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A given amount of gas occupies 1000cc at ${27^ \circ }\,C$ and $1200\,cc$ and ${87^ \circ }\,C$ . What is its volume coefficient of expansion
(A) ${\dfrac{1}{{273}}^0}\,{C^{ - 1}}$
(B) ${\dfrac{1}{{173}}^0}\,{C^{ - 1}}$
(C) ${173^0}\,{C^{ - 1}}$
(D) ${273^0}\,{C^{ - 1}}$

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Last updated date: 29th May 2024
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Answer
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Hint: Use the formula of the volume coefficient of expansion given below, and substitute the value of the temperature and the volume before expansion and after expansion in it. The simplification of the obtained equation provides the answer.

Formula used:
The formula of the volume coefficient of expansion is given by
$\alpha = \dfrac{{{V_2} - {V_1}}}{{{V_1}{t_2} - {V_2}{t_1}}}$
Where $\alpha $ is the volume coefficient of expansion, ${V_2}$ is the volume of the gas after expansion, ${V_1}$ is the volume of the gas before expansion, ${t_1}$ is the first temperature of the gas and ${t_2}$ is the second temperature of the gas.

Complete step by step solution
It is given that the
Initial volume of the gas, ${V_1} = 1000\,cc$
Final volume of the gas, ${V_2} = 1200\,cc$
Initial temperature of the gas, ${t_1} = {27^ \circ }\,C$
Final temperature of the gas, ${t_2} = {87^ \circ }\,C$
By using the formula of the volume coefficient of expansion,
$\alpha = \dfrac{{{V_2} - {V_1}}}{{{V_1}{t_2} - {V_2}{t_1}}}$
Substituting the values of the initial and the final temperature and also the initial and the final volume of the gas.
$\alpha = \dfrac{{1200 - 1000}}{{\left( {1000 \times 87} \right) - \left( {1200 \times 27} \right)}}$
By simplifying the above equation, we get
$\alpha = \dfrac{{200}}{{87000 - 32400}}$
By doing basic arithmetic operation, we get
$\alpha = \dfrac{{200}}{{54600}}$
By further simplification,
$\alpha = \dfrac{1}{{273}}{\,^0}\,{C^{ - 1}}$
Hence the value of the coefficient of expansion is obtained as $\dfrac{1}{{273}}{\,^0}\,{C^{ - 1}}$ .

Thus the option (A) is correct.

Note: The basic concept behind this question is the gas molecules occupy greater space on heating. Hence when the temperature increases, the volume occupied by the gas also increases. This is because heating causes the molecules in the gas to move further apart.