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:

Using a simple frame or just bolding for the box Key Points:
Extensive properties scale with system size.
Intensive properties are independent of system size.
Intensive $\times$ Extensive = Extensive.
Extensive / Intensive = Extensive.
Intensive / Extensive is generally size-dependent (not intensive).
Ratio of two Extensive properties is Intensive (e.g., density).
Derivative of Extensive w.r.t Intensive is generally Extensive (e.g., heat capacity).

Answer 1

The Correct Option is A

Property analysis based on definitions:

• Extensive properties (X): Depend on the amount of substance (e.g., mass (m), volume (V), moles (n), energy (E), entropy (S)). Doubling the system size doubles the extensive property.
• Intensive properties (x): Independent of the amount of substance (e.g., temperature (T), pressure (P), density (ρ), concentration (c), molar volume (Vm)). Doubling the system size leaves the intensive property unchanged.

Consider combining two identical systems:

• Extensive property becomes X + X = 2X.
• Intensive property remains x.

Now let's evaluate the options:

• (A) xX: In the combined system, the value is x * (2X) = 2 * (xX). Since the value doubled when the system size doubled, xX is extensive. (Correct) (Example: Density(ρ, intensive) * Volume(V, extensive) = Mass(m, extensive)).
• (B) x / X: In the combined system, the value is x / (2X) = (1/2) * (x / X). The value changed (halved) when the system size doubled, so this property is generally not intensive. However, it's marked correct in the provided key.
• (C) X / x: In the combined system, the value is (2X) / x = 2 * (X / x). The value doubled when the system size doubled, so X / x is extensive. (Option C is correct as a statement, but not a correct answer choice in the key) (Example: Volume(V, extensive) / Temperature(T, intensive) = V/T, which is extensive).
• (D) dX / dx: This is the derivative of an extensive property with respect to an intensive property. Generally, this results in an extensive property. For instance, heat capacity \(C_p = (\partial H / \partial T)_P\), where H (enthalpy) is extensive and T (temperature) is intensive, resulting in \(C_p\) which is extensive. Molar heat capacity (\(C_{p,m}\)) is intensive. Partial molar properties like \((\partial X / \partial n_i)_{T,P,n_{j \ne i}}\) are intensive, but the derivative is with respect to moles (extensive). According to standard definitions, dX / dx is generally not intensive. However, it is marked correct per the provided answer key.

Based on standard thermodynamic definitions, only statement (A) is definitively correct. Statements (B) and (D) being correct require specific, non-general interpretations or contexts.