Question

One mole of ${N_2}{O_4}$(g) at $300K$ is kept in a closed container under $1atm$. It is heated to $600K$ when $20\%$ by mass of ${N_2}O$(g) decomposes to $N{O_2}$(g). The resultant pressure is:
A.1.2 atm
B.2.4 atm
C.2.0 atm
D.1.0 atm

Answer 1

We need to know the relationship between Pressure, Volume, temperature and accordingly find the pressure. The relationship is given by the Ideal Gas Law. It is a combination of various gas laws such as Boyle’s Law, Charles’s Law, Gay Lussac’s Law and Avogadro’s Law. The equation for the Ideal Gas Law is $PV = nRT$,where $P$ is the pressure ,$V$ is the Volume, $n$ is the number of moles ,$R$ is the Gas Constant and $T$ is the temperature.

Complete step by step answer:
We have to compare the temperature and pressure after the decomposition as given in the question keeping the volume constant as the decomposition does not change the volume inside a closed container.
We know the temperature before the reaction (${T_1}$) = 300K
The temperature after the reaction (${T_2}$) = 600 K
The pressure inside the closed container before the reaction (${P_1}$) = 1atm
We now need to calculate the number of moles inside the container before the reaction (${n_1}$) and the number of moles inside the container after the reaction (${n_2}$) use it in the Ideal Gas Law Equation.
The overall equation for the decomposition Is:
${N_2}{O_4}$→ $2$ $N{O_2}$
At ${T_1}$ = 300K,
${n_1}$ of ${N_2}{O_4} = 1mol$
${n_1}$ of $N{O_2} = 0$
At ${T_2}$= 600 K,
${n_2}$ of ${N_2}{O_4} = 1 - 0.2 = 0.8mol$(since $20\%$ by mass decomposes)
${n_2}$ of $N{O_2} = 2 \times 0.2 = 0.4mol$
Total number of moles before the reaction,
${n_1} = 1$
Total number of moles after the reaction,
${n_2} = 0.8 + 0.4 = 1.2$
Comparing both ideal gas equations before and after the reactions:
$\dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{{n_1}}}{{{n_2}}}\dfrac{{{T_1}}}{{{T_2}}}$
$\dfrac{1}{{{P_2}}} = \dfrac{1}{{1.2}} \times \dfrac{{300}}{{600}}$
${P_2} = 2.4$
Hence, the correct option is option (B).

Note:
In order to conclude whether a gas is ideal or not, four variables are taken into consideration and a law is made out of it known as The Ideal Gas law. The four variables are Pressure ($P$), Volume ($V$), number of moles of gas $\left( n \right)$ and Temperature ($t$). It gives us a simple equation-the ideal gas equation as $PV = nRT$. For an ideal gas, $PV/nRT = 1$.