Integral to Log?

[Ashley Barrera 1J]

Hi, in recent lecture we see the internal sign a lot with final volume on top and initial volume on the bottom. I do not remember how to do calculations with the integral sign at all. How does an integral turn into a logarithmic term in our final equation of work=-nRTln(v2/v1)

[Anna_Sang_1K]

For an integral, you are finding the area under a curve. You are subtracting the initial amount from the final amount. When you minus a natural log from another, it can be written as a fraction such as ln7-ln3 = ln7/3.

[Brendan Quan 2J]

For the integral of -PdV, we first rearrange the ideal gas law PV = nRT to P = nRT/V. Because n, R, and T are constant, we can bring it outside the integral and we are left with -nRT times the integral of 1/V dV. The integral of 1/V dV is ln V, but because it is a definite integral, we subtract the upper bound (Vf) from the lower bound (Vi). Combining ln Vf - ln Vi = ln (Vf/Vi), we are left with w = -nRT ln (Vf/Vi).