Question

The root mean square velocity of an ideal gas at constant pressure varies with density (d) as:
A. \[{d^2}\]
B. d
C. \[\sqrt d \]
D. \[\dfrac{1}{{\sqrt d }}\]

Answer 1

Hint: We know that the formula of density is mass/volume. Here, we have to identify the relation between density and the RMS speed for an ideal gas. So, first we have to rearrange the RMS speed formula using the gas equation of an ideal gas, that is, PV=nRT and after that we have to rearrange the RMS formula in terms of density.

Complete Step by Step Solution:
Here, we have to identify how rms velocity and density are related in an ideal gas.
Let's understand what an ideal gas is. Ideal gas is described as a gas that comprises a set of point particles that are moving randomly. The interactions between the particles happen through elastic collisions. The equation for an ideal gas is PV=nRT, Where, P stands for pressure, V stands for volume, n is mole number of electrons, R stands for gas constant and T stands for temperature.

For n=1, the equation for ideal gas is,
\[PV = RT\]

Now, we have put PV in place of RT in the formula of rms velocity.
\[{v_{rms}} = \sqrt {\dfrac{{3PV}}{M}} \]
And Density (d)=Mass (M)/Volume(V)

So, the above equation becomes,
\[{v_{rms}} = \sqrt {\dfrac{{3P}}{d}} \]
Therefore,
\[{v_{rms}} = \sqrt {\dfrac{1}{d}} \]
So, relation is inversely proportional.
Hence, option D is right.

Note: There is the absence of an attractive force among molecules in the case of an ideal gas. Therefore, its liquefaction is not possible by applying low temperature and high pressure. All the molecules of an ideal gas move at different speeds. It is to be noted that the ideal gas equation follows the absolute pressure of 105 Pa and the temperature of 273.15 K.