To supplement what StallionSA wrote above, you do need a second audio device, but it does not have to be a sound card in the physical sense of a card which you insert into a slot on your mainboard.
If you do a search on the web you will find many USB "sound card" devices which could be used to connect your headphones and these range from budget to some quite expensive USB DACs specifically designed for quality headphone listening.
Once you have your USB sound card plugged in (which should in most cases be plug and play to use so long as it conforms to the USB audio conventions) you then would then need to configure
to set the USB device to default ALSA sound card 0, and the other as sound card 1, and then use pavucontrol to direct the application streaming the radio station to the sound card to which the speakers are connected.
It is also possible to do via the environmental variable PULSE_SINK at the time of launching the application, eg in a Bourne shell
PULSE_SINK="alsa_output.usb-Creative_Technology_SB_X-Fi_Surround_5.1_000004it-00-S51.analog-stereo" mplayer "http://mozart.wkar.msu.edu/wkar-fm-mp3" &
The list of available sinks (Pulseaudio term for output device) can be found using
pactl list short sinks