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This makes absolutely no sense to me. My physics book states the following

(University Physics 12th ed, Pg 650) Suppose we want to change the volume of a certain quantity of an ideal gas from 2.0 L to 5.0 L while keeping the temperature constant at $T = 300 K$. Figure 19.8 shows two different ways in which we can do this. In Fig. 19.8a the gas is contained in a cylinder with a piston, with an initial volume of 2.0 L. we let the gas expand slowly, supplying heat from the electric heater to keep the temperature at 300 K. After expanding in this slow, controlled, isothermal manner, the gas reaches its final volume of 5.0 L; it absorbs a definite amount of heat in the process.

I'm not quite sure how the temperature can be kept constant while adding heat to the system. Exactly what is going on in each step of the path from the first state to the final state?

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    $\begingroup$ The gas is expanding while being heated. Because of the expansion it works against the outside pressure and that work would cool it if not for the heating. $\endgroup$
    – hyportnex
    Jul 7, 2015 at 21:34

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The first thing to realize is: Temperature and heat are not synonyms. Young and Freedman cover that I'm pretty sure.

Briefly, temperature is a measure of average internal energy contained by the gas where heat is a measure of the total internal energy contained by the gas, and the capacity of the gas to do work.

Since we are talking about ideal gases, the energy carried by the gas is carried by the kinetic energy of moving gas molecules. If the gas is in a container, some of the molecules collide with the container walls and transfer energy out of the gas and into the container. If one of the walls is attached to a piston, some of that energy could transfer into motion of the piston.

So what happens in a process where we add heat, but keep temperature constant? If we didn't change the number of molecules of gas, and we add the heat slowly so that the temperature - and pressure -remains constant, what parameters can we change to account for the increased energy?

The ideal gas law gives us a clue. Remember it is

$$ PV = nRT$$

and, since we stipulated that this is an isothermal process, the right hand side of the equation is a constant.

So we have

$$P=\dfrac{nRT}{V}$$

Now where is the heat going that we added to the system, if the temperature doesn't change? It's going into the work done to push the piston and the volume of the container will expand.

Remember I said that heat is the capacity of the energy in the gas to do work? So how do we calculate that work?

Work is defined as

$$W=\int PdV$$

If we substitute our second equation into that integral and do the integral (the limits of integration are the final and initial volume of the container), we find that the work done is:

$$W=nRT\ln (\frac{V_{final}}{V_{initial}})$$

So to sum up the answer to your question - How can we add heat to the gas and the temperature remains the same? The energy due to the added heat was used to do the work of expanding the piston.

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