It is not essential to read any book between completing a Part IA course and starting Part I Chemical Engineering. The most important thing is to be keen and ready to learn in October.
It is useful to go through the following lecture notes in the week before starting Part I Chemical Engineering:
ex-ET: read thermofluids notes
ex-NST: read energetics/equilibria/kinetics notes
Chemical engineers need to be competent at maths. In particular, it is important that that they are comfortable with calculus - so you might need to revise differentiation and integration of simple functions if you can't easily remember how to do this.
The following two books on Chemical Engineering may be of interest:
K. A. Solen and J. N. Harb (2010) Introduction to Chemical Engineering: tools for today and tomorrow: Wiley, ISBN 9780470885727
This book is fairly short (227 pages) and is easy to read. It describes the discipline and gives examples of the different types of calculations that are performed by chemical engineers. Don't be put off by the use of old-fashioned American units throughout the book. The book should be available in your College Library, possibly under its former title namely Introduction to Chemical Process Fundamentals & Design.
M. M. Denn (2012) Chemical Engineering: an introduction: Cambridge University Press, ISBN 9781107669376
This book has similar aims to the above book. It describes the discipline well and gives examples of applications of chemical engineering, including mathematical analysis of them. This makes it less user-friendly as an introduction then the above book, but it is still worth a look. It's not too long (about 250 pages).