PA38 INSTRUCTOR BECOMES BUSY WITH LNDG ACTIVITIES AND LANDS ON THE WRONG RWY AT FAT.

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LNDG RWY 29R; CLRED TO LAND RWY 29L. I WAS THE INSTRUCTOR; BUT WAS FLYING THE ACFT ON THE APCH AND LNDG. ALL INSTRUCTIONS HAD BEEN TO 'FOLLOW TFC; R TFC FOR RWY 29R.' THE VISIBILITY WAS RPTING 2 SM; THE ARPT AND PRECEDING TFC WERE IN SIGHT RIGHT UP UNTIL THE TURN TO BASE. AT THIS POINT; I WAS GIVEN THE INSTRUCTIONS 'FOLLOW THE CESSNA; CLRED TO LAND RWY 29L.' A FEW MOMENTS LATER; I SIGHTED THE RWY AND HEARD TFC CLRING THE RWY; SO KNEW I WAS SAFE TO LAND. THE PROB AROSE DUE TO THE FACT THAT I WAS GIVEN THE CLRNC TO LAND JUST AS MY WORKLOAD WAS INCREASED AND I WAS PREOCCUPIED WITH LOOKING FOR THE RWY. THE TURN TO BASE AND THEN TO FINAL PUT THE ARPT RIGHT IN THE SUN FROM WHERE I WAS LOOKING AND WITH HAZE/FOG ADDING TO THIS; MADE IT VERY HARD TO SEE. WITH MY MIND ALREADY SET TO LNDG ON THE R RWY; I DIDN'T PICK UP ON THE DIFFERENT CLRNC WHEN CLRED TO LAND. THINGS THAT WOULD HAVE INCREASED THE CHANCES OF THIS HAPPENING: 1) A LOT OF THE TIME WHEN A CLRNC FOR A DIFFERENT RWY FROM ONE PREVIOUSLY ASSIGNED IS ISSUED; THE CTLR WOULD SAY 'CHANGE RWY CLRED TO LAND RWY 29L.' THIS WAS NOT SAID. 2) CLRNC TO LAND IS NORMALLY GIVEN SOONER; ED; DOWNWIND; WHEN WORKLOAD ISN'T AS HIGH. 3) FAMILIARITY WITH THE ARPT; R TFC IS NORMALLY RWY 29R; L TFC NORMALLY RWY 29L. ALTHOUGH I BELIEVE DOING EITHER OF THESE FIRST 2 THINGS WOULD HAVE ALERTED ME TO THE FACT I WAS CLRED TO LAND ON A DIFFERENT RWY; WHICH OCCURS ON A DAILY BASIS WITHOUT ANY PROBS; I; OF COURSE; SHOULD HAVE PAID MORE ATTN TO THE CLRNC ISSUED AND ACTED ACCORDING TO THE INSTRUCTIONS GIVEN. I ALSO BELIEVE THAT THE FACT I WAS FLYING THE AIRPLANE; WHEREAS USUALLY THE STUDENT IS WHILE I'M INSTRUCTING; THIS ADDED TO THE WORKLOAD. FURTHER; ALTHOUGH I HAVE OVER 1200+ HRS; I AM PERFECTLY CAPABLE OF HANDLING THE ADDITIONAL WORKLOAD. THIS WAS PROBABLY WHAT CAUSED ME TO NOT PAY THE ATTN REQUIRED TO THE CTLR. I ALSO HAD '29R' WRITTEN DOWN FROM THE PREVIOUS CLRNC AND I HADN'T CHANGED THIS TO RWY 29L; AS I WAS TOO BUSY FLYING THE APCH. I HAVE LEARNED FROM THIS -- ESPECIALLY IN SITS WHERE WORKLOAD IS INCREASED -- TO STILL PAY ATTN TO DETAILS WHICH CAN OFTEN GET OVERLOOKED WHEN YOUR MIND IS PREOCCUPIED. ALTHOUGH THIS HAS NEVER HAPPENED BEFORE; AND I DIDN'T THINK I WOULD DO SOMETHING LIKE THIS AFTER 1200 HRS; I AM SURE I WILL BE PAYING MUCH MORE ATTN TO THINGS LIKE THIS IN THE FUTURE.