Answer
Verified
393k+ views
Hint: Use the ideal gas equation to find the density of the gas. You can assume the mass of the gas and the molecular weight. Express the ideal gas equation in terms of Avogadro’s number and Boltzman constant.
Formula Used:
The density of a gas is given by,
$\rho =\dfrac{{{m}^{'}}}{v}$
Where,
${{m}^{'}}$ is the mass of the gas molecule
$v$ is the volume
The Ideal Gas equation is given by,
$PV=nRT$
Where,
P is the pressure of the gas
V is the volume of the gas
n is the number of moles
T is the temperature
Complete step by step answer:
We can write the ideal gas equation,
$PV=nRT$………………….(1)
Where,
P is the pressure of the gas
V is the volume of the gas
n is the number of moles
T is the temperature
$R=NK$ , N = Avogadro’s number and K = Boltzman Constant
We can replace ‘n’ with the following:
$n=\dfrac{{{m}^{'}}}{M}$
Where,
${{m}^{'}}$ is the mass of the gas
$M$ is the molecular weight of the gas.
Hence, we can write equation (1) in the following way,
$PV=(\dfrac{{{m}^{'}}}{M})RT$
$\Rightarrow P=(\dfrac{{{m}^{'}}}{V})(\dfrac{RT}{M})$
We can write the density of the gas as,
$\rho =\dfrac{{{m}^{'}}}{V}$
Hence, putting this expression we get,
$\Rightarrow P=\dfrac{\rho RT}{M}$
$\Rightarrow \dfrac{PM}{RT}=\rho $
$\Rightarrow \rho =\dfrac{PM}{NKT}$(As, R=NK)
Here, N is Avogadro's number and K is the Boltzmann Constant.
So, finally, we can write,
$\Rightarrow \rho =\dfrac{Pm}{KT}$(As, $m=\dfrac{M}{N}$)
So, the density of the gas is given by,
$\dfrac{Pm}{KT}$
Hence, the correct answer is - (A).
Note: Here, we have assumed that there is an N number of molecules in the gas. Hence, the mass of each molecule is given by,
$m=\dfrac{M}{N}$
This expression also shows that the ideal gas will always have a finite density until it reaches absolute zero temperature.
Formula Used:
The density of a gas is given by,
$\rho =\dfrac{{{m}^{'}}}{v}$
Where,
${{m}^{'}}$ is the mass of the gas molecule
$v$ is the volume
The Ideal Gas equation is given by,
$PV=nRT$
Where,
P is the pressure of the gas
V is the volume of the gas
n is the number of moles
T is the temperature
Complete step by step answer:
We can write the ideal gas equation,
$PV=nRT$………………….(1)
Where,
P is the pressure of the gas
V is the volume of the gas
n is the number of moles
T is the temperature
$R=NK$ , N = Avogadro’s number and K = Boltzman Constant
We can replace ‘n’ with the following:
$n=\dfrac{{{m}^{'}}}{M}$
Where,
${{m}^{'}}$ is the mass of the gas
$M$ is the molecular weight of the gas.
Hence, we can write equation (1) in the following way,
$PV=(\dfrac{{{m}^{'}}}{M})RT$
$\Rightarrow P=(\dfrac{{{m}^{'}}}{V})(\dfrac{RT}{M})$
We can write the density of the gas as,
$\rho =\dfrac{{{m}^{'}}}{V}$
Hence, putting this expression we get,
$\Rightarrow P=\dfrac{\rho RT}{M}$
$\Rightarrow \dfrac{PM}{RT}=\rho $
$\Rightarrow \rho =\dfrac{PM}{NKT}$(As, R=NK)
Here, N is Avogadro's number and K is the Boltzmann Constant.
So, finally, we can write,
$\Rightarrow \rho =\dfrac{Pm}{KT}$(As, $m=\dfrac{M}{N}$)
So, the density of the gas is given by,
$\dfrac{Pm}{KT}$
Hence, the correct answer is - (A).
Note: Here, we have assumed that there is an N number of molecules in the gas. Hence, the mass of each molecule is given by,
$m=\dfrac{M}{N}$
This expression also shows that the ideal gas will always have a finite density until it reaches absolute zero temperature.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Which type of bond is stronger ionic or covalent class 12 chemistry CBSE
What organs are located on the left side of your body class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
When people say No pun intended what does that mea class 8 english CBSE