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Chapter 17: Problem 30

How does the density of copper that is just above its melting temperature of \(1356 \mathrm{~K}\) compare to that of copper at room temperature?

Short Answer

Expert verified
Answer: The density of copper just above its melting temperature (8.86 g/cm³) is slightly lower than its density at room temperature (8.92 g/cm³).

Step by step solution

01

Review the formula for density changes with temperature

The formula for estimating the change in density of a given substance due to a change in temperature is given by: Density at temperature \(T_2\) = Density at temperature \(T_1\) × \((1 - \beta \times \Delta T)\) Here, \(\beta\) is the coefficient of volume expansion, and \(\Delta T = T_2 - T_1\) is the difference in temperature.
02

Gather the necessary values

To apply the formula, we need the following values: 1. Density of copper at room temperature: Given as \(\rho_1 = 8.92 g/cm^3\) 2. Coefficient of volume expansion of copper: \(\beta = 5 \times 10^{-5} K^{-1}\) 3. Room temperature: \(T_1 = 298K\) (approximately 25 °C) 4. Copper's melting temperature: \(T_2 = 1356 K\)
03

Calculate the change in density

Using the values above, we can calculate the density of copper just above its melting temperature (\(\rho_2\)): \(\Delta T = T_2 - T_1 = 1356K - 298K = 1058K\) \(\rho_2 = \rho_1 × (1 - \beta \times \Delta T)\) \(\rho_2 = 8.92 g/cm^3 × (1 - (5 × 10^{-5} K^{-1}) × (1058K))\)
04

Find the density of copper just above its melting temperature

Now, we can calculate the value of \(\rho_2\): \(\rho_2 = 8.92 g/cm^3 × (1 - (5 × 10^{-5} K^{-1}) × (1058K)) \approx 8.86 g/cm^3\)
05

Compare the densities

Comparing the two densities, we observe that the density of copper just above its melting temperature (\(\rho_2 = 8.86 g/cm^3\)) is slightly lower than its density at room temperature (\(\rho_1 = 8.92 g/cm^3\)). This is expected as substances typically expand when heated, causing a decrease in density.

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Most popular questions from this chapter

Express each of the following temperatures in degrees Celsius and in kelvins. a) \(-19^{\circ} \mathrm{F}\) b) \(98.6^{\circ} \mathrm{F}\) c) \(52^{\circ} \mathrm{F}\)

a) Suppose a bimetallic strip is constructed of copper and steel strips of thickness \(1.0 \mathrm{~mm}\) and length \(25 \mathrm{~mm},\) and the temperature of the strip is reduced by \(5.0 \mathrm{~K}\). Determine the radius of curvature of the cooled strip (the radius of curvature of the interface between the two strips). b) If the strip is \(25 \mathrm{~mm}\) long, how far is the maximum deviation of the strip from the straight orientation?

Suppose a bimetallic strip is constructed of two strips of metals with linear expansion coefficients \(\alpha_{1}\) and \(\alpha_{2}\), where \(\alpha_{1}>\alpha_{2}\) a) If the temperature of the bimetallic strip is reduced by \(\Delta T\), what way will the strip bend (toward the side made of metal 1 or the side made of metal 2)? Briefly explain. b) If the temperature is increased by \(\Delta T\), which way will the strip bend?

You are designing a precision mercury thermometer based on the thermal expansion of mercury \(\left(\beta=1.8 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\right)\) which causes the mercury to expand up a thin capillary as the temperature increases. The equation for the change in volume of the mercury as a function of temperature is \(\Delta V=\beta V_{0} \Delta T\) where \(V_{0}\) is the initial volume of the mercury and \(\Delta V\) is the change in volume due to a change in temperature, \(\Delta T .\) In response to a temperature change of \(1.0^{\circ} \mathrm{C}\), the column of mercury in your precision thermometer should move a distance \(D=1.0 \mathrm{~cm}\) up a cylindrical capillary of radius \(r=0.10 \mathrm{~mm} .\) Determine the initial volume of mercury that allows this change. Then find the radius of a spherical bulb that contains this volume of mercury.

A solid cylinder and a cylindrical shell, of identical radius and length and made of the same material, experience the same temperature increase \(\Delta T .\) Which of the two will expand to a larger outer radius?
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