This is probably one of the two or three questions I get asked most often for this course: How can one use C_V in equations that compute some quantity for a process where volume does not remain constant? I usually answer that one should recall that C_V is just a number (possibly varying as a function of temperature). Thus, when considering paths B+C in video 5.5, it is "obvious" why the heat change for path C involves C_V - the volume does remain constant along path C. Its appearance in the integral for work along path B, by contrast, is not because there is some deep connection involving changes at constant volume, but instead that we need the work integral to be the opposite of the heat integral in order to satisfy the net internal energy change for B+C having to be equal to zero, i.e., we need the "number" (because an integral is, after all, just a number) to be the same. In many places in this course, we see processes where C_V appears in the expression for some state function change that, again, is NOT associated with a constant volume process. However, in many instances one can think of its appearance as being associated with the idea of: IF one were to follow a path that FIRST varied something other than volume, and THEN varied the volume to get to the final state point, the PORTION of the TOTAL expression for the state function change involving C_V would be associated with that FIRST part of the path. As the equation IS for a state function, it holds INDEPENDENT of path -- so we can indeed evaluate that "special" route if we find it computationally convenient, and the "numbers" will be universal for ALL paths with the same starting and ending points, i.e., C_V will appear. And, just as a final note, let's say that for some REAL gas, C_V was actually a function of V. Even in that case, one could certainly include C_V in an integral over a volume change -- it is simply that it could not be removed from the integral as a constant. Instead, it would need to be integrated (perhaps through representation as a polynomial in V) along with any other functions of V when evaluating the integral.
During adiabatic steps, why work expression contains heat capacity at constant volume although volume is changing?
This is probably one of the two or three questions I get asked most often for this course: How can one use C_V in equations that compute some quantity for a process where volume does not remain constant? I usually answer that one should recall that C_V is just a number (possibly varying as a function of temperature). Thus, when considering paths B+C in video 5.5, it is "obvious" why the heat change for path C involves C_V - the volume does remain constant along path C. Its appearance in the integral for work along path B, by contrast, is not because there is some deep connection involving changes at constant volume, but instead that we need the work integral to be the opposite of the heat integral in order to satisfy the net internal energy change for B+C having to be equal to zero, i.e., we need the "number" (because an integral is, after all, just a number) to be the same.
In many places in this course, we see processes where C_V appears in the expression for some state function change that, again, is NOT associated with a constant volume process. However, in many instances one can think of its appearance as being associated with the idea of: IF one were to follow a path that FIRST varied something other than volume, and THEN varied the volume to get to the final state point, the PORTION of the TOTAL expression for the state function change involving C_V would be associated with that FIRST part of the path. As the equation IS for a state function, it holds INDEPENDENT of path -- so we can indeed evaluate that "special" route if we find it computationally convenient, and the "numbers" will be universal for ALL paths with the same starting and ending points, i.e., C_V will appear.
And, just as a final note, let's say that for some REAL gas, C_V was actually a function of V. Even in that case, one could certainly include C_V in an integral over a volume change -- it is simply that it could not be removed from the integral as a constant. Instead, it would need to be integrated (perhaps through representation as a polynomial in V) along with any other functions of V when evaluating the integral.
@@ChemProfCramer Thanks a lot sir