# What percent of a sample of nitrogen must be allowed to escape if its temperature, pressure and volume are to be changed from $220{}^\circ C$, 3 atm and 1.65 litres to $110{}^\circ C$, 0.7 atm and 1.00 litre respectively?(A)- 81.8%(B)- 71.8%(C)- 76.8%(D)- 86.8%

Verified
118.8k+ views
Hint: Combining Boyle’s law, Charle’s law, Gay-Lussac’s law, and Avagadro’s law gives us the Combined Gas law which can combine into one proportion as-
$V\propto \dfrac{T}{P}$
Removing the proportionality and inserting a constant,
$\dfrac{PV}{T}=C$
This clearly says that as the pressure rises, the temperature also rises and vice versa.
Therefore, the ideal gas equation is given as-
$PV=nRT$
Where P = pressure of the gas;
V = volume of the gas;
n = number of moles;
T = absolute temperature; and
R = Ideal gas constant, also known as Boltzmann constant $=0.082057\text{ L atm K}{{\text{ }}^{-1}}\text{ mo}{{\text{l}}^{-1}}$

-According to question,
The initial condition of the system has-
\begin{align} & {{T}_{1}}=220{}^\circ C=493K \\ & {{P}_{1}}=3atm \\ & {{V}_{1}}=1.65l \\ \end{align}
The final condition of the system has-
\begin{align} & {{T}_{2}}=110{}^\circ C=383K \\ & {{P}_{2}}=0.07atm \\ & {{V}_{2}}=1.00l \\ \end{align}
-Using the Ideal Gas equation, let us calculate the number of moles in initial and final conditions,
${{n}_{1}}=\dfrac{{{P}_{1}}{{V}_{1}}}{R{{T}_{1}}}=\dfrac{3\times 1.65}{R\times 493}$
${{n}_{2}}=\dfrac{{{P}_{2}}{{V}_{2}}}{R{{T}_{2}}}=\dfrac{0.7\times 1}{R\times 383}$

-Thus, the fraction of the gas remaining can be given as-
$\dfrac{{{n}_{2}}}{{{n}_{1}}}=\dfrac{0.7\times 1\times 493}{3\times 1.65\times 393}=0.18$
-Converting the fraction into a percentage, we will get
Percentage of the fraction of moles = $0.18\times 100=18%$
-Now, the percent of nitrogen sample left =(100-18)%=82%
Therefore, 82% the sample of nitrogen must be allowed to escape if the temperature, pressure and volume of the system must be changed from $220{}^\circ C$, 3 atm and 1.65 l to $110{}^\circ C$, 0.7 atm and 1 l respectively.

Hence, the correct answer is option A.

Note:
The application of Combined Gas law is that it can be used to explain to the mechanics where pressure, temperature, and volume are affected. For example, in air conditioners and refrigerators, and the formation of clouds.