The time constant(sec) of an RC circuit is equal to the product of the resistance and the capacitance of the circuit.
It is the time required to charge the capacitor through the resistor to 63. 2% of full charge,or to discharge it to 36.8% of its initial voltage.
The voltage of the RC circuit is measured using adc of the ATmega8, input voltage for RC circuit is given from PB0. The timer is started at the time of the PB0 making 1 .
The adc of ATmega8(ADCH) is 8 bit long so corresponding to 5V get 255 in ADCH. The TCNT1 value is taken to a variable when the output voltage of the RC circuit become 63.2% of the input voltage.That is 3.16 v corresponding to these voltage ADCH show 161(appr).
Using an LCD can show the TCNT1 value. TCNT1 is 16 bit long.Here ATmega8 running in 8MHz clock,timer prescaled by 1024.
So if you get the real time multiply the TCNT1 value to (1024/8000000).
Some test examples: