Exhibit 5.1

March 21, 2022

Adit EdTech Acquisition Corp.

1345 Avenue of the Americas

33^rd Floor

New York, NY 10105

Ladies and Gentlemen:

We have acted as counsel to Adit EdTech Acquisition Corp., a Delaware corporation (the “Company”), in connection with the Company’s Registration Statement on Form S-4 (File No. 333-261880) (as amended to the date hereof, the “Registration Statement”) filed pursuant to the Securities Act of 1933, as amended (the “Securities Act”). The Registration Statement registers an aggregate of 308,100,000 shares of the Company’s common stock, par value $0.0001 per share (the “Shares”), to be issued as consideration under that certain Agreement and Plan of Merger, by and among the Company, ADEX Merger Sub, LLC, a Delaware limited liability company and a wholly owned direct subsidiary of the Company (“Merger Sub”), and Griid Holdco LLC, a Delaware limited liability company (“GRIID”), dated as of November 29, 2021 (as amended from time to time, the “Merger Agreement”). Pursuant to the Merger Agreement, among other things, Merger Sub will merge with and into GRIID (the “Merger”) at the effective time of the Merger (the “Effective Time”).

We have reviewed the (a) Merger Agreement, (b) the Amended and Restated Certificate of Incorporation of the Company and the Bylaws of the Company, each as currently in effect, (c) the forms of the Amended and Restated Certificate of Incorporation (the “New Charter”) and the Amended and Restated Bylaws of the Company, filed as Exhibits 3.1 and 3.2 to the Registration Statement, respectively, each of which is to be in effect upon the closing of the Merger, and such other corporate records, certificates and other documents, and such questions of law, as we have considered necessary for the purposes of this opinion. We have assumed that all signatures are genuine, that all documents submitted to us as originals are authentic and that all copies of such documents submitted to us conform to the originals. We have relied as to certain matters on information obtained from public officials, officers of the Company and other sources believed by us to be responsible.

We have assumed that, prior to the issuance of any of the Shares, (1) all approvals and actions for the issuance of the Shares referred to in the Registration Statement will have become effective, including the adoption of the Merger Agreement and the transactions contemplated thereby and approval of the New Charter by the stockholders of the Company and the filing of the New Charter with the Secretary of State of the State of Delaware, (2) to the extent required under the laws of Delaware, the members of GRIID will have adopted the

Adit EdTech Acquisition Corp.

March 21, 2022

Merger Agreement, including the transactions contemplated thereby, (3) the transactions contemplated by the Merger Agreement will have been consummated in accordance with the Merger Agreement and (4) no shares of capital stock of the Company will have been issued or commitments to issue capital stock made by the Company prior to the Effective Time, other than as expressly permitted by the Merger Agreement.

Based upon the foregoing, we are of the opinion that the Shares have been duly authorized and, when the Registration Statement has become effective under the Securities Act, the Shares, when duly issued in accordance with the Merger Agreement, will be validly issued, fully paid and nonassessable.

We are members of the bar of the District of Columbia. We do not express any opinion herein on any laws other than the Delaware General Corporation Law and applicable provisions of the Delaware Constitution and reported judicial decisions interpreting such laws.

We hereby consent to the filing of this opinion as Exhibit 5.1 to the Registration Statement. We also hereby consent to the reference to our firm under the heading “Legal Matters” in the proxy statement/prospectus constituting part of the Registration Statement. In giving such consent, we do not thereby admit that we are in the category of persons whose consent is required under Section 7 of the Securities Act.

Very truly yours,

/s/ Covington & Burling LLP