How do I start from a stock solution to prepare an aqueous solution of #"HNO"_3# that is #5"% w/w"# if the starting concentration is unknown and presented in a range?

It's hard to say. If you have access to a UV-Vis spectrometer, that would be great... Then you could do this with relative ease.

It's strange that you have an unknown concentration that is given in a range, since the stock chemists are obligated to be accurate, but OK, since it's nitric acid... that's fine.

Here's a UV for it in the sodium nitrate form:

Now, the only way I know of that you can figure out your concentration is #5% "w/w"# is to measure the absorbance of the solution you make until your calculations give you the molar concentration you expect.

#0.05xx"18.015 g" = "0.90075 g"#

That would be #5%# of the mass of #"1 mol"# of water. Now, just convert this to #"mol/L"#:

#["0.90075" cancel"g"xx("1 mol HNO"_3)/("63.0119" cancel"g")]/("18.015" cancel"g" xx ("1" cancel"mL")/("0.9970749" cancel"g" 25^o"C H"_2"O") xx "1 L"/("1000" cancel"mL")#

#= color(blue)("0.7912 M HNO"_3)#

at #5% "w/w"#.

The upper limit of a UV-Vis spectrometer is about #"1 M"#. All that really matters is that you figure out the original concentration, right? Then just make sure that you dilute to below #"1 M"# to keep the UV-Vis... usable.

You can make any magnitude of dilution you want, so long as it makes the solution low enough in concentration that you get an accurate reading on the UV-Vis. Just know what the magnitude of dilution is. You can work backwards here.

Suppose the dilution you did was a 1 in 20 dilution. Then suppose you got an absorbance that corresponds to #"0.4 M"#. Let's say your final volume right after making the solution was #"100 mL"#. Reporting it back into a #% "w/w"#, that would mean that you have diluted it to:

#["0.1" cancel"L" xx ("0.4" cancel("mol HNO"_3))/cancel"L" xx "63.0119 g"/cancel("mol HNO"_3)]/["100" cancel"mL" xx ("0.9970749 g 2"5^o"C H"_2"O")/("1" cancel"mL")] xx 100%#

#= 2.53% "w/w"#

If this happens, that means you diluted it too much and you have to start over. You probably should dilute it so that you get near #5% "w/w"#. Get an absorbance corresponding to near #"0.79 M"#. That's when you know you are near enough to #5% "w/w"#.

Even then, let's suppose you really did do a 1-in-20 dilution and you got that #"0.4 M"# solution. Then, you would know that the original concentration is about 20 times what you got, which is:

#2.53% xx 20 = 50.6%#