EXHIBIT 4.4(b)




                                                                  CONFORMED COPY


                        LYONDELL PETROCHEMICAL COMPANY,

                            EQUISTAR CHEMICALS, LP

                                      AND

                   TEXAS COMMERCE BANK NATIONAL ASSOCIATION,

                                    TRUSTEE



                         SECOND SUPPLEMENTAL INDENTURE

                                  DATED AS OF

                               DECEMBER 1, 1997

                                      TO

                                   INDENTURE

                                  DATED AS OF

                                 MAY 31, 1989

             (AS SUPPLEMENTED BY THE FIRST SUPPLEMENTAL INDENTURE
                           DATED AS OF MAY 31, 1989)

 
          THIS SECOND SUPPLEMENTAL INDENTURE (this "Supplement"), dated as of
December 1, 1997, between Lyondell Petrochemical Company, a Delaware corporation
("Lyondell"), Equistar Chemicals, LP, a Delaware limited partnership
("Equistar") and Texas Commerce Bank National Association, as Trustee (the
"Trustee"), supplements the Indenture dated as of May 31, 1989 (the
"Indenture"), between Lyondell and the Trustee, as supplemented by the First
Supplemental Indenture dated as of May 31, 1989 (the "First Supplemental
Indenture"), pursuant to which the Company's 10.00% Notes Due 1999 (the "Notes")
were issued and are outstanding.

                                   RECITALS

          WHEREAS, Lyondell has executed and delivered to the Trustee the
Indenture, providing for the issuance from time to time of Lyondell's unsecured
debentures, notes or other evidences of indebtedness, issuable in one or more
series (the "Securities"), and Lyondell has executed and delivered to the
Trustee the First Supplemental Indenture, providing for the issuance of the
Notes, which are Securities under the Indenture;

          WHEREAS, Lyondell and Millennium Chemicals Inc., a Delaware
corporation ("Millennium"), have entered into a Master Transaction Agreement
dated July 25, 1997, as amended (the "Master Transaction Agreement"), which
provides that, on the Closing Date (as defined in the Master Transaction
Agreement) each of Lyondell and Millennium will contribute or cause to be
contributed certain assets to a joint venture partnership and that such joint
venture partnership will assume certain liabilities of each of Lyondell and
Millennium;

          WHEREAS, Lyondell has caused two of its wholly-owned subsidiaries,
Lyondell Petrochemical L.P. Inc. ("Lyondell LP") and Lyondell Petrochemical G.P.
Inc., each a Delaware corporation, and Millennium has caused two of its wholly-
owned subsidiaries, to execute and deliver the Limited Partnership Agreement of
Equistar dated as of October 10, 1997, and Equistar has been organized by the
partners thereof to serve as the joint venture partnership contemplated by the
Master Transaction Agreement;

          WHEREAS, the Master Transaction Agreement provides that Lyondell will
contribute certain specified assets (the "Assets") to Equistar and that Equistar
will assume certain specified liabilities of Lyondell, including the Notes,
pursuant to an Asset Contribution Agreement to be entered into on the Closing
Date between Lyondell, Lyondell LP and Equistar (the "Asset Contribution
Agreement"), the form of which is attached as an exhibit  to the Master
Transaction Agreement;

          WHEREAS, pursuant to the Asset Contribution Agreement, on the Closing
Date, Lyondell will contribute the Assets to Equistar and Equistar will assume
the Notes;

 
          WHEREAS, Section 12.01 of the Indenture provides that nothing
contained in the Indenture or in any of the Securities shall prevent any sale or
conveyance of all or substantially all the property of Lyondell to any other
corporation, provided that upon any such sale or conveyance the due and punctual
payment of the principal of and premium, if any, and interest, if any, on all of
the Securities, according to their tenor, and the due and punctual performance
and observance of all of the covenants and conditions of the Indenture and in
such series to be performed by Lyondell shall be expressly assumed, by
supplemental indenture, by the corporation which shall have acquired such
property;

          WHEREAS, for purposes of Section 12.01 of the Indenture, the Assets
constitute substantially all of the assets of Lyondell;

          WHEREAS, pursuant to Section 12.03 of the Indenture, upon such
assumption by supplemental indenture as specified in the foregoing paragraph,
the transferee shall succeed to and be substituted for Lyondell, with the same
effect as if it had been named in the Indenture;

          WHEREAS, Section 11.01 of the Indenture provides that under certain
conditions, Lyondell and the Trustee may, from time to time and at any time
enter into an indenture or indentures supplemental to the Indenture, inter alia,
to evidence the succession of another corporation to the Company and the
assumption by any such successor, pursuant to Article 12 of the Indenture of the
covenants, agreements and obligations of Lyondell contained in the Indenture and
the Securities; and

          WHEREAS, in connection with the contribution of the Assets to Equistar
by Lyondell and the assumption of the Notes by Equistar, Lyondell and Equistar
have duly determined to make, execute and deliver to the Trustee this Supplement
pursuant to the Indenture;

          NOW, THEREFORE, THIS SUPPLEMENT WITNESSETH:

          In consideration of the premises and other good and valuable
consideration, the receipt and sufficiency of which are hereby acknowledged, and
to comply with Sections 12.01 and 12.03 of the Indenture, the parties hereto
hereby agree, for the equal and proportionate benefit of the respective Holders
from time to time of the Securities, as follows:

                                  SECTION ONE

                                  DEFINITIONS

          Capitalized terms used and not otherwise defined herein have the
respective meanings assigned to such terms in the Indenture.

                                      -3-

 
                                  SECTION TWO

                       SUCCESSION BY TRANSFER OF ASSETS

          On the Closing Date, the Assets will be transferred to Equistar, and
effective upon such transfer, (a) Equistar hereby expressly assumes the due and
punctual payment of the principal of and premium, if any, and interest, if any,
on all of the Securities of each series and the due and punctual performance of
all of the covenants and conditions of the Indenture, as supplemented by the
First Supplemental Indenture and this Supplement, and in such series to be
performed by Lyondell; and (b) Equistar will succeed to and be substituted for
Lyondell as the "Company" for purposes of the Indenture, with the same effect as
if Equistar had been named as the "Company" in the Indenture, as supplemented;
provided, however, that Lyondell shall not be released from any of its
obligations under the Indenture and under the Securities of each series,
including the obligation to pay the principal of and premium, if any, and
interest, if any, on the Securities.

          After the Closing Date, for purposes of the Indenture, the term
"Company" shall mean and include both Equistar and Lyondell, and Equistar shall
not be a "Subsidiary" of Lyondell.

                                 SECTION THREE

                                 RATIFICATION

          Except as expressly amended and supplemented on this Supplement, the
Indenture shall remain unchanged and in full force and effect.  This Supplement
shall be construed as supplemental to the Indenture and shall form a part
thereof.

                                 SECTION FOUR

                                 GOVERNING LAW

          This Supplement shall be governed by and construed in accordance with
the laws of the State of New York applicable to agreements made and to be
performed therein.

                                 SECTION FIVE

                                 COUNTERPARTS

          This instrument may be executed in any number of counterparts, each of
which so executed shall be deemed to be an original, but all such counterparts
shall together constitute but one and the same instrument.

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          IN WITNESS WHEREOF, each of Lyondell Petrochemical Company and
Equistar Chemicals, LP have caused this Second Supplemental Indenture to be duly
executed and its seal to be affixed hereunto and the same to be attested by its
Secretary or an Assistant Secretary, and Texas Commerce Bank National
Association as Trustee, has caused this Second Supplemental Indenture to be
signed by one of its Vice Presidents or Assistant Vice Presidents as of the day
and year first above written.

                                   LYONDELL PETROCHEMICAL COMPANY


[SEAL]                             By   /s/ Russell S. Young
                                      ---------------------------------------
                                       Name:  Russell S. Young
                                       Title: Senior Vice President, Chief
                                               Financial Officer and Treasurer

Attest:

     /s/ Kerry A. Galvin
- -------------------------------
Name:  Kerry A. Galvin
Title: Assistant Secretary


                                   EQUISTAR CHEMICALS, LP


[SEAL]                             By   /s/ Joseph M. Putz
                                      ---------------------------------------
                                       Name:  Joseph M. Putz
                                       Title: Senior Vice President, Finance
                                               and Administration

Attest:

     /s/ Gerald A. O'Brien
- -------------------------------
Name:  Gerald A. O'Brien
Title: Vice President and Secretary


                                   TEXAS COMMERCE BANK NATIONAL
                                   ASSOCIATION, Trustee


                                   By   /s/ Mauri J. Cowen
                                      ---------------------------------------
                                       Name:  Mauri J. Cowen
                                       Title: Vice President and Trust Officer

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