if it is showing up all messed up when you preview your post, you can try enclosing it in triple backticks (e.g. ```) before and after the code. This should help fix the formatting.