The page linked to above is pretty good but it doesn't account for the LED in the circuit. What the OP has is close to the constant resistance discharge but with an added drop (similar in effect to the internal resistance curve at the bottom of the page). You can use the basic RC (only) equation from Wikipedia and modify it :
The tau0 above is the time constant which is the value of R (ohms) times the value of C (farads). t is the elapsed time since the discharge started, in seconds. V
0 is the starting value of the cap's voltage but for use in the current equation you want to subtract out the drop across the LED. Let's be simple and use the V
f for that LED and assume it's constant despite the varying current that will be flowing through it. So where you see V
0, put in V
0 - V
f. If you put the equations into a spreadsheet you'll be able to play with them to find a set that (potentially) works.
There are practical constraints on what you can use though. A cap will have a voltage rating, the max voltage you can charge it up to. Large caps, like the type you need, will have low voltage ratings. That in turn means you need an ever larger cap which in turn means a lower voltage, so on and so forth. Also (as noted on the above page) caps have an internal leakage path so the voltage will decrease over time even when not supplying current to the "timer circuit". Also LEDs have a max continuous current that you don't want to exceed right at the start, when the cap's voltage is at it's max. And lastly there's some min current through the LED such that it's still visibly on.
Can you find a cap and resistor and charge it to some voltage such that 10 minutes later the LED is still on ? Perhaps but it's not a sure thing. One minute is easy I think. 10 ... that might be hard. Just for fun I put the equations into a spreadsheet and found that a super cap = 5F, charged to V
0 = 5V and discharging through a 160 ohm resistor and a typical red LED (V
f = 1.8V) draws 20 mA initially (which should be OK) and should still have 10 mA flowing through at the 10 min mark. Is 10 mA still visible ? Perhaps. I didn't take into account any self discharge of the cap (which surely will exist) and be aware that a cap like that has a pretty big tolerance, a big size and a big price. Search DigiKey and see what's available.
A plot of the above results.
(click on to enlarge)
TenMinRC.jpg
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