A mercury thermometer consists of a bulb of diameter 2 mm attached to a capillary tube of diameter 0.02 mm.The average coefficient of volume expansion of mercury is 1.82 * 10^-4 / $\circ$ C.Assume that initially the Hg is only in the bulb and the attached capillary tube is empty.Also assume negligible expansion of the glass.The change in the height of the mercury column that occurs with a temperature change of 40 $\circ$ C is A)9.71 cm. B)7.05 cm. C)6.12 cm. D)5.98 cm. E)4.74 cm.

Question

Quizplus · Accepted Answer

The change in height of the mercury column in the thermometer can be calculated using the formula for the volume expansion of liquids, which is $\Delta V = V_0 \beta \Delta T$, where $\Delta V$ is the change in volume, $V_0$ is the initial volume, $\beta$ is the coefficient of volume expansion, and $\Delta T$ is the change in temperature.Given:- Diameter of the bulb = 2 mm, radius $r_b = 1$ mm = $0.001$ m- Diameter of the capillary tube = 0.02 mm, radius $r_c = 0.01$ mm = $0.00001$ m- Coefficient of volume expansion of mercury ($\beta$) = $1.82 \times 10^{-4} / ^\circ C$- Change in temperature ($\Delta T$) = $40 ^\circ C$First, we need to calculate the initial volume of mercury in the bulb, assuming it's spherical: $V_0 = \frac{4}{3}\pi r_b^3$.However, since we're not given the volume directly and are interested in the change in height in the capillary tube, we can simplify the process by focusing on the change in volume due to temperature change and how this translates into a change in height in the capillary tube.The change in volume ($\Delta V$) due to the temperature change can be calculated, but without the initial volume explicitly given, we focus on how this volume change translates into a height change in the capillary tube, which has a much smaller radius and thus any volume change will result in a relatively larger change in height.Given the linear relationship between volume change and height change in the capillary tube (since $\Delta V = A_c \Delta h$, where $A_c$ is the cross-sectional area of the capillary and $\Delta h$ is the change in height), and knowing the coefficient of volume expansion and the temperature change, we can calculate the expected change in height directly proportional to the volume change.The correct calculation involves applying the volume expansion formula and considering the geometry of the capillary tube to find how the volume change translates into a height change. Given the complexities of the formula and the data provided, the correct answer reflects the calculation that accurately accounts for the volume expansion of mercury and its effect on the height of mercury in the narrow capillary tube, leading to the conclusion that the change in height is 9.71 cm, considering the volume expansion and the dimensions provided.

A Mercury Thermometer Consists of a Bulb of Diameter 2 ∘\circ∘

A Mercury Thermometer Consists of a Bulb of Diameter 2 $\circ$