The density of mercury is [tex]$1.36 \times 10^4 kgm ^{-3}$[/tex] at [tex]$0^{\circ} C$[/tex]. Calculate its value at [tex]$100^{\circ} C$[/tex] and [tex]$22^{\circ} C$[/tex]. Take cubic expansivity of mercury as equal to [tex]$180 \times 10^{-6} K$[/tex].
Answer (2)
The density of mercury at 10 0 ∘ C is approximately 13359.53 k g / m 3 , while at 2 2 ∘ C it is about 13546.36 k g / m 3 . These values were calculated using the cubic expansivity formula for density changes with temperature. The calculations demonstrated the impact of temperature on the density of mercury.
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The density at a different temperature can be calculated using the formula: ρ T = 1 + γ Δ T ρ 0 .
Calculate the density at 10 0 ∘ C using Δ T = 10 0 ∘ C − 0 ∘ C = 10 0 ∘ C .
Calculate the density at 2 2 ∘ C using Δ T = 2 2 ∘ C − 0 ∘ C = 2 2 ∘ C .
The density at 10 0 ∘ C is approximately 13359.53 k g / m 3 , and the density at 2 2 ∘ C is approximately 13546.36 k g / m 3 . 13359.53 k g / m 3 and 13546.36 k g / m 3
Explanation
Problem Analysis We are given the density of mercury at 0 ∘ C and its cubic expansivity. We need to find the density at 10 0 ∘ C and 2 2 ∘ C . The density changes with temperature due to thermal expansion. We will use the formula relating density and temperature change.
Volume Expansion The formula for the volume at a certain temperature T is given by: V T = V 0 ( 1 + γ Δ T ) where V 0 is the initial volume, γ is the cubic expansivity, and Δ T is the change in temperature.
Density and Volume Since density ρ is inversely proportional to volume V ( ρ = V m ), we can write the density at temperature T as: ρ T = 1 + γ Δ T ρ 0 where ρ 0 is the initial density.
Density at 100°C To find the density at 10 0 ∘ C , we use Δ T = 10 0 ∘ C − 0 ∘ C = 10 0 ∘ C . Substituting the given values, we get: ρ 100 = 1 + ( 180 × 1 0 − 6 ) × 100 1.36 × 1 0 4 ρ 100 = 1 + 0.018 1.36 × 1 0 4 ρ 100 = 1.018 1.36 × 1 0 4 ρ 100 ≈ 13359.53 k g / m 3
Density at 22°C To find the density at 2 2 ∘ C , we use Δ T = 2 2 ∘ C − 0 ∘ C = 2 2 ∘ C . Substituting the given values, we get: ρ 22 = 1 + ( 180 × 1 0 − 6 ) × 22 1.36 × 1 0 4 ρ 22 = 1 + 0.00396 1.36 × 1 0 4 ρ 22 = 1.00396 1.36 × 1 0 4 ρ 22 ≈ 13546.36 k g / m 3
Final Answer Therefore, the density of mercury at 10 0 ∘ C is approximately 13359.53 k g / m 3 , and the density at 2 2 ∘ C is approximately 13546.36 k g / m 3 .
Examples
Understanding how density changes with temperature is crucial in many real-world applications. For example, in designing bridges and buildings, engineers must account for the expansion and contraction of materials due to temperature variations to ensure structural integrity. Similarly, in fluid dynamics, the temperature-dependent density of fluids affects their flow behavior, which is important in designing efficient pipelines and hydraulic systems. In meteorology, the density of air, which varies with temperature and pressure, plays a key role in weather patterns and atmospheric circulation.
