# Setting up integration, thermodynamics

• Markel
In summary, when dealing with integrations in thermodynamics, it is important to consider the nature of the process and choose the appropriate independent and dependent variables to set the differential and relate the two quantities. Additionally, using triple backticks can help fix formatting issues with LaTeX code.
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 (VF/ Vi)

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

W= $$\int$$Vdp

W= nRT ln (Pf/ Pi)
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

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.

## 1. What is integration in thermodynamics?

Integration in thermodynamics refers to the mathematical process of finding the total change in a system by integrating over a range of variables. In thermodynamics, this is typically done to find the total internal energy, entropy, or work done on a system.

## 2. Why is integration important in thermodynamics?

Integration is important in thermodynamics because it allows us to find the total change in a system without having to know the exact details of every infinitesimal change. This makes it a powerful tool for analyzing complex systems and understanding their overall behavior.

## 3. How do you set up an integration in thermodynamics?

To set up an integration in thermodynamics, you first need to identify the variables that are changing in the system. Then, you need to set up the appropriate integral, depending on the specific quantity you are trying to find. Finally, you need to determine the limits of integration, which correspond to the initial and final states of the system.

## 4. What are some common applications of integration in thermodynamics?

Integration is used in a variety of applications in thermodynamics. Some common examples include calculating the heat transfer in a heat engine, determining the change in entropy during a phase transition, and finding the total work done in a thermodynamic cycle.

## 5. How does integration relate to the laws of thermodynamics?

Integration is a fundamental tool in thermodynamics and is closely related to the laws of thermodynamics. The first law, which states that energy cannot be created or destroyed, is used to set up integrals to calculate the total energy change in a system. The second law, which deals with the direction of heat flow, can also be expressed as an integral. Finally, the third law, which deals with the behavior of systems at absolute zero, can also be analyzed using integration.

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