All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
First, we would like to advise you to guard your oaths and not to hasten to swearing oaths all the time unnecessarily. Allaah The Exalted Says (what means): {...But guard your oaths…} [Quran 5:89] As for your question, if you took an oath not to do certain acts during working hours such as checking your email or sending a Fatwa, and did not specify a particular time frame or length for those rest periods in your intention while taking the oath, then each time you do such acts, your oath is considered broken.
If you break you oath, you should pay one expiation, and the oath becomes dissolved unless you make a new oath after breaking the first. In this case, you should expiate for the new oath and so on. The expiation for breaking an oath is to choose to feed ten needy people, clothe them or free a believing slave. Doing any such action is sufficient for clearing one of liability. If the oath-taker cannot afford any of that, he should fast three days. There is no difference of opinion among scholars in this regard. In principle, the expiation for breaking an oath is as explained in the following verse. Allaah The Exalted Says (what means): {Allaah Will not Impose blame upon you for what is meaningless in your oaths, but He Will Impose blame upon you for [breaking] what you intended of oaths. So its expiation is the feeding of ten needy people from the average of that which you feed your [own] families or clothing them or the freeing of a slave. But whoever cannot find [or afford it] – then a fast of three days [is required]. That is the expiation for oaths when you have sworn. But guard your oaths. Thus does Allaah Make clear to you His verses that you may be grateful.} [Quran 5:89] As for how to expiate through feeding, the person should give food to ten needy people; half a Saa’ [volumetric measure (double handful)] for each, equaling approximately 1500 grams. It is also said that one should feed each of the ten needy people one Mudd [volumetric measure (handful)], equaling 750 grams, but the first opinion is more prudent so as to be on the safe side (in terms of clearing liability). Whoever could not find poor people in the country where he lives, can entrust any charitable society to pay the expiation on his behalf to the eligible recipients.
Allaah Knows best.