Question

At what temperature, the sample of neon gas would be heated to double its pressure initial volume of gas is reduced by $15%$ at ${{75}^{\circ }}C$ ?
(A)- ${{592}^{\circ }}C$
(B)- ${{319}^{\circ }}C$
(C)- ${{789}^{\circ }}C$
(D)- None

Answer 1

Hint: In order to find the temperature, given the relation of the pressure and volume of the final and initial state of the neon gas in the system, the ideal gas equation is used.

Complete step by step answer:
It is given that, the neon gas present at temperature ${{T}_{1}}={{75}^{\circ }}C=75+273=348K$ has pressure ${{P}_{1}}$ and volume ${{V}_{1}}$ , is heated such that, the volume of the gas decreases by $15%$ of the initial volume. So, the final volume is ${{V}_{2}}={{V}_{1}}-15%\,{{V}_{1}}={{V}_{1}}-\dfrac{15}{100}\,{{V}_{1}}=\dfrac{85}{100}{{V}_{1}}$
Also, it is given that on heating the gas, the pressure is doubled. So, the final pressure ${{P}_{2}}=2{{P}_{1}}$.
Then, in order to find the increased temperature, that is, the final temperature, ${{T}_{2}}$ , we will make use of the ideal gas equation. It is given as: $PV=nRT$ , where R is the gas constant and n is the number of moles of gas, are constants.
So, for the neon gas having two different temperature, pressure and volume will be related as: $\dfrac{{{P}_{1}}{{V}_{1}}}{{{T}_{1}}}=\dfrac{{{P}_{2}}{{V}_{2}}}{{{T}_{2}}}$
Substituting the values of ${{P}_{1}},{{V}_{1}},{{T}_{1}}\,\,and\,\,{{P}_{2}},{{V}_{2}},\,{{T}_{2}}$ in the above equation, we will get:
$\dfrac{{{P}_{1}}{{V}_{1}}}{348}=\dfrac{2{{P}_{1}}}{{{T}_{2}}}\times \left( \dfrac{85{{V}_{1}}}{100} \right)$
${{T}_{2}}=\dfrac{348\times 2{{P}_{1}}\times 85{{V}_{1}}}{{{P}_{1}}{{V}_{1}}\times 100}$
${{T}_{2}}=\dfrac{348\times 2\times 85}{100}=591.6\,K$
${{T}_{2}}=591.6\,-273={{318.6}^{\circ }}C\approx {{319}^{\circ }}C$

Therefore, the temperature at which neon gas must be heated is option (B)- ${{319}^{\circ }}C$.

Additional information: From the gas equation, various other relationships can be obtained like Boyle's law, Charles’s law and Avogadro's law.

Note: The temperature obtained from the gas equation is converted from the standard unit of Kelvin to Celsius. Also, the ideal gas equation used, usually predicts the approximate behaviour of the real gases. But, at low temperature and higher pressure, it deviates from the ideal gas behaviour.