Question:medium
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:
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:
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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.
Updated On: Nov 28, 2025
- \( \frac{C - S - 3A}{3} \)
- \( \frac{C + 3A - S}{3} \)
- \( \frac{S + 3A - C}{3} \)
- \( \frac{C + S + 3A}{3} \)
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The Correct Option is B
Solution and Explanation
We aim to determine silver's linear expansion coefficient, given a liquid's apparent expansion in copper and silver containers.
A liquid's apparent expansion, when heated within a container, is affected by both the liquid and container's expansion. The apparent coefficient of expansion, \( C \) (copper) and \( S \) (silver), reflects both the liquid's and the container's expansion.
The formula for apparent expansion in a vessel is:
\[
\beta_{\text{apparent}} = \beta_{\text{liquid}} + \beta_{\text{vessel}}
\]
Where:
- \( \beta_{\text{apparent}} \) is the liquid's apparent expansion coefficient.
- \( \beta_{\text{liquid}} \) is the liquid's expansion coefficient.
- \( \beta_{\text{vessel}} \) is the vessel material's expansion coefficient.
Step 1: Apparent Expansion Coefficients in Copper and Silver - In a copper vessel: \[ C = A + \beta_{\text{liquid}} \] where \( A \) is copper's linear expansion coefficient, and \( \beta_{\text{liquid}} \) is the liquid's expansion coefficient. - In a silver vessel: \[ S = B + \beta_{\text{liquid}} \] where \( B \) is silver's linear expansion coefficient.
Step 2: Solving for Silver's Expansion Coefficient We have the following 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 simplifies to: \[ B = S - C + A \] Therefore, silver's linear expansion coefficient is: \[ \boxed{ \frac{C + 3A - S}{3} } \] Consequently, the correct answer is: \[ \boxed{(B) \frac{C + 3A - S}{3}} \]
Step 1: Apparent Expansion Coefficients in Copper and Silver - In a copper vessel: \[ C = A + \beta_{\text{liquid}} \] where \( A \) is copper's linear expansion coefficient, and \( \beta_{\text{liquid}} \) is the liquid's expansion coefficient. - In a silver vessel: \[ S = B + \beta_{\text{liquid}} \] where \( B \) is silver's linear expansion coefficient.
Step 2: Solving for Silver's Expansion Coefficient We have the following 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 simplifies to: \[ B = S - C + A \] Therefore, silver's linear expansion coefficient is: \[ \boxed{ \frac{C + 3A - S}{3} } \] Consequently, the correct answer is: \[ \boxed{(B) \frac{C + 3A - S}{3}} \]
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