Q199
General SciencePhysics
The density of water at 20°C is 998 kg m$^{-3}$ and that at 40°C is 992 kg m$^{-3}$. The coefficient of the cubical expansion of water is
- a.$2 \times 10^{-4}\ ^\circ C^{-1}$
- b.$6 \times 10^{-4}\ ^\circ C^{-1}$
- c.$3.00 \times 10^{-4}\ ^\circ C^{-1}$
- d.$4 \times 10^{-4}\ ^\circ C^{-1}$
Answer: (C) $3.00 \times 10^{-4}\ ^\circ C^{-1}$
Mass is constant, so density falls as volume grows: $\rho_{20}=\rho_{40}(1+\gamma\Delta T)$. Hence $\gamma=\dfrac{\rho_{20}-\rho_{40}}{\rho_{40}\,\Delta T}=\dfrac{998-992}{992\times20}=\dfrac{6}{19840}\approx3.0\times10^{-4}\ {}^\circ C^{-1}$. Using $\rho_{20}$ instead of $\rho_{40}$ in the denominator changes the result only in the third significant figure. Option (B) merely echoes the density difference of 6 as a distractor. Water is unusual in that it contracts when heated from 0°C to 4°C and has its maximum density (1000 kg m$^{-3}$) at 4°C.