(D) Statement $(i)$ is false: Pressure $P = \frac{1}{3} \rho v_{rms}^2$. It is proportional to the square of the root mean square speed,not the mean s

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(D) Statement $(i)$ is false: Pressure $P = \frac{1}{3} \rho v_{rms}^2$. It is proportional to the square of the root mean square speed,not the mean speed.Statement $(ii)$ is false: $v_{rms} = \sqrt{\frac{3RT}{M}}$. Since $P \propto T$,$v_{rms} \propto \sqrt{P}$,not $P$.Statement $(iii)$ is true: The rate of diffusion is directly proportional to the average speed of the gas molecules.Statement $(iv)$ is true: The mean translational kinetic energy per molecule is given by $E = \frac{3}{2} k_B T$,which is directly proportional to the absolute temperature $T$.Therefore,statements $(iii)$ and $(iv)$ are correct. The correct option is $(D)$.

Class 11 Physics Kinetic Theory of Gases Mix Examples-Kinetic Theory of Gases

Medium

In the kinetic theory of gases,which of these statements is/are true?
$(i)$ The pressure of a gas is proportional to the mean speed of the molecules.
$(ii)$ The root mean square speed of the molecules is proportional to the pressure.
$(iii)$ The rate of diffusion is proportional to the mean speed of the molecules.
$(iv)$ The mean translational kinetic energy of a gas is proportional to its kelvin temperature.

A
$(ii)$ and $(iii)$ only
B
$(i), (ii)$ and $(iv)$ only
C
$(i)$ and $(iii)$ only
D
$(iii)$ and $(iv)$ only

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Kinetic Theory of Gases

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Mix Examples-Kinetic Theory of Gases

During an experiment,an ideal gas is found to obey a condition $\frac{P^2}{\rho} = \text{constant}$ [$\rho = \text{density of the gas}$]. The gas is initially at temperature $T$,pressure $P$,and density $\rho$. The gas expands such that density changes to $\rho/2$.

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$CO_2$ $(O-C-O)$ is a triatomic gas. The mean kinetic energy of one gram of the gas will be (where $N$ is Avogadro's number,$k$ is Boltzmann's constant,and the molecular weight of $CO_2 = 44$).

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$A$ vessel contains $14 \, g$ ($7$ moles) of hydrogen and $96 \, g$ ($3$ moles) of oxygen at $STP$. $A$ chemical reaction is induced by passing an electric spark in the vessel until one of the gases is consumed. The temperature is brought back to its starting value of $273 \, K$. The pressure in the vessel is ...... $atm$.

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If the potential energy of a gas molecule is $U = \frac{M}{r^6} - \frac{N}{r^{12}}$,where $M$ and $N$ are positive constants,then the potential energy at equilibrium must be

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The density of water is $1000 \; kg \; m^{-3}$. The density of water vapour at $100 \; ^{\circ}C$ and $1 \; atm$ pressure is $0.6 \; kg \; m^{-3}$. The volume of a molecule multiplied by the total number gives what is called molecular volume. Estimate the ratio (or fraction) of the molecular volume to the total volume occupied by the water vapour under the above conditions of temperature and pressure.

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During an experiment,an ideal gas is found to obey a condition $\frac{P^2}{\rho} = \text{constant}$ [$\rho = \text{density of the gas}$]. The gas is initially at temperature $T$,pressure $P$,and density $\rho$. The gas expands such that density changes to $\rho/2$.

$CO_2$ $(O-C-O)$ is a triatomic gas. The mean kinetic energy of one gram of the gas will be (where $N$ is Avogadro's number,$k$ is Boltzmann's constant,and the molecular weight of $CO_2 = 44$).

If the potential energy of a gas molecule is $U = \frac{M}{r^6} - \frac{N}{r^{12}}$,where $M$ and $N$ are positive constants,then the potential energy at equilibrium must be

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