Question

What is the density of \[{N_2}\] gas at \[227^\circ C\] and 5.00 atm pressure ? (\[R = 0.0821atm{K^{ - 1}}mo{l^{ - 1}}\])
A) \[0.29\dfrac{g}{{ml}}\]
B) \[1.40\dfrac{g}{{ml}}\]
C) \[2.81\dfrac{g}{{ml}}\]
D) \[3.41\dfrac{g}{{ml}}\]

Answer 1

Hint:The density of a gas is dependent on the temperature and pressure of the surroundings, as the temperature of the surroundings increases keeping other factors constant the density will decrease and if the pressure is increasing keeping other factors constant then the density increases. There is a relation \[\rho = \dfrac{{Pm}}{{RT}}\], to find the density of a gas at a given temperature and pressure .

Step by step solution :-
From the ideal gas equation we write that \[PV = nRT\]
Where ; P = pressure of the gas = 5.00 atm
V = volume of the gas
n = number of moles ; \[n = \dfrac{{Mass\;of\;{N_2}\;}}{{molar\;mass\;of{N_2}}}\]
R = gas constant = \[0.0821atm{K^{ - 1}}mo{l^{ - 1}}\]
T = temperature on Kelvin scale. ; \[T = (273 + 227)\;K\]= \[500\;K\]
\[density\;of\;gas\;(\rho ) = \dfrac{{mass\;ofgas}}{{volume\;of\;gas}}\]
Density of the gas (\[\rho \]) = \[\dfrac{{mass}}{{volume}}\]= \[\dfrac{M}{V}\]
\[ \Rightarrow \rho = \dfrac{M}{V}\]……..eq. (1)
From ideal gas equation \[PV = nRT\]
\[ \Rightarrow PV = \dfrac{M}{m}RT\]
Sending the volume term to right hand side , we get
\[ \Rightarrow Pm = \dfrac{M}{V}RT\]
Putting the value obtained from eq.(1) , we get
\[ \Rightarrow Pm = \rho RT\]
Equating it with density term , we get
\[ \Rightarrow \rho = \dfrac{{Pm}}{{RT}}\]
Putting all the values , we get
\[ \Rightarrow \rho = \dfrac{{5 \times 28}}{{0.0821 \times 500}}\dfrac{g}{{ml}}\]
Simplifying the terms we get
\[ \Rightarrow \rho = 3.41\;\dfrac{g}{{ml}}\]
It means that , the density of \[{N_2}\] gas at \[227^\circ C\] and 5.00 atm pressure is \[3.41\dfrac{g}{{ml}}\].

Hence , option ( D ) is the correct answer.

Note:- In order to find the volume of the gas take the volume of the container in which gas is kept.
One of the properties of the gases is “ gases take the volume of the container in which they are kept.”

The molecules of the gases are continuously in motion as molecules constantly collide with the walls of the container due to which their momentum changes , this change in momentum is transferred to the walls of the container , consequently pressure is exerted by gas molecules on the walls of the container.