Question

The invariable volume of a brass sphere is \[1000\]\[cc\] at \[{0^0}C\]. Its volume at \[{100^0}C\] is :
\[\left( {\alpha = 18 \times {{10}^{ - 6}}{/^0}C} \right)\]
A) \[1000\]\[cc\]
B) \[994.6\]\[cc\]
C) \[1005.4\]\[cc\]
D) \[100.54\]\[cc\]

Answer 1

Hint: We know that material is a good conductor of heat and electricity. When temperature increases, then conductivity also changes. When temperature changes, the volume of material will change.

Formula used: We are calculating volume at \[{100^0}C\] by this formula:
Final volume = initial volume \[\times \left( {1 + \gamma \Delta T} \right)\]. In question, \[\alpha \] is given. The relation between \[\alpha \] and \[\gamma \] is given as \[\gamma = 3\alpha \].

Complete step by step solution:
Given: Volume of a brass sphere at \[{0^0}C\] =\[1000\]\[cc\],\[\alpha = 18 \times {10^{ - 6}}{/^0}C\]
At \[{100^0}C\], volume of a brass sphere is given by following formula
Final volume = initial volume \[\times \left( {1 + \gamma \Delta T} \right)\]
Final volume = initial volume \[\times \left( {1 + 3\alpha \Delta T} \right)\]
Here: Initial volume is define as volume at \[{0^0}C\]=\[1000\]\[cc\]
Final volume is define as volume at \[{100^0}C\]
So, we can calculate, Final volume = \[1000 \times \left( {1 + 3 \times 8 \times {{10}^{ - 6}} \times 100} \right)\]
\[ \Rightarrow Final{\text{ }}volume = 1005.4\]
Hence, volume at \[{100^0}C\] = \[1005.4\] \[cc\]

Hence, the correct option is (C).

Additional information: Thermal expansion refers to a fractional change in size of a material to a change in temperature. This fractional change of size may be one of the following type:
(1) change in length compared to original length is called linear expansion
(2) change in the area compared to its original area is called areal expansion.
(3) change in volume compared to its original volume is called volumetric expansion. It is also called cubical expansion.

A coefficient of thermal expansion is a ratio. This coefficient is given as, the ratio of the fractional change in length, area or original volume of a material to its change in temperature.

Note: Students must consider that here the value of linear expansion, \[\alpha \] is given .In formula cubic expansion, \[\gamma \] is used. We know that the relation between cubic and linear expansion is given by Cubical expansion = 3 \[\times\] linear expansion. Students first find the value of cubic expansion. Then put this value in a formula to find the final volume.