Normally we don't answer homework questions directly, but in case I was obscure in my "hint", here is my take on it. I don't want to limit my reply to just about the fridge. Since you have the first law and work in your question, I want to generalise the fridge to any thermodynamic system involving work, heat and efficiency.
So excuse me if I seem to getting into silly minor details, I am taking a chance that this will make sense to you.
When you buy the fridge, it is obviously frost free and the system works at peak efficiency to remove heat. It does this because the cooling fins and pipes are directly in contact with the air in the fridge, and every time you open the door, warmer air from your kitchen drifts in, so the fridge has to remove that heat every time.
As the frost builds up, the warm air from the kitchen that enters your fridge is no longer in direct contact with the cooling fin/panels, so the efficiency of the fridge drops and it needs to run longer (and use more power) to cool down the air, as the ice the air is in contact with is not as cold as the fins are, (although it does seem that way when you stick your hand in, to take out cold food).
The reason I said look at the definition of power is because it is energy over time, so in the long run, yes it is energy saving to defrost regularly. Over time less work by the system is required to obtain the highest efficiency.
Now take the short term, yes it is also energy saving to watch tv instead and not bother to defrost the fridge. But then the icebox fills with ice, and eventually you have to clear it out. I included you in the system as having to do the mechanical work to remove the ice (I appreciate in terms of a fridge you can turn it off and the ice will melt, but you may still have to do work in removing the water.)
As I say, I really wanted to generalise the fridge to be any system, where you have to do work. Real work, not just pulling the plug out and putting it back in. Say the plug was hidden somewhere behind the fridge (as mine is) and you had to haul the fridge around to switch it off, plus you do work in taking all the food out to someone's fridge (and putting it all back) and then mop up the defrosted ice that ends up on the floor.
If I were to defrost my freezer would I save energy?
From what I have read I would expect the answer to this question to be no, however it would be great to hear if Im wrong.
So in the short term, you would save energy, because you would not have to go to all the bother of defrosting (and all the work that could involve).
In the long run though, if you did not defrost it, it would become less and less efficient and need more energy to run it, because the cooling fins are insulated by the ice and so not as efficient.
So that why I said there is more than one answer, short term yes, energy savings but long term no, energy is wasted.
I am sorry if you just wanted to know simply about the fridge, and not any more than that :)
But I thought it better that you think about all similiar systems involving energy, work and efficiency, as this is the stuff that you need to think about for more complicated system, where the laws of thermodynamics need to include everything to do with the system.
