Question

$X$ is an extensive property and $x$ is an intensive property of a thermodynamic system. Which of the following statement(s) is/are correct?

• A. $Xx$ is extensive.
• B. $\frac{X}{x}$ is intensive.
• C. $\frac{X}{x}$ is extensive.
• D. $\frac{dX}{dx}$ is intensive.

Hint:

Scaling trick: \begin{itemize} \item Multiply by intensive $\Rightarrow$ remains extensive \item Divide extensive by intensive $\Rightarrow$ still extensive \item Ratio of two extensive properties $\Rightarrow$ intensive \end{itemize}

Answer 1

The Correct Option is A

Concept: \begin{itemize} \item Extensive property depends on system size \item Intensive property independent of size \end{itemize} Let system size scale by factor $\lambda$: \[ X \rightarrow \lambda X, \quad x \rightarrow x \] (A) $Xx$ \[ Xx \rightarrow (\lambda X)(x) = \lambda (Xx) \] Thus extensive. Correct. (B) $\frac{X{x}$} \[ \frac{X}{x} \rightarrow \frac{\lambda X}{x} = \lambda \frac{X}{x} \] So extensive, not intensive. False. (C) $\frac{X{x}$ is extensive} True. (D) $\frac{dX{dx}$} Derivative of extensive w.r.t intensive typically remains extensive (depends on scaling behavior), not necessarily intensive. Thus false. Conclusion: Correct statements: A and C.