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Consider a sample of an equimolar mixture of ${\text{He}}$ gas and ${\text{Ne}}$ gas both at $300{\text{ K}}$ and $1{\text{ atmosphere}}$. Assuming ideal behaviour, which of the following quantities are equal for two samples?
A.Root mean square speed of molecules
B.Mean translational kinetic energy of molecules
C.Number density of molecules
D.Kinetic energy of molecules

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Last updated date: 28th Feb 2024
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IVSAT 2024
Answer
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Hint: To solve this we must know the kinetic molecular theory of gases. The kinetic molecular theory explains the properties of gases like compressibility, liquefaction, low domestic, diffusion, effusion, etc. The theory explains that gas is made up of a large number of molecules, and they are continually in random motion.

Complete step-by-step answer:Let us discuss the given option one by one and find relation between them.
A.We are given a sample of an equimolar mixture of ${\text{He}}$ gas and ${\text{Ne}}$ gas both at $300{\text{ K}}$ and $1{\text{ atmosphere}}$
The molecular mass of ${\text{He}}$ gas is $4.00{\text{ u}}$ and the molecular mass of ${\text{Ne}}$ gas is $20.17{\text{ u}}$. Temperature of the gas mixture is $300{\text{ K}}$.
We know the expression for the root mean square speed of gas molecules is as follows:
$u = \sqrt {\dfrac{{3RT}}{M}} $
Where, $u$ is the root mean square speed,
$R$ is the universal gas constant,
$T$ is the absolute temperature,
$M$ is the molecular mass of the gas.
Thus,
$u \propto \dfrac{1}{{\sqrt M }}$
The root mean square of the gas particle depends on the molecular mass of the gas. The molecular masses of ${\text{He}}$ gas and ${\text{Ne}}$ gas are different.
Thus, the root mean square speed of the molecules of ${\text{He}}$ gas and ${\text{Ne}}$ gas is not equal.
Thus, option (A) is not correct.
B.We know the expression for the mean translational kinetic energy of gas molecules is as follows:
$\overline K = \dfrac{3}{2}kT$
Where $\overline K $ is the mean translational kinetic energy,
$k$ is the Boltzmann constant,
$T$ is the temperature.
Thus,
$\overline K \propto T$
The mean translational kinetic energy depends on the temperature of the gas. The temperature of ${\text{He}}$ gas and ${\text{Ne}}$ gas is same i.e. $300{\text{ K}}$.
Thus, the mean translational kinetic energy of ${\text{He}}$ gas and ${\text{Ne}}$ gas is equal.
Thus, option (B) is correct.
C.We know the expression for the number density of gas molecules is as follows:
$d = \dfrac{m}{V}$
Where $d$ is the density of the gas,
$m$ is the molar mass of the gas,
$V$ is the volume of the gas.
For equimolar mixture of ${\text{He}}$ gas and ${\text{Ne}}$ gas the number of gas molecules of ${\text{He}}$ gas and ${\text{Ne}}$ gas are equal.
Thus, the number density of ${\text{He}}$ gas and ${\text{Ne}}$ gas is equal.
Thus, option (C) is correct.
D.We know the expression for the kinetic energy of gas molecules is as follows:
\[KE = \dfrac{1}{2}m{v^2}\]
Where $KE$ is the kinetic energy,
$m$ is the mass of the gas molecules,
\[v\] is the velocity of the gas molecules.
For equimolar mixture of ${\text{He}}$ gas and ${\text{Ne}}$ gas the mass of gas molecules of ${\text{He}}$ gas and ${\text{Ne}}$ gas are equal.
Thus, the kinetic energy of ${\text{He}}$ gas and ${\text{Ne}}$ gas is equal.
Thus, option (D) is correct.
Thus, the quantities that are equal for two samples are mean translational kinetic energy of molecules, number density of molecules and kinetic energy of molecules.

Thus, the correct options are (B), (C) and (D).

Note:We know that the kinetic theory of matter depends on the two fundamental factors that are the molecular constitution of matter and the identification of the heat with the motion of molecules. It depends on the way the particles of the matter move in space. The kinetic theory of matter is also known as the kinetic particle theory.
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