Question

The coefficient of volume expansion of glycerine is \( 49 \times 10^{-5} \, \text{K}^{-1} \). What is the fractional change in its density for a \( 30^\circ \text{C} \) rise in temperature?

• A. \( 1.5 \times 10^{-2} \)
• B. \( 2 \times 10^{-4} \)
• C. \( 3.5 \times 10^{-3} \)
• D. \( 2.5 \times 10^{-2} \)

Hint:

The fractional change in density is negative, indicating a decrease in density with increasing temperature due to volume expansion.

Answer 1

The Correct Option is D

The fractional change in the volume \( \Delta V \) is related to the coefficient of volume expansion \( \beta \) and the change in temperature \( \Delta T \) by the equation: \[ \Delta V = \beta V_0 \Delta T \] The fractional change in density \( \Delta \rho \) is related to the fractional change in volume as: \[ \frac{\Delta \rho}{\rho} = -\frac{\Delta V}{V_0} \] Substituting \( \beta = 49 \times 10^{-5} \, \text{K}^{-1} \) and \( \Delta T = 30^\circ \text{C} \), we get: \[ \frac{\Delta \rho}{\rho} = -\beta \Delta T = -49 \times 10^{-5} \times 30 = -2.5 \times 10^{-2} \]
Thus, the correct answer is (d).