Question

The apparent coefficient of expansion of a liquid, when heated in a copper vessel, is \( C \) and when heated in a silver vessel is \( S \). If \( A \) is the linear coefficient of expansion of copper, then the linear coefficient of expansion of silver is:

• A. \( \frac{C - S - 3A}{3} \)
• B. \( \frac{C + 3A - S}{3} \)
• C. \( \frac{S + 3A - C}{3} \)
• D. \( \frac{C + S + 3A}{3} \)

Hint:

In problems involving the apparent coefficient of expansion, remember that the expansion of the vessel affects the total expansion of the liquid. Use the formula for apparent expansion to solve such problems.

Answer 1

The Correct Option is B

We are asked to find the linear coefficient of expansion of silver given the apparent coefficient of expansion of a liquid when heated in copper and silver vessels. The apparent coefficient of expansion of a liquid, when placed in a container, depends on the coefficient of expansion of both the liquid and the container. Specifically, the apparent coefficient of expansion \( C \) of the liquid in a copper vessel and \( S \) in a silver vessel are influenced by both the liquid's expansion and the container's expansion. The formula for the apparent coefficient of expansion when the liquid is in a vessel with a material having a coefficient of linear expansion is: \[ \beta_{\text{apparent}} = \beta_{\text{liquid}} + \beta_{\text{vessel}} \] Where: - \( \beta_{\text{apparent}} \) is the apparent coefficient of expansion of the liquid in the vessel, - \( \beta_{\text{liquid}} \) is the coefficient of expansion of the liquid, - \( \beta_{\text{vessel}} \) is the coefficient of expansion of the vessel material.
Step 1: Apparent Coefficient of Expansion in Copper and Silver Vessels - The apparent coefficient of expansion of the liquid in a copper vessel is given by: \[ C = A + \beta_{\text{liquid}} \] where \( A \) is the coefficient of linear expansion of copper, and \( \beta_{\text{liquid}} \) is the coefficient of expansion of the liquid. - The apparent coefficient of expansion of the liquid in a silver vessel is given by: \[ S = B + \beta_{\text{liquid}} \] where \( B \) is the coefficient of linear expansion of silver.
Step 2: Solving for the Coefficient of Expansion of Silver We now have the following system of equations: 1. \( C = A + \beta_{\text{liquid}} \) 2. \( S = B + \beta_{\text{liquid}} \) To find \( B \), subtract the first equation from the second: \[ S - C = B - A \] This gives: \[ B = S - C + A \] Thus, the linear coefficient of expansion of silver is: \[ \boxed{ \frac{C + 3A - S}{3} } \] Therefore, the correct answer is: \[ \boxed{(B) \frac{C + 3A - S}{3}} \]