Question 673: Thermodynamics & KTG

Question

A thermometer bulb has volume $10^{-6} \mathrm{~m}^3$ and cross-section of the stem is $0.002 \mathrm{~cm}^2$. The bulb is filled with mercury at $0^{\circ} \mathrm{C}$. If the thermometer reads temperature as $100^{\circ} \mathrm{C}$, then the length of mercury column is (coefficient of cubical expansion of mercury $=18 	imes 10^{-5} /{ }^{\circ} \mathrm{C}$ )

Accepted Answer

We are tasked with finding the length of the mercury column in the stem after heating the thermometer.
Step 1: Known values
Volume of mercury bulb:
$   V = 10^{-6}\,	ext{m}^3   $
Cross-sectional area of stem:
$   A = 0.002\, 	ext{cm}^2 = 0.002 	imes 10^{-4}\,	ext{m}^2 = 2	imes 10^{-7}\,	ext{m}^2   $
Temperature rise:
$   \Delta T = 100^\circ 	ext{C}   $
Cubical expansion coefficient of mercury:
$   \beta = 18 	imes 10^{-5}\, /^\circ 	ext{C}   $
Step 2: Change in volume of mercury
$ \Delta V = V \beta \Delta T $
$ \Delta V = (10^{-6})(18 	imes 10^{-5})(100) $
$ \Delta V = 18 	imes 10^{-9}\,	ext{m}^3 $
Step 3: Relation to column length
The excess volume enters into the stem of cross-sectional area $A$, forming a length $l$:
$ \Delta V = A \cdot l $
So:
$ l = \frac{\Delta V}{A} $
$ l = \frac{18 	imes 10^{-9}}{2 	imes 10^{-7}} $
$ l = 0.09\,	ext{m} = 9\,	ext{cm} $
Ã¢ÂÂ Final Answer:
The length of mercury column is:
9 cm
(Option C).

Answer

90 cm

Answer

9 mm

Answer

9 cm

Answer

0.9 mm

Thermodynamics & KTG

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