Here is how I'd do it (I skip the part when you parse the weight table and prompt the user):

#!/usr/bin/perl
use strict;
use warnings;

# whatever you need to fill %weights
...

sub mol_form {
    my $formula = shift;
    my $mass;
    for ($formula =~ /[A-Za-z]\d*/g) {
        /(?<element>[A-Za-z])(?<amount>\d+)?/;
        $mass += ($+{amount} // 1) * $weights{$+{element}};
    }
    return $mass;
}

say mol_form "H2O";
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The most tricky part is to separate each group of element+optionalquantity, and then to process them. That's why I ended up with two regexes, not just one.