Gemara

(Beraisa): The Torah discusses giving "Terumah" from Lachmei Todah, like it says regarding Terumas Ma'aser, Just like there one part in 10 is Terumah, also here.

Question: Why don't we learn from Terumah taken from the spoils that Bnei Yisrael took from Midyan?

Answer: We learn Terumas Lachmei Todah from Terumas Ma'aser, for both of these apply to all generations.

Question: Why don't we learn from Chalah, which is called Terumah?

Answer (Tana d'Vei R. Yishmael): We learn Terumas Lachmei Todah from Terumas Ma'aser, for regarding both of these it says "Mimenu Terumas Hash-m".

Bechoros 27a (Shmuel): A Tamei Kohen may eat Terumas Chutz la'Aretz, except for one whose Tum'ah comes from his own body (e.g. a Zav or Nidah). Even such Teme'im are forbidden only to eat it, but they may touch it.

(Ravina): Therefore, a Nidah may separate Chalah for a young Kohen (under nine, for he has never had emissions) to eat. If no young Kohen is around, she takes it with a roasted and burns it.

She then separates a second Chalah, and gives it to any adult Kohen to eat, lest people forget the (primary) Mitzvah of Chalah.

Chalah 4:8 (Mishnah - R. Gamliel): In Eretz Yisrael until Keziv, we separate one Chalah. From Keziv until the river and Emanah, we separate two Chalos. One is burned; it has a Shi'ur. One is for a Kohen; it has no Shi'ur. From the river until Emanah and inward, we separate two Chalos. One is burned; it has no Shi'ur. One is for a Kohen; it has a Shi'ur. A Tevul Yom may eat it;

R. Yosi says, he need not immerse.

It is forbidden to a Zav, Zavah, Nidah or Yoledes. One may give it to any Kohen.

Rishonim

Rambam (Hilchos Bikurim 5:9): Nowadays that no dough is Tahor, in Eretz Yisrael we separate one part in 48 for Chalah and burn it. It is Chalah mid'Oraisa. From Keziv until Emanah we separate (also) a second Chalah for a Kohen to eat. There is no Shi'ur for it, like it used to be.

Rambam (10): Kohanim may eat Chalah of Chutz la'Aretz even if it is Tamei, unless Tum'ah exudes from the Kohen's body, e.g. a Ba'al Keri, Nidah or Metzora. Therefore, where there is a young Kohen who did not yet see Keri or Nidah, or an adult who immersed from Keri, one may separate just one Chalah of one part in 48 for the child.

Rosh (Chulin 8:4): Near Eretz Yisrael, where the Peros are of Eretz Yisrael but everything is Tamei, one must separate Chalah mid'Oraisa, with a Shi'ur. It is burned. We also separate a second Chalah mid'Rabanan, for Kohanim to eat. It has no Shi'ur. Chachamim permit a Tevul Yom from Keri to eat it. R. Yosi permits a Ba'al Keri even without Tevilah. They are lenient, for this Chalah is merely lest people forget the law of Chalah. The Seifa forbids to a Zav, Zavah, Nidah or Yoledes. It teaches that even though R. Yosi permits a Ba'al Keri without Tevilah, he does not permit these. Even a Tahor Kohen may not eat Chalah of places near Eretz Yisrael, for the Chalah is Tamei, and it could be confused with Chalah of Eretz Yisrael. In places far from Eretz Yisrael, e.g. Bavel, it suffices to separate one Chalah; a young Kohen eats it. Bechoros 27a teaches that where there is no young Kohen, a Nidah burns one Chalah, and separates another for an adult Kohen (who immersed from Keri) to eat. It seems that such an adult has the same law as a child. We are no more stringent about this Chalah than about Chalah of places near Eretz Yisrael. Also Bahag says so. The Gemara mentions giving it to a child, for it is a burden for an adult to immerse for it. Even though he immerses for Terumah, Chalah of Chutz la'Aretz is a small amount; it has no Shi'ur. The Gemara says that she takes it on the end of a Mapah (our text is Masa, i.e. stick); this shows that it is a small amount. Nowadays, in most places we separate only one Chalah even though we burn it! In the days of the Amora'im, they had Mei Chatas, and they ate Tahor Chalah in Eretz Yisrael. Nowadays that Chalah is not eaten in Eretz Yisrael, even in Chutz la'Aretz one need not separate (another) Chalah for the Kohen (to eat). Even in Eretz Yisrael, we separate just one Chalah and burn it. It is a good custom to separate (a second Chalah for the Kohen), lest people forget that Chalah is eaten even when (there is Taharah, e.g.) the Mikdash will be rebuilt.

Rashba (Piskei Chalah 4, cited in Beis Yosef YD 322 DH u'Chlal): The Mishnah connotes that in Chutz la'Aretz we should separate one Chalah without a Shi'ur, to be burned, and one Chalah for a Kohen, with a Shi'ur one part in 48. I heard that some places still do so, but here and in France, people separate only one Chalah without a Shi'ur and burn it.

Tosfos (Bechoros 27b DH v'Chi): When Chachamim enacted Chalah of Chutz la'Aretz, presumably they did not require Kohanim to go to Eretz Yisrael to be sprinkled due to Tum'as Mes before eating it. Rather, it suffices to immerse from all (other) Torah Tum'os. Because Chalah in Eretz Yisrael must be burned, near Eretz Yisrael (where the Peros are of Eretz Yisrael) we also obligate separating a Chalah that is burned, in addition to the Chalah that Kohanim eat. It has a Shi'ur, for it is mid'Oraisa. We also separate a Chalah for a Kohen to eat, lest people forget the law of Chalah. It has no Shi'ur, for it is mid'Rabanan. Further from Eretz Yisrael (where the Peros are not of Eretz Yisrael) both Chalos are mid'Rabanan. The one that is burned has no Shi'ur (for no one benefits); the Chalah for a Kohen has a Shi'ur.

Rashi (77b DH v'Neilaf): The Gemara suggested that we learn one part in 50 (from what Bnei Yisrael gave to the Leviyim).

Rashi (77b DH v'Neilaf): The Gemara suggested that we learn from Chalah one part in 48 (the Shi'ur for bakers).

Poskim

Shulchan Aruch (YD 322:4): Nowadays that no dough is Tahor, in Eretz Yisrael we separate one part in 48 for Chalah and burn it. From Keziv until Emanah we separate (also) a second Chalah for a Kohen to eat. There is no Shi'ur for it.

Beis Yosef (DH u'Chlal): Sefer ha'Terumah says that we do not separate Chalah for a Kohen because one would need to guard it from Zarim and Nochrim, and sometimes one puts it in a hot pot, and it forbids all the contrents to Zarim.

Taz (2): Tosfos and the Rosh hold that near Eretz Yisrael, the primary Chalah is mid'Oraisa. It has a Shi'ur, and we burn it. The other Chalah is only lest people forget that Chalah is eaten, so it has no Shi'ur. Further away, both are mid'Rabanan, so the one that is eaten is primary. It has a Shi'ur. Since the purpose is lest people forget the law of eating Chalah, we are lenient to allow eating it without ha'Arev Shemesh, and without even Tevilah for Tum'os that did not leave the body. The Rambam does not distinguish between places far from or close to Eretz Yisrael. He holds that places conquered by Olei Mitzrayim, but not by Olei Bavel, did not get Kedushah. Chalah there was only mid'Rabanan even in the days of Ezra. We separate Chalah and burn it, for it is Eretz Yisrael. We separate a second Chalah for a Kohen to eat, lest people say that we burn Tahor Chalah. In places that even Olei Mitzrayim did not conquer, a second Chalah is needed only if there is no Tahor Kohen to eat the first. The Tur is difficult. He initially explains like the Rosh, but concludes 'from the Rambam, it seems that only in Chutz la'Aretz (a young or Tahor Kohen may eat Chalah).' This implies that the first opinion permits them even in Eretz Yisrael. Surely no one permits this! With difficulty, we can say that he means 'even according to the Rambam, who does not distinguish far or near places, Eretz Yisrael is unlike Chutz la'Aretz.' Why did the Shulchan Aruch copy the Rambam's opinion? It seems that the Beis Yosef did not notice that it is unlike the Rosh!

Shach (8): The Maharil says that R. Tam holds that we burn the Chalah (that must be burned) in the fire of a male or female Kohen. Maharam agrees. It is proper (for a Yisrael who burns it) to make a separate fire on which he does not cook, or in the oven used to bake Matzah before the Matzah is put in, for then surely he will not use the Chalah to cook (he willl wait until the Chalah is totally burned, due to concern for Chametz). It seems that only Hana'ah or burning it is forbidden to a Zar. Other Hana'os are permitted, like the Rema (331:19) says about Terumah.

Gra (4): The first Chalah is forbidden to Zavim, like Perush ha'Mishnayos says. The second is permitted. A Nidah may separate it. If there is no child, an adult may eat it. Tosfos holds that we always require two Chalos (in Chutz la'Aretz). The Heter for a Tevul Yom is in places far from Eretz Yisrael.

Pischei Teshuvah (2): If a woman set aside Chalah and a bird ate it, Birkei Yosef said that nowadays that we do not give Chalah to a Kohen, she is exempt. When it was given to Kohanim, she would be liable, like regarding Reishis ha'Gez (333:5).

Rema (5): Some say that nowadays that Chalah is not eaten in Eretz Yisrael, in all places we separate only one Chalah, and burn it. This is the custom is in all these lands. There is no Shi'ur. In any case the custom is to take a k'Zayis. Some say that when we burn the Chalah, we make a separate fire, for a Yisrael may not benefit from it. The custom is to put it in the oven before baking bread.

Taz (5): The Maharshal says that nowadays we do not give even to a young Kohen, and all the more so to an old Kohen who immersed, for we lack Kohanim of definite lineage. R. Yerucham says that women who bake Matzah on Yom Tov of Pesach make a small cake, and do not call it Chalah until after baking. (Since no one may eat Chalah, one may not bake it on Yom Tov.) After calling it Chalah, it is Muktzah and one may not move it, even if there is a young Kohen, for we do not consider him to be a definite Kohen.

Gra (11): The Rema is like the Rosh, who holds that the Chiyuv to take a second Chalah was only when there was Mei Chatas and they ate Chalah and Terumah (in Eretz Yisrael).

Pischei Teshuvah (3): The Rivash (94) says that we are unsure of Kohanim's lineage nowadays. Maharashdam says so regarding a woman how was captured or did Chalitzah and wants to marry a Kohen. The Maharit and Chut ha'Shani refuted him. Shevus Yakov supports him.

Question: Why did Rashi explain the Hava Amina to learn one part in 48 from Chalah? The Shi'ur of Chalah is mid'Rabanan! Mid'Oraisa, it has no Shi'ur! Perhaps the Hava Amina was to learn from Chalah that Terumah has no Shi'ur. Also, why did Rashi say one part in 48? A Ba'al ha'Bayis gives one part in 24! Perhaps we suggested learning the smaller Chidush, i.e. the smaller Shi'ur. However, Rashi explained that the Hava Amina to learn from Terumas Midyan was one part in 50. He did not suggest learning the smaller Shi'ur, i.e. one part in 500, from the Terumah given to Elazar!

Hagahas Rav Daitsh: Perhaps the Gemara asked according to the opinion (Kidushin 58b) that all disagree with Shmuel, and hold that the Torah gave a Shi'ur (for Terumah, and likewise) for Chalah!

Answer (Noda bi'Yehudah DH u'Mah): There are two reasons for the Mitzvah of Chalah. One is to uproot the Isur Tevel from the dough. Separating Chalah permits the dough, even if he never gives it to a Kohen. The primary Mitzvah is to give it to a Kohen. The Torah gave no Shi'ur for the first Mitzvah. There is a Torah Shi'ur for the latter Mitzvah. "Titnu" teaches a proper amount. The Shi'ur, one part in 48 or 24, is partially mid'Oraisa and partially mid'Rabanan. When the dough is an Isaron (43.2 eggs), one part in 24 (1.8 eggs) is a proper Nesinah. The Torah obligates all dough this size or bigger. Chachamim enacted to give one part in 24 even if the dough is bigger than this. In Rashi's Perush on Chumash, on the verse "ki'Serumas Goren" he said that the Torah gave no Shi'ur, but Chachamim fixed one part in 24 (or 48) for a Ba'al ha'Bayis (or baker). Then he explained "Titnu la'Shem" to obligated giving a proper amount, like the Sifri. This is like I explained.