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During an all-night cram session, a student heats up a one-half liter (0.50 x 10- 3 m3) glass (Pyrex) beaker of cold coffee. Initially, the temperature is 18 °C, and the beaker is filled to the brim. A short time later when the student returns, the temperature has risen to 92 °C. The coefficient of volume expansion of coffee is the same as that of water. How much coffee (in cubic meters) has spilled out of the beak?

Respuesta :

Answer:

Δ[tex]V_{c}[/tex] = 7.415 10⁻⁶ m³   (7.415 10⁻³ L)

Explanation:

This problem must find the change in volume of coffee and pyrex, the expression

                 ΔV = V₀ β ΔT

The expansion coefficients are

Water Volumetry        β = 207 10⁻⁶ C⁻¹

Pyrex Linear                α = 3.2 10⁻⁶ C⁻¹

In general, for solids the solids find the linear expansion coefficient tabulated, but the linear volumetric coefficient is related

                  β = 3 α

Let's calm the volume change for the pyrex glass

                Δ[tex]V_{p}[/tex] = V₀ 3α ΔT

                 Δ[tex]V_{p}[/tex]= 0.50 10⁻³ 3 3.2 10⁻⁶ (92-18)

                Δ[tex]V_{p}[/tex] = 3.552 10⁻⁷ m³

We look for the change in volume for coffee that is equal to that of water

                Δ[tex]V_{w}[/tex] = 0.50 10⁻³ 210 10⁻⁶ (92-18)

                Δ[tex]V_{w}[/tex] = 7.77 10⁻⁶ m³

The amount of spilled coffee is the difference in volumes

              Δ[tex]V_{c}[/tex] =  Δ[tex]V_{w}[/tex] -  Δ[tex]V_{p}[/tex]

             Δ[tex]V_{c}[/tex] = 7.77 10⁻⁶ - 3.552 10⁻⁷

              Δ[tex]V_{c}[/tex] = 7.415 10⁻⁶ m³

Let's reduce to liters

            Δ[tex]V_{c}[/tex] = 7.415 10⁻⁶ m³ (1000 L / 1 m3)

             ΔΔ[tex]V_{c}[/tex] = 7.415 10⁻³ L

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