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.

Step by step solution

01

Calculate total change in volume

We are given that the mercury column rises 1 cm (\(D = 1.0\) cm) up a cylindrical capillary of radius \(r = 0.1\) mm. We can use this to find the total change in volume due to the temperature change. The formula for the volume of a cylinder is \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height. For our situation, the height is the distance the mercury rises, which is \(D\). So, \(\Delta V = \pi r^2 D\). \(\Delta V = \pi (0.1 \cdot 10^{-1}\mathrm{~cm})^2 (1.0\mathrm{~cm}) = \pi \cdot 10^{-3}\mathrm{~cm}^3\)

02

Calculate initial volume of mercury, \(V_0\)

Now we want to find the initial volume of mercury, \(V_0\). We can use the equation provided for change in volume: \(\Delta V = \beta V_0 \Delta T\). We are given that \(\Delta T = 1.0 ^\circ\mathrm{C}\) Solving for \(V_0\), we get \(V_0 =\frac{\Delta V}{\beta \Delta T} =\frac{\pi \cdot 10^{-3}\mathrm{~cm}^3}{(1.8 \cdot 10^{-4}\ ^{\circ}\mathrm{C}^{-1})(1.0 ^\circ\mathrm{C})} = \frac{5.55\ldots}{1.8 \cdot 10^{-4} }\mathrm{~cm}^3\) \(V_0 \approx 30.83\mathrm{~cm}^3\) So, the initial volume of mercury is approximately \(30.83\mathrm{~cm}^3\).

03

Calculate the radius of spherical bulb

Finally, we want to find the radius of the spherical bulb that contains this volume of mercury. The volumne of a sphere is given by the formula, \(V = \frac{4}{3}\pi r^3\). We can solve for \(r\), the radius of the sphere as follows: \(r = \sqrt[3]{\frac{3V}{4\pi}} = \sqrt[3]{\frac{3 \cdot 30.83\mathrm{~cm}^3}{4\pi}} \approx \sqrt[3]{7.74\ldots} \approx 1.98\mathrm{~cm}\) So, the radius of the spherical bulb that contains the initial volume of mercury is approximately \(1.98\mathrm{~cm}\).

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