To calculate the density of mercury at 70 °C, we can use the formula for the density change due to thermal expansion. The density of a material changes with temperature according to the following relationship:
\[ \rho_T = \rho_0 (1 - \beta(T - T_0)) \]
Where:
- \(\rho_T\) is the density at temperature \(T\),
- \(\rho_0\) is the initial density at the initial temperature \(T_0\),
- \(\beta\) is the volumetric coefficient of expansion,
- \(T\) is the final temperature,
- \(T_0\) is the initial temperature.
In this case:
- The initial density \(\rho_0 = 1.07 \times 10^4 , \text{kg/m}^3\),
- The volumetric coefficient of expansion \(\beta = 200 \times 10^{-6} , \text{°C}^{-1}\),
- The initial temperature \(T_0 = 0 , \text{°C}\),
- The final temperature \(T = 70 , \text{°C}\).
Now we can plug these values into the equation:
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Calculate the temperature difference: \[ T - T_0 = 70 , \text{°C} - 0 , \text{°C} = 70 , \text{°C} \]
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Substitute the values into the density formula: \[ \rho_{70} = 1.07 \times 10^4 , \text{kg/m}^3 \times \left(1 - 200 \times 10^{-6} \times 70\right) \]
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Calculate \(200 \times 10^{-6} \times 70\): \[ 200 \times 10^{-6} \times 70 = 0.014 \]
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Substitute back into the density equation: \[ \rho_{70} = 1.07 \times 10^4 , \text{kg/m}^3 \times (1 - 0.014) \] \[ \rho_{70} = 1.07 \times 10^4 , \text{kg/m}^3 \times 0.986 \]
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Calculate the density: \[ \rho_{70} = 1.07 \times 10^4 \times 0.986 \approx 10553.62 , \text{kg/m}^3 \]
Thus, the density of mercury at 70 °C is approximately: \[ \boxed{10553.62 , \text{kg/m}^3} \]