Scholars are almost unanimous that Paul did not write Second Timothy. The Second Epistle ot Timothy is one of the epistles called the 'Pastorals' and was written early in the second century by one of Paul's followers.

Paul and Timothy were both dead long before this epistle was written.

There are two letters to Timothy. I Timothy has 6 chapters and II Timothy has 3 chapters.

The First Epistle to Timothy is traditionally thought to have been written by Paul from Laodicea. The Second Epistle to Timothy is traditionally thought to have been written by Paul from Rome. However, 1 Timothy and 2 Timothy, known (along with Titus) as the Pastorals because of their concern for the pastoral role of bishops in the second-century Church, are almost unanimously regarded by critical scholars as not written by Paul. We do not really know who wrote them, or from whence they were written.

The thirteen epistles that are almost certainly written by Paul, in order, are:

Epistle to the Romans

First Epistle to the Corinthians

Second Epistle to the Corinthians

Epistle to the Galatians

Epistle to the Ephesians

Epistle to the Philippians

Epistle to the Colossians

First Epistle to the Thessalonians

Second Epistle to the Thessalonians

First Epistle to Timothy

Second Epistle to Timothy

Epistle to Titus

Epistle to Philemon

although some theologians doubt that some letters (eg Ephesians) are written by him as the style of writing is different from the others.

The fourteenth letter, the Epistle to the Hebrews - is unlikely to have been written by Paul. He doesn't claim to have written it nor is the style of writing his. Pauline authorship is rejected by modern scholarship and even in the early church its authorship was debated.