chmod a freshly mounted external drive to set up writing access

Try this first,

umount /dev/sdb1

chmod -R 0777 /mnt/external   

then mount with

mount /dev/sdb1 /mnt/external

or try

chmod -R 0777 /mnt/external

chmod -R 777 /mnt/external

No need to specify the device. You chmod the directory recursively.

However, usually external drives are formatted with FAT32 or some sort of Windows-compatible file system, which does not have POSIX / UNIX permissions. So this step may be redundant.

How is your drive formatted?

Have you tried writing to it as a non-root user?