The mitzvah framework for purification and the laws of ritual immersion provide conceptual background to why a chatzitzah invalidates tevillah. The exact mitzvah numbering should be verified before use in a practical halakhic discussion.
וְדִינֵי חֲצִיצָה שֶׁאָמְרוּ זִכְרוֹנָם לִבְרָכָה (עירובין ד ב) כִּי דְּבַר תּוֹרָה אִם הָיָה דָּבָר חוֹצֵץ חוֹפֶה אֶת רֹב הָאָדָם אוֹ רֹב הַכְּלִי, לֹא עָלְתָה לָהֶם טְבִילָה, וְהוּא שֶׁיַּקְפִּיד הָאָדָם בַּדָּבָר, אֲבָל אִם אֵינוֹ מַקְפִּיד, עָלְתָה לוֹ טְבִילָה, וְאִם הַדָּבָר הַחוֹצֵץ אֵינוֹ אֶלָּא עַל מִעוּטוֹ, אֲפִלּוּ מַקְפִּיד עָלְתָה לוֹ טְבִילָה דִּין תּוֹרָה, אֲבָל מִדִּבְרֵי סוֹפְרִים שֶׁכָּל דָּבָר הַחוֹצֵץ וַאֲפִלּוּ כָּל שֶׁהוּא אִם מַקְפִּיד עָלָיו לֹא עָלְתָה לוֹ טְבִילָה, גְּזֵרָה מִעוּטוֹ מִשּׁוּם רֻבּוֹ שֶׁהוּא אָסוּר מִן הַתּוֹרָה כְּשֶׁמַּקְפִּיד, וְאִם הָיָה דָּבָר הַחוֹצֵץ חוֹפֶה רֻבּוֹ, אַף עַל פִּי שֶׁאֵינוֹ מַקְפִּיד, לֹא עָלְתָה לוֹ טְבִילָה, גְּזֵרָה רֹב שֶׁאֵינוֹ מַקְפִּיד מִשּׁוּם רֹב הַמַּקְפִּיד שֶׁאָסוּר דְּבַר תּוֹרָה. נִמְצָא לְפִי זֶה שֶׁאִם הָיָה דָּבָר חוֹצֵץ עַל מִעוּטוֹ וְאֵינוֹ מַקְפִּיד, עָלְתָה לוֹ טְבִילָה אֲפִלּוּ מִדְּרַבָּנָן, דְּהַשְׁתָּא יֵשׁ בַּדָּבָר תַּרְתֵּי לְטֵיבוּתָא. וַחֲפִיפָה זוֹ צְרִיכָה סָמוּךְ לַטְּבִילָה, כְּלוֹמַר שֶׁלֹּא תִּתְעַסֵּק בְּדָבָר אַחֵר בֵּינְתַיִם. וּמִכָּל מָקוֹם אִם נִזְדַּמְּנָה לְאִשָּׁה טְבִילָה בְּלֵילֵי שַׁבָּת וְיוֹם טוֹב הִתִּירוּ אֲפִלּוּ לְכַתְּחִלָּה שֶׁתְּהֵא חוֹפֶפֶת בַּיּוֹם וְטוֹבֶלֶת בַּלַּיְלָה, וְכֵן מִפְּנֵי חֲשַׁשׁ שֶׁיִּכָּנְסוּ הַמַּיִם בְּכָל מָקוֹם שֶׁבְּגוּפָה אָמְרוּ חֲכָמֵינוּ זִכְרוֹנָם לִבְרָכָה (נדה סז א), שֶׁצְּרִיכָה הָאִשָּׁה לַעֲמֹד בִּשְׁעַת טְבִילָה בְּעִנְיָן שֶׁתְּהֵא נִרְאֵית כְּאוֹרֶגֶת וּכְמֵינֶקֶת אֶת בְּנָהּ שֶׁעוֹמֵד תַּחַת הַדַּד, וּצְרִיכָה לְהִזָּהֵר גַּם כֵּן בַּטַּבָּעוֹת שֶׁבְּיָדָהּ וְהַצְּמִידִים אִם הֵם מֵהַדַּקִּים לִטְּלָם מִיָּדֶיהָ בִּשְׁעַת טְבִילָה, וְכֵן הָרְצוּעוֹת שֶׁבְּרֹאשָׁהּ לְהַתִּירָן, וְתָדִיחַ אֶת פִּיהָ וְשִׁנֶּיהָ, שֶׁאַף עַל פִּי שֶׁכָּל הַמְּקוֹמוֹת שֶׁבְּבֵית הַסְּתָרִים אֵין צְרִיכִין שֶׁיָּבוֹאוּ בָּהֶם מַיִם, צְרִיכִין הֵם לִהְיוֹתָם רְאוּיִים לְבִיאַת מַיִם, וּמַעֲשֶׂה הָיָה (שם סו ב) בְּאִשָּׁה שֶׁטָּבְלָה וְנִמְצָא לָהּ עֶצֶם בֵּין שִׁנֶּיהָ וְהִצְרִיכוּהָ חֲכָמִים טְבִילָה אַחֶרֶת. טָבְלָה וְנִמְצָא עָלֶיהָ שׁוּם דָּבָר חוֹצֵץ כְּגוֹן בָּצֵק אוֹ זֶפֶת אוֹ דָּם יָבֵשׁ וְכַיּוֹצֵא בָּהֶן, מִכָּל הַדְּבָרִים שֶׁמָּנוּ חֲכָמִים שֶׁחוֹצְצִין אֲפִילּוּ כְּטִפַּת חַרְדָּל, כָּל זְמַן שֶׁמַּקְפֶּדֶת עָלָיו וְדַרְכָּהּ לְהַעֲבִירוֹ, לֹא עָלְתָה לָהּ טְבִילָה, וַאֲפִלּוּ הָיָה הַדָּבָר הַחוֹצֵץ תַּחַת הַצִּפֹּרֶן וַאֲפִלּוּ כְּנֶגֶד הַבָּשָׂר, מִכֵּיוָן שֶׁמַּקְפֶּדֶת, חוֹצֵץ. וּלְפִיכָךְ נָהֲגוּ בְּנוֹת יִשְׂרָאֵל לִטֹּל צִפָּרְנָן בִּשְׁעַת טְבִילָה. וְיֶתֶר רֻבֵּי פְּרָטֶיהָ שֶׁל דִּינֵי חֲצִיצוֹת וְדִינֵי מִקְוָאוֹת וּטְבוּל יוֹם, מְבֹאָרִים בְּמַסֶּכֶת מִקְוָאוֹת וּבְמַסֶּכֶת טְבוּל יוֹם.
The laws of separations [from the water] that they, may their memory be blessed, said (Eruvin 4b) is that it is the word of the Torah if there was something separating, covering most of the person or the vessel, the immersion would not count for them — and that is when the person is concerned about the thing. But if he is not concerned, the immersion counts. But if the thing that separated was only on its lesser part, the immersion counts for him from Torah writ, even if he is concerned about it. But [rabbinically, with] anything — even the smallest amount — that separates that he is concerned about, his immersion does not count; [as a] decree of its lesser part, on account of its majority, which is forbidden from the Torah when he is concerned. And if the thing that separated covered most of him — even if he is not concerned — his immersion does not count; [as a] decree of the majority about which he is not concerned, on account of the majority about which he is concerned, which is forbidden from Torah writ. It comes out that if there was something on his lesser part and he is not concerned, the immersion counted for him — even rabbinically — as now there are two advantages to the thing. And this scrubbing needs to be proximate to her immersion — meaning to say that she not be involved in anything in between. And nonetheless if the immersion of a woman happened to be on Shabbat or a holiday, they permitted even from the outset that she scrub during the day and immerse at night. And likewise the Sages, may their memory be blessed, said (Niddah 67a) that on account of the concern that water should enter onto all parts of her body, a woman needs to stand at the time of the immersion in such a way that she appear like a weaver or one nursing her child standing under her breast. And she also needs to be careful about rings on her hand and bracelets — and if they are tight, she must take them off her hands at the time of immersion; and so [too,] to untie the bands on her hair, and to rinse her mouth and teeth. As even though water does not need to come into any of the hidden places, they need to be fit for the water to come in. And there was a case (Niddah 66b) of a woman who immersed and a bone was found between her teeth, and the Sages required her another immersion. If she immersed and then something else separating was found upon her, such as dough or tar or dry blood and what is similar to them from all of the things that the Sages enumerated as separating — even it if is as small as a mustard seed, so long as she is concerned about it and her way is to remove it — her immersion did not count for her. And even if the thing was under the fingernail, and even if it was [impressed on] the flesh — since she is concerned — it is [considered to be] separating. And therefore the daughters of Israel have been accustomed to cut their nails at the time of immersion. And the rest of its many details of the laws of separations and the laws of mikvaot and one who has immersed on that day are elucidated in Tractate Mikvaot and in Tractate Tevul Yom.