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Zorluk: ZorThermal Expansion of Liquids and Anomalous Expansion of Water

A glass flask with an internal volume of 400 cm3400\text{ cm}^3 is filled to the brim with a liquid at 25∘C25^\circ\text{C}. The system is heated to 75∘C75^\circ\text{C}, causing 9.0 cm39.0\text{ cm}^3 of liquid to overflow. Given that the linear expansivity of glass is 1.0×10−5 K−11.0 \times 10^{-5}\text{ K}^{-1}, what is the real cubic expansivity of the liquid in units of 10−4 K−110^{-4}\text{ K}^{-1}?

Cevap: 4.8 10^-4 K^-1

Cevap

The real cubic expansivity of the liquid is 4.8×10−4 K−14.8 \times 10^{-4}\text{ K}^{-1}, which equals 4.84.8 in units of 10−4 K−110^{-4}\text{ K}^{-1}.
The real cubic expansivity of a liquid equals the sum of its apparent cubic expansivity and the volumetric expansivity of the vessel (γr=γa+γv\gamma_r = \gamma_a + \gamma_v). Calculating the apparent cubic expansivity yields γa=9.0400×50=4.5×10−4 K−1\gamma_a = \frac{9.0}{400 \times 50} = 4.5 \times 10^{-4}\text{ K}^{-1}. The cubic expansivity of the glass container is γv=3×1.0×10−5=0.3×10−4 K−1\gamma_v = 3 \times 1.0 \times 10^{-5} = 0.3 \times 10^{-4}\text{ K}^{-1}. Adding these values gives a real cubic expansivity of γr=4.8×10−4 K−1\gamma_r = 4.8 \times 10^{-4}\text{ K}^{-1}, which equals 4.84.8 in units of 10−4 K−110^{-4}\text{ K}^{-1}.

Adım Adım Çözüm

1
Calculate the temperature change
ΔT=75∘C−25∘C=50 K\Delta T = 75^\circ\text{C} - 25^\circ\text{C} = 50\text{ K}
Thermal expansion depends directly on the change in temperature.
2
Determine the apparent cubic expansivity of the liquid
γa=ΔVaV0ΔT=9.0 cm3400 cm3×50 K=4.5×10−4 K−1\gamma_a = \frac{\Delta V_a}{V_0 \Delta T} = \frac{9.0\text{ cm}^3}{400\text{ cm}^3 \times 50\text{ K}} = 4.5 \times 10^{-4}\text{ K}^{-1}
Apparent cubic expansivity is determined from the volume of liquid that overflows relative to the initial volume and temperature increase.
3
Calculate the cubic expansivity of the glass container
γv=3α=3×1.0×10−5 K−1=0.3×10−4 K−1\gamma_v = 3\alpha = 3 \times 1.0 \times 10^{-5}\text{ K}^{-1} = 0.3 \times 10^{-4}\text{ K}^{-1}
The volumetric expansivity of an isotropic solid container is three times its linear expansivity.
4
Compute the real cubic expansivity of the liquid
γr=γa+γv=4.5×10−4 K−1+0.3×10−4 K−1=4.8×10−4 K−1\gamma_r = \gamma_a + \gamma_v = 4.5 \times 10^{-4}\text{ K}^{-1} + 0.3 \times 10^{-4}\text{ K}^{-1} = 4.8 \times 10^{-4}\text{ K}^{-1}
Real expansivity accounts for both the observed apparent expansion of the liquid and the expansion of the vessel containing it.

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