# What is the density of ${{N}_{2}}$ gas at $227{}^\circ C$and 5.00 atm pressure?$(R=0.082\text{ }L\text{ }atm\text{ }{{K}^{-1}}\text{ }mo{{l}^{-1}})$ (A)- 1.40 g/mL(B)- 2.81 g/mL(C)- 3.41 g/mL(D)- 0.29 g/mL

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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 $=0.082057\text{ L atm K}{{\text{ }}^{-1}}\text{ mo}{{\text{l}}^{-1}}$

-As we know that number of moles (n) $=\dfrac{\text{given mass (w)}}{\text{molar mass(M)}}$ , so substituting this value in the ideal gas equation we get,
$PV=\dfrac{w}{M}(RT)$
Rearranging the equation to get the value of PM,
$PM=\dfrac{w}{V}(RT)$
As we know, $Density(d)=\dfrac{mass(m)}{volume(V)}$
The equation can now be written as,
\begin{align} & PM=dRT \\ & d=\dfrac{PM}{RT} \\ \end{align}
According to question,
P = 5 atm
T = 500K
M = 28 g/mol
-Inserting these values in the density formula, we get
$d=\dfrac{PM}{RT}=\dfrac{5\times 28}{0.0821\times 500}=3.41g/ml$

So, the correct answer is option D.

Note:
Nitrogen which is the principal component of the earth’s atmosphere 78.084% percent by volume is a colourless, odourless, tasteless and relatively inert gas. Nitrogen gas does not support combustion nor reacts with most of the substances. Nitrogen gas being stable has many industrial uses. Since it is chemically inactive with most of the elements, hence is used as a preservative to prevent oxidation. When chilled to its liquid state, is widely used as a refrigerant in medical, chemical and manufacturing industries. Nitrogen is also an important element in the synthesis of many organic compounds, as it serves as a limiting nutrient in many ecosystems.