Kinetic Theory of Gases

7 mol of a certain monoatomic ideal gas undergoes a temperature increase of 40K at constant pressure. The increase in the internal energy of the gas in this process is :

(Given R = 8.3 JK$^{-1}$ mol$^{-1}$)

A gas has $n$ degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be :

Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is 1:4, then

A. The r.m.s. velocity of gas molecules in two vessels will be the same.

B. The ratio of pressure in these vessels will be 1:4.

C. The ratio of pressure will be 1:1.

D. The r.m.s. velocity of gas molecules in two vessels will be in the ratio of 1:4.

Choose the correct answer from the options given below :

Which statements are correct about degrees of freedom ?

(A) A molecule with n degrees of freedom has $n^2$ different ways of storing energy.

(B) Each degree of freedom is associated with $\dfrac12RT$ average energy per mole.

(C) A monatomic gas molecule has 1 rotational degree of freedom where as diatomic molecule has 2 rotational degrees of freedom.

(D) $\mathrm{CH_4}$ has a total of 6 degrees of freedom.

Choose the correct answer from the options given below :

Given below are two statements :

Statement I : The average momentum of a molecule in a sample of an ideal gas depends on temperature.

Statement II : The rms speed of oxygen molecules in a gas is $v$. If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become $2v$.

In the light of the above statements, choose the correct answer from the options given below :

A vessel contains 14 g of nitrogen gas at a temperature of 27°C. The amount of heat to be transferred to the gas to double the r.m.s speed of its molecules will be :

Take R = 8.32 J mol$^{-1}$ k$^{-1}$.

At a certain temperature, the degrees of freedom per molecule for gas is 8. The gas performs 150 J of work when it expands under constant pressure. The amount of heat absorbed by the gas will be ______ J.

One mole of a monoatomic gas is mixed with three moles of a diatomic gas. The molecular specific heat of mixture at constant volume is $\dfrac{\alpha^2}4R$ J/mol K; then the value of α will be _______. (Assume that the given diatomic gas has no vibrational mode).

The root mean square speed of smoke particles of mass $5\times10^{-17}$ kg in their Brownian motion in air at NTP is approximately. [Given k = $1.38\times10^{-23}$ J K$^{-1}$]

If the r.m.s speed of oxygen at room temperature is $\sqrt{56}$ m/s. The r.m.s speed of nitrogen at the same temperature is _______.

The value of $X$ is ____.

The value of $Y$ is ____.

The value of $\dfrac{Q}{RT_0}$ is

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$) of potassium permanganate ($\ce{KMnO4}$) of different concentrations $n_1$ and $n_2\ (n_1 > n_2)$ molecules per unit volume with $\Delta n = (n_1 - n_2) \ll n_1$. When they are connected by a tube of small length $l$ and cross-sectional area $S$, $\ce{KMnO4}$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order $(\Delta n)^2$, which of the following is/are correct? ($k_B$ is the Boltzmann constant)

A thermally isolated cylindrical closed vessel of height 8 m is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass 8.3 kg. Thus the partition is held initially at a distance of 4 m from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains 0.1 mole of an ideal gas at temperature 300 K. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in m) will be ______ (take the acceleration due to gravity $= 10\ \mathrm{m\,s^{-2}}$ and the universal gas constant $= 8.3\ \mathrm{J\,mol^{-1}K^{-1}}$).

One mole of a monatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature $(V$-$T)$ diagram. The correct statement(s) is/are:

[$R$ is the gas constant]

A small particle of mass $m$ moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is $L = L_0$ the particle speed is $v = v_0$. The piston is moved inward at a very low speed $V$ such that $V \ll \dfrac{dL}{L}v_0$, where $dL$ is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

A mixture of ideal gas containing 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially at pressure $P_0$, volume $V_0$, and temperature $T_0$. If the gas mixture is adiabatically compressed to a volume $V_0/4$, then the correct statement(s) is/are,

(Given $2^{1.2} = 2.3$; $2^{3.2} = 9.2$; $R$ is gas constant)

A flat plate is moving normal to its plane through a gas under the action of a constant force $F$. The gas is kept at a very low pressure. The speed of the plate $v$ is much less than the average speed $u$ of the gas molecules. Which of the following options is/are true?

$C_p$ and $C_v$ are specific heats at constant pressure and constant volume respectively. It is observed that

$C_p-C_v=a$ for hydrogen gas

$C_p-C_v=b$ for nitrogen gas

The correct relation between a and b is :