关键词: 链式关系, 隐函数定理, 焦耳气体自由膨胀实验

Abstract: This paper, through proving the chain relation, elaborates on the differences in the conditions required for the establishment of its two distinct forms, examines the considerations for applying chain relation, and highlights a prevalent error in textbooks when using chain relation to analyze Joules gas free expansion experiment.

Key words: chain relation, implicit function theorem, Joules experiment of free expansion of gases