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:

Extensive properties scale with the amount of matter. Intensive properties do not. Check how each expression changes when the system is doubled.

Answer 1

The Correct Option is A, C

Step 1: Understanding the Concept:
An extensive property such as volume or energy doubles when the system is doubled. An intensive property such as temperature or pressure stays the same.

Step 2: Key Approach:
Imagine the system is made \( \lambda \) times bigger. Then \( X \) becomes \( \lambda X \) and \( x \) stays \( x \). We see how each expression changes.

Step 3: Check statement (A):
\( xX \) becomes \( x \cdot \lambda X = \lambda\, xX \). It scales with size, so it is extensive. (A) is TRUE.

Step 4: Check statement (B):
\( x/X \) becomes \( x / (\lambda X) = \tfrac{1}{\lambda}\,(x/X) \). It shrinks as the system grows, so it is neither intensive nor extensive. (B) is FALSE.

Step 5: Check statement (C):
\( X/x \) becomes \( \lambda X / x = \lambda\,(X/x) \). It scales with size, so it is extensive. (C) is TRUE.

Step 6: Check statement (D):
In \( dX/dx \) the top changes by a factor \( \lambda \) and the bottom does not change. So the result is also multiplied by \( \lambda \) and is extensive. A real example is heat capacity \( C_V = dU/dT \), which depends on how much substance there is. (D) is FALSE.

Final Answer:
Statements (A) and (C) are correct. \[ \boxed{\text{(A) and (C)}} \]