5GHz antenna with 2.4GHz Wi-Fi adapter
5GHz antenna with 2.4GHz Wi-Fi adapter
Your existing Wi-Fi adapter is only on the 2.4GHz band, but it has a detachable antenna. If you replace it with one that supports 5GHz, the faster speeds should work as long as your PC is compatible with the 5GHz frequency.
You'll need a Wi-Fi adapter for 5GHz support. The antenna is irrelevant if the chip on the card can't utilize it.
The adapter isn't compatible with 5GHz Wi-Fi, meaning you can't achieve that unless you replace it with a compatible model. Your router must support 5GHz for this to work. Antennas don't change the functionality.