The density of mercury is 13,600 kg/m^3 at 0 °C. Assuming thermal expansion causes a decrease in density with increasing temperature, which of the following is the most reasonable estimate for the density of mercury at 200 °C?

• A. About 12,000 kg/m^3
• B. About 13,600 kg/m^3
• C. About 14,000 kg/m^3
• D. About 13,000 kg/m^3

Answer: (D)

Verified step by step guidance

1. Understand that density (\(\rho\)) is defined as mass (\(m\)) divided by volume (\(V\)), i.e., \(\rho = \frac{m}{V}\). Since the mass of mercury does not change with temperature, any change in density is due to a change in volume caused by thermal expansion.
2. Recall that thermal expansion causes the volume of a substance to increase as temperature increases. This means the volume at 200 °C will be larger than at 0 °C, leading to a decrease in density because density is inversely proportional to volume.
3. Use the concept of volumetric thermal expansion, where the change in volume \(\Delta V\) can be approximated by \(\Delta V = V_0 \beta \Delta T\), where \(V_0\) is the initial volume, \(\beta\) is the volumetric thermal expansion coefficient, and \(\Delta T\) is the change in temperature.
4. Express the new volume at 200 °C as \(V = V_0 (1 + \beta \Delta T)\), and then calculate the new density using \(\rho = \frac{m}{V} = \frac{m}{V_0 (1 + \beta \Delta T)} = \frac{\rho_0}{1 + \beta \Delta T}\), where \(\rho_0\) is the initial density at 0 °C.
5. Since \(\beta\) is positive, the denominator increases with temperature, so the density decreases. Therefore, the density at 200 °C will be somewhat less than 13,600 kg/m\(^3\), making an estimate around 13,000 kg/m\(^3\) reasonable.