Capacitance calculation

For the 555 circuit, look at the applications section of the 555 data sheet. Transpose the free-running oscillator equation to solve for C as a function of Freq.
For the other circuit, I would have to know what the code is doing with AN0, RA2 and RA3? Likely, it is measuring the time it takes to charge/discharge the capacitor to a given threshold as defined by the internal comparator.

I just looked at the data sheet. That is a rather poor sensor. Its capacitance is effected by the measurement frequency, the ambient temperature, and by-the-way, slightly by changes in humidity. Good Luck...

The pinouts of the 555 (5,6,7,8) in figure 1. is definitely wrong.

Allen

Both designs require a microcontroller ; although, one might use an off-the-shelf frequency meter for the 555-based circuit (with appropriate re-numbering of the pins). If you are comfortable with microcontrollers, then there is no advantage to the 555-based design.

Microchip has a application notes on capacitance measurement that include the code. A newer chip could be substituted for the 16C family used in this early note:
http://ww1.microchip.com/downloads/en/AppNotes/00611b.pdf
Here is a note on the theory:
http://ww1.microchip.com/downloads/en/AppNotes/01014a.pdf
A newer note utilizes the charge-time measurement unit (CTMU) included in some chips:
**broken link removed**

John

Edit: It took me awhile to find a link to this series of capacitance measurement chips. I have used the CAV424 several years ago. It might be a simpler solution, if you do not want to use a microcontroller. There is also the CAV414 and CAV444.
http://www.analogmicro.de/english/index.html

Yes the pin numbers for 5678 are the wrong way around, pictorially however not looking at the pin numbers it does look correct.