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What load current will you be trying to control with the transistor ?
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“ An IR led with 1.3V forward voltage and 100mA forward current”
What is the LED supply voltage ?
Assume the load current is 100mA.
This will be the saturation current Ic(sat)
Ib(sat) should be 10% Ic(sat) = .1 - 10% = 10mA
Rb = (Vcc - .6) / Ib(sat) = (5v - .6v) / .01A = 440 ohms
Assume V(supply led) = 5V
Rc = (V(supply led) - V(led) - Vce(sat)) / Ic(sat)
Rc = (5v - 1.3v - .5) / 100mA
Rc = 3.2V / .1A = 32 ohms
Rc(W) = V(Rc) * Ic(sat) = 3.2V * .1A = 320 mW use 1/2W
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Ic(sat) is the LED current you want to flow through the LED when the transistor is turned ON.
Vce(sat) is from the data sheet: VCE(sat) Collector-Emitter Saturation Voltage . . . .5v
NOTE:
You can confirm Vce(sat) by measuring Vce when the transistor is turned ON (use a DVM)
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32 vs 39 here nor there 
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R2 (Rb) is 440 Ω as per post #3. Use 390 or 430 or 470 Ω
R1 (Rc) is 32 Ω (39) as per post #4.
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The 10% ensures the transistor is in full saturation for switching applications (transistor is in saturation mode).
Hfe or beta will have no effect, which is used when designing for linear operation.
Ib(sat) is the base current necessary to achieve the the Ic(sat) current of 100mA
When the GPIO pin goes HIGH, that i/o pin is at Vcc, 5V.
The Vbe (voltage from the base to emitter) of a silicon transistor is .6V
The voltage across the base resistor Rb is Vcc - Vbe = 5v - .6v = 4.4v
“ Ahh, I missed post #3... ” :o
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Typically Vbe for silicon is .6v to .7v (for 99% of the time, we use .6v, but you can always confirm what ‘your’ transistor Vbe is with a DVM).
Vbe ‘maximum’ is a value that the transistor ‘should not’ be run at (when base currents are really large and could damage the device).
Saturating a transistor is of course not so good if you're switching high frequencies,
like 38kHz remote control signals.
Could be better to use a 2-transistor current source.
Leo..
Better, use a logic MOSFET.
At high frequencies, there should be a base to ground discharge resistor too.
Every IR remote control I ever opened (thousands of them) use normal NPN transistors.
Basic diagram here.
Leo..
Yes, must point out that there is a Vce drop that raises the power expended in the transistor.
The cautionary note to the schematic is new people sometimes take a cct. as applicable to all situations.