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A mercury thermometer is constructed as
shown. The capillary tube has a diameter
of 0.0045 cm, and the bulb has a diameter of
0.24 cm. Neglecting the expansion of the glass, find
the change in height of the mercury column
for a temperature change of 36 ◦C. The
volume expansion coefficient for mercury is
0.000182 (◦C)^−1
Answer in units of cm.

A mercury thermometer is constructed as shown The capillary tube has a diameter of 00045 cm and the bulb has a diameter of 024 cm Neglecting the expansion of th class=

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mberisso

Answer:

The change in height of the mercury is approximately  2.981 cm

Explanation:

Recall that the formula for thermal expansion in volume is:

[tex]\frac{\Delta V}{V_0} =\alpha_V\, \Delta\, T\\\Delta V = V_0\,\, \alpha_V\,\,\Delta C[/tex]

from which we solved for the change in volume [tex]\Delta V[/tex] due to a given change in temperature [tex]\Delta T[/tex]

We can estimate the initial volume of the mercury in the spherical bulb of diameter 0.24 cm ( radius R = 0.12 cm) using the formula for the volume of a sphere:

[tex]V_0=\frac{4}{3} \pi \, R^3\\V_0=\frac{4}{3} \pi \, (0.12\,cm)^3\\V_0=0.007238\,cm^3[/tex]

Therefore, the change in volume with a change in temperature of 36°C becomes:

[tex]\Delta V = V_0\,\, \alpha_V\,\,\Delta C\\\Delta V = 0.007238229\, cm^3\,(0.000182)\,(36)\\\Delta V=0.0000474248\, cm^3[/tex]

Now, we can use this difference in volume, to estimate the height of the cylinder of mercury with diameter 0.0045 cm (radius r= 0.00225 cm):

[tex]V_{cyl}=\pi r^2\,h\\h =\frac{V_{cyl}}{\pi r^2} \\h=\frac{0.0000474248\, cm^3}{\pi \, (0.00225\,cm)^2} \\h=2.98188 \,cm[/tex]

The change in height of the mercury column for a temperature change of 36°C will be  2.981 cm.

What is thermal expansion?

Thermal expansion iis the tendency of matter to alter its form, area, volume, and density in response to a change in temperature.

The given data in the problem is;

d is the diameter of capillary tube = 0.0045 cm

D is the diameter of an bulb=0.24 cm.

h is the change in height of the mercury column=?

h[tex]\triangle t[/tex] is the temperature change = 36 ◦C.

[tex]\triangle v[/tex] is the volume expansion =0.000182 (◦C)⁻¹

The formula for the thermal expansion is given as;

[tex]\rm \frac{\triangle v }{v_0} = \alpha v \triangle T \\\\ \triangle v = V_0 \alpha v \triangle T \\\\[/tex]

The initial volume of the mercury in the spherical bulb  will be;

[tex]\rm v_0 = \frac{4}{3}\pi r^3 \\\\ \rm v_0 = \frac{4}{3}\pi (0.12)^3 \\\\ \rm v_0 = 0.007298 \ cm^3[/tex]

The change in the volume will be;

[tex]\triangle v = 0.007238 \times (0.000182)(36) \\\\ \triangle v = 0.000047428 \ cm^3[/tex]

Now, we can use this difference in volume, to estimate the height of the cylinder of mercury with diameter 0.0045 cm (radius r= 0.00225 cm.

[tex]\rm V_{cycle }= \pi r^2 h \\\\ \rm h = \frac{v_{cycle}}{\pi r^2h} \\\\ \rm h= 2.98188 cm[/tex]

Hence the change in height of the mercury column for a temperature change of 36°C will be  2.981 cm.

To learn more about the thermal expansion refer to the lnk;

https://brainly.com/question/26046548