Ideal gas concept Questions in English

(C) The density of a substance is defined as the ratio of its mass to its volume,given by $\rho = \frac{m}{V}$.
Since the gas is in an airtight container,the total mass $(m)$ of the gas remains constant.
Because the container is airtight and rigid,the volume $(V)$ occupied by the gas also remains constant.
Since both mass and volume remain constant,the density $(\rho)$ of the gas will remain the same.

(C) According to the kinetic theory of gases,gas molecules are considered to be point masses or rigid spheres.
Collisions between molecules and between molecules and the walls of the container are considered to be perfectly elastic.
In a perfectly elastic collision,both momentum and kinetic energy are conserved.
Therefore,molecules of a gas behave like perfectly elastic rigid spheres.

(B) An ideal gas can never be liquefied because the intermolecular forces of attraction among the ideal gas molecules are negligible.
For an ideal gas,the intermolecular forces are considered to be zero.
Since liquefaction requires strong intermolecular forces to bring molecules together into a liquid state,a gas that perfectly obeys the ideal gas equation $PV = nRT$ at all temperatures and pressures cannot be liquefied.
Correct choice - option $B$.

(D) An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to inter-particle interactions and obey the ideal gas law $PV = nRT$.
Van der Waals equation is given by $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$. This equation is used to describe the behavior of real gases by accounting for the finite size of molecules and intermolecular forces.
Since an ideal gas assumes no intermolecular forces and negligible molecular volume,it does not follow the Van der Waals equation.
Therefore,the statement that is not true for an ideal gas is that it follows the Van der Waals equation.
Correct choice - $D$.

(C) The kinetic theory of gases was primarily developed by James Clerk Maxwell and Ludwig Boltzmann in the $19^{th}$ century.
This theory is based on a simplified molecular or particle description of a gas,from which many macroscopic properties of the gas can be derived.
Therefore,the correct choice is option-$C$.

(C) According to the kinetic theory of gases,the root mean square velocity of gas molecules is given by the formula $v_{rms} = \sqrt{\frac{3RT}{M}}$.
At absolute temperature,$T = 0 \ K$.
Substituting this value into the formula,we get $v_{rms} = \sqrt{\frac{3R(0)}{M}} = 0$.
Since the root mean square velocity represents the average motion of the molecules,a velocity of $0$ implies that all molecular motion ceases at absolute zero.

(B) By definition,an ideal gas consists of point-like particles that do not exert any intermolecular forces on each other.
Since potential energy arises from the work done against or by intermolecular forces,the potential energy of an ideal gas is zero.
Therefore,the internal energy of an ideal gas is entirely due to the kinetic energy of its molecules,which depends only on the temperature of the gas.

(C) The temperature of an ideal gas is related to the average kinetic energy of its molecules due to their random thermal motion.
When a vessel containing an ideal gas is placed in a moving train or car,the entire vessel (and the gas inside it) moves with a uniform velocity $v$ relative to the ground.
This bulk motion of the gas does not affect the random thermal motion of the gas molecules relative to the centre of mass of the gas.
Since temperature depends only on the random kinetic energy of the molecules,the temperature of the gas remains constant regardless of the uniform motion of the vessel.
Therefore,the temperature does not increase,and the centre of mass moves with the velocity of the train,not randomly.

(C) For an ideal gas,the internal energy $(U)$ is a function of temperature $(T)$ only. This is because,in an ideal gas model,there are no intermolecular forces of attraction or repulsion between the gas molecules. Therefore,the potential energy of the gas is zero,and the total internal energy consists solely of the kinetic energy of the molecules,which is directly proportional to the absolute temperature of the gas. Thus,$U = f(T)$.

(A) In an ideal gas,the molecules are assumed to be point masses with no intermolecular forces of attraction or repulsion.
Since internal energy is the sum of kinetic and potential energy,and the potential energy due to intermolecular forces is zero for an ideal gas,the internal energy consists solely of the kinetic energy of the gas molecules.
Therefore,the internal energy of an ideal gas is a function of temperature only.

(C) According to the kinetic theory of gases,an ideal gas is defined as a gas in which there are no intermolecular forces of attraction or repulsion between the molecules. The pressure exerted by an ideal gas is given by the formula $P = \frac{1}{3} \rho v_{rms}^2$. Since the assumption of 'no intermolecular forces' is the fundamental postulate of the kinetic theory of gases for an ideal gas,the pressure exerted by such a gas remains consistent with the ideal gas equation $PV = nRT$. Therefore,if there are no intermolecular forces,the pressure exerted by the gas remains unchanged compared to the theoretical model of an ideal gas.

(D) According to the Kinetic Theory of Gases,the following are the key postulates:
$1$. The volume occupied by the gas molecules is negligible compared to the total volume of the gas.
$2$. Gas molecules are in constant,random motion with all possible velocities.
$3$. There are no intermolecular forces of attraction or repulsion between the molecules.
$4$. Collisions between molecules and with the walls of the container are perfectly elastic.
Since all the statements given in options $A$,$B$,and $C$ are standard postulates of the Kinetic Theory of Gases,none of them is incorrect. Therefore,the correct choice is $D$.

(C) The temperature of an ideal gas is related to the average kinetic energy of its molecules,which depends on the internal energy of the system.
Since the cylinder is moving at a uniform speed,there is no change in the internal state (pressure,volume,or internal energy) of the gas relative to the cylinder.
Therefore,the temperature of the gas molecules remains the same.

(A) When intermolecular forces are removed,water behaves as an ideal gas.
At $STP$,the volume of $1 \ mole$ of an ideal gas is $22.4 \ L$.
The molar mass of water $(H_2O)$ is $18 \ g/mol = 0.018 \ kg/mol$.
The number of moles in $4.5 \ kg$ of water is $n = \frac{4.5 \ kg}{0.018 \ kg/mol} = 250 \ moles$.
The volume $V$ at $STP$ is given by $V = n \times 22.4 \ L/mol$.
$V = 250 \times 22.4 \ L = 5600 \ L$.
Since $1000 \ L = 1 \ m^3$,the volume is $V = 5.6 \ m^3$.

(A) According to the kinetic theory of gases,an ideal gas is defined as a gas that follows the ideal gas equation $PV = nRT$ under all conditions of temperature and pressure.
One of the fundamental assumptions of the kinetic theory is that there are no intermolecular forces of attraction or repulsion between gas molecules.
Since liquefaction of a gas requires the presence of intermolecular forces to bring molecules together into a liquid state,ideal gases cannot be liquefied.
Therefore,option $A$ is the correct statement.
Option $B$ is incorrect because gas molecules obey Newton's laws of motion.
Option $C$ is incorrect because pressure is inversely proportional to volume only at constant temperature (Boyle's Law).
Option $D$ is incorrect because gas molecules move in straight lines between successive collisions.

(C) According to the kinetic theory of gases,molecules are in constant random motion and undergo elastic collisions.
Option $A$ is an assumption: the time taken for a collision is very small compared to the time between two successive collisions.
Option $B$ is an assumption: there are no intermolecular forces of attraction or repulsion between gas molecules.
Option $D$ is an assumption: collisions between molecules and with the walls are perfectly elastic,meaning there is no loss in total kinetic energy.
Option $C$ is $NOT$ an assumption: when a molecule collides with the container wall,it undergoes a change in momentum (it bounces back),which is the origin of gas pressure. Therefore,the momentum change is not negligible.

$(B)$ is the incorrect statement. An ideal gas is considered isotropic, meaning its properties are independent of direction. The constant $\left(\frac{1}{3}\right)$ in the pressure equation $P = \frac{1}{3}\rho v^2_{rms}$ arises because the mean square velocity is distributed equally among the three spatial dimensions $(x, y, z)$, such that $\langle v^2_x \rangle = \langle v^2_y \rangle = \langle v^2_z \rangle = \frac{1}{3} \langle v^2 \rangle$.

(D) Generally,a gas behaves more like an ideal gas at higher temperature and low pressure.
At high temperatures,the kinetic energy of the molecules is very high,making the potential energy due to intermolecular forces negligible.
At low pressures,the volume occupied by the gas molecules is negligible compared to the total volume of the container.
Therefore,real gases approach ideal gas behavior under conditions of high temperature and low pressure.

(B) The mean distance $D$ between gas molecules is related to the number density $n$ by the relation $D \approx n^{-1/3}$.
From the ideal gas law,$PV = n_{mol}RT$,where $n_{mol} = N/N_A$.
Thus,$P = (N/V) \cdot (R/N_A) \cdot T = n \cdot k_B \cdot T$,where $n = N/V$ is the number density.
Given $P = 4.0 \times 10^{-15} \, atm = 4.0 \times 10^{-15} \times 10^5 \, Pa = 4.0 \times 10^{-10} \, Pa$.
Using $n = P / (k_B T)$,where $k_B = R/N_A = 8.0 / (6 \times 10^{23}) \approx 1.33 \times 10^{-23} \, J/K$.
$n = (4.0 \times 10^{-10}) / (1.33 \times 10^{-23} \times 300) = (4.0 \times 10^{-10}) / (4.0 \times 10^{-21}) = 10^{11} \, molecules/m^3$.
The mean distance $D \approx n^{-1/3} = (10^{11})^{-1/3} \approx 10^{-3.66} \, m \approx 2.15 \times 10^{-4} \, m = 0.215 \, mm$.
Thus,the order of magnitude is $0.2 \, mm$.

(A) An ideal gas is a theoretical gas composed of a set of randomly-moving,non-interacting point particles.
It is defined as a gas that obeys the ideal gas law,$PV = nRT$,under all conditions of temperature and pressure.
In this model,the particles of the gas are assumed to have no volume and there are no intermolecular forces of attraction or repulsion between them.
While no real gas is perfectly ideal,many gases behave like ideal gases at low pressures and high temperatures.

(N/A) Macroscopic quantities are those that describe the bulk properties of a system as a whole,without considering the individual behavior of its constituent particles. Examples include pressure $(P)$,volume $(V)$,temperature $(T)$,and internal energy $(U)$. These are directly measurable in a laboratory.
Microscopic quantities are those that describe the properties of individual particles (atoms or molecules) within the system,such as their position $(r)$,velocity $(v)$,momentum $(p)$,and kinetic energy $(k)$. These quantities are not directly measurable for a large number of particles and are instead handled using statistical methods.

(N/A) Gases are composed of a large number of atoms or molecules.
The theoretical framework used to explain the macroscopic properties of gases based on their microscopic motion is known as the kinetic theory of gases.
The kinetic theory of gases is based on the assumption that gases consist of atoms and molecules that are in constant,random motion at very high speeds.
In gases,the interatomic (or intermolecular) forces are negligible compared to those in solids or liquids because the particles are far apart,allowing these forces to be ignored in the ideal gas model.

(N/A) The approach by which the macroscopic properties of gases can be explained at a microscopic level is called the kinetic theory of gases. In the $19^{th}$ century,scientists like Maxwell,Boltzmann,and others developed this theory.
Kinetic theory of gases explains the pressure and temperature of gases in terms of the motion and collisions of molecules.
It incorporates the gas laws and Avogadro's hypothesis.
Kinetic theory of gases relates measurable macroscopic properties of a gas,such as viscosity,pressure,temperature,internal energy,heat capacity,specific heat,conduction,and diffusion,with molecular parameters. From this,the estimation of molecular size and mass can be performed.

(N/A) The kinetic theory of gases is a theoretical model that describes the macroscopic properties of gases (such as pressure,temperature,and volume) by considering their molecular composition and motion.
It is based on the following fundamental assumptions:
$1$. $A$ gas consists of a large number of identical particles (atoms or molecules) that are in constant,random motion.
$2$. The volume occupied by the gas particles themselves is negligible compared to the total volume of the container.
$3$. The particles exert no forces on each other except during elastic collisions.
$4$. All collisions between particles or with the walls of the container are perfectly elastic,meaning kinetic energy is conserved.
$5$. The average kinetic energy of the particles is directly proportional to the absolute temperature of the gas.

(N/A) The kinetic theory of gases is fundamental to understanding the macroscopic properties of matter from a microscopic perspective. Its importance includes:
$1$. It provides a molecular explanation for macroscopic properties like pressure,temperature,and internal energy.
$2$. It successfully explains the gas laws,such as Boyle's Law,Charles's Law,and Avogadro's Law,based on the motion of molecules.
$3$. It helps in understanding the concept of temperature as a measure of the average kinetic energy of gas molecules $(KE_{avg} = \frac{3}{2} k_B T)$.
$4$. It explains the phenomenon of diffusion,effusion,and Brownian motion in gases.
$5$. It provides a basis for calculating the specific heat capacities of gases and understanding the degrees of freedom of molecules.

(C) According to the kinetic theory of gases and experimental evidence from atomic physics, the size of an atom is approximately $1 \, \text{Å}$ $(\mathring{A})$.
Since $1 \, \text{Å} = 10^{-10} \, m$, the order of magnitude of the dimension of an atom is $10^{-10} \, m$.
Therefore, the correct option is $C$.

(N/A) The three fundamental states of matter are:
$1$. Solid: Particles are closely packed in a fixed structure,having a definite shape and volume.
$2$. Liquid: Particles are less tightly packed than in solids,having a definite volume but no fixed shape; they take the shape of the container.
$3$. Gas: Particles are far apart and move randomly,having neither a definite shape nor a definite volume.

(B) The Boltzmann constant,denoted by $k_B$,is a fundamental physical constant that relates the average kinetic energy of particles in a gas to the thermodynamic temperature of the gas.
Its value is defined as $k_B = R/N_A$,where $R$ is the universal gas constant and $N_A$ is the Avogadro constant.
Since both $R$ and $N_A$ are universal constants,the Boltzmann constant $k_B$ is also a universal constant.
Therefore,the value of the Boltzmann constant is the same for all gases and is approximately $1.38 \times 10^{-23} \ J/K$.
Thus,the statement is false.

(A) An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to inter-particle interactions.
It obeys the ideal gas law $PV = nRT$ at all temperatures and pressures.
In this equation,$P$ is the pressure,$V$ is the volume,$n$ is the number of moles,$R$ is the universal gas constant,and $T$ is the absolute temperature.
Ideal gases follow the assumptions of the kinetic molecular theory,which include that the gas particles have negligible volume and there are no attractive or repulsive forces between them.

(N/A) The ideal gas model is based on the microscopic behavior of gas particles,defined by the following postulates:
$(1)$ $A$ gas consists of a large number of microscopic particles called molecules. These molecules may be monoatomic,diatomic,or polyatomic. For a chemically stable gas,all molecules are identical.
$(2)$ Molecules are considered as perfectly rigid spheres or point particles with no internal structure.
$(3)$ Molecules are in constant,random motion,colliding with each other and the walls of the container.
$(4)$ The motion of gas molecules follows Newton's laws of motion.
$(5)$ The total number of molecules in a given sample is extremely large.
$(6)$ The actual volume occupied by the molecules is negligible compared to the total volume of the container.
$(7)$ Intermolecular forces are negligible except during collisions.
$(8)$ Collisions between molecules and between molecules and the container walls are perfectly elastic. The duration of a collision is negligible compared to the time between successive collisions.

(A) The kinetic theory of gases is a theoretical model that describes the macroscopic properties of gases (such as pressure,temperature,and volume) by considering their molecular composition and motion.
$1$. The fundamental assumption of the kinetic theory is that a gas consists of a large number of small particles (molecules) that are in constant,random motion.
$2$. These molecules frequently collide with each other and with the walls of the container.
$3$. The pressure exerted by the gas is a direct result of these collisions with the container walls.
$4$. The theory assumes that intermolecular forces are negligible (except during collisions) and that the potential energy of the molecules is negligible compared to their kinetic energy.
Therefore,the theory is primarily based on the dynamics of molecular collisions.

(D) The pressure of an ideal gas is a macroscopic property that depends on the kinetic energy of the gas molecules and the volume of the container. According to the kinetic theory of gases,the pressure exerted by an ideal gas is given by the formula $P = \frac{1}{3} \frac{M}{V} v_{rms}^2$. This derivation assumes that the gas is in equilibrium and the collisions with the walls are elastic. The derivation holds true regardless of the shape of the container,as long as the volume $V$ is well-defined and the gas is uniformly distributed. Therefore,the shape of the container does not affect the pressure calculation for an ideal gas.

(N/A) During the elastic collision of gas molecules,both the total linear momentum and the total kinetic energy of the system are conserved.
$1$. Conservation of Linear Momentum: According to Newton's laws,in the absence of any external force,the total momentum of the colliding particles remains constant.
$2$. Conservation of Kinetic Energy: In an elastic collision,there is no loss of energy to other forms such as heat,sound,or deformation. Therefore,the sum of the kinetic energies of the molecules before the collision is equal to the sum of the kinetic energies after the collision.
These conservation laws are fundamental to the Kinetic Theory of Gases,which assumes that gas molecules behave as hard,perfectly elastic spheres.

(N/A) Yes,we can consider a container of irregular shape while calculating the pressure of an ideal gas.
According to the kinetic theory of gases,the pressure exerted by an ideal gas is due to the continuous collisions of gas molecules with the walls of the container.
Since the gas molecules move randomly in all directions,they strike every part of the container's surface regardless of its shape.
Pascal's law states that in a fluid at rest,the pressure at any point is the same in all directions.
Therefore,the pressure exerted by an ideal gas is uniform throughout the container and acts perpendicular to the surface at every point,irrespective of the shape of the container.

(A) $(i)$ False. By definition,there are no intermolecular forces of attraction or repulsion in an ideal gas.
$(ii)$ True. Since there are no intermolecular forces,the potential energy associated with the interaction between molecules is zero.
$(iii)$ False. Different gases have molecules of different sizes and structures.
$(iv)$ True. According to the kinetic theory of gases,ideal gas molecules are considered point masses or perfectly elastic spheres that undergo elastic collisions.

(A) At $STP$,$1 \,mole$ of an ideal gas occupies a volume $V = 22.4 \times 10^{-3} \,m^3$.
The number of molecules in $1 \,mole$ is given by Avogadro's number,$N_A \approx 6.022 \times 10^{23}$.
The volume available per molecule is $v = \frac{V}{N_A} = \frac{22.4 \times 10^{-3}}{6.022 \times 10^{23}} \approx 3.72 \times 10^{-26} \,m^3$.
The average distance $d$ between molecules is approximately the cube root of the volume per molecule:
$d \approx (v)^{1/3} = (3.72 \times 10^{-26})^{1/3} \,m$.
$d \approx 3.34 \times 10^{-9} \,m = 3.34 \,nm$.
Thus,the order of magnitude of the average distance between molecules is $1 \,nm$.

(D) An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions.
$1$. The molecules of an ideal gas are considered to be point masses,meaning their volume is infinitesimally small compared to the volume of the container.
$2$. There are no intermolecular forces of attraction or repulsion between the molecules of an ideal gas.
$3$. An ideal gas strictly obeys the ideal gas equation $PV = nRT$ under all conditions of temperature and pressure.
Since all the given statements are correct,the correct option is $D$.

(B) An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to inter-particle interactions. It obeys the ideal gas law $PV = nRT$ and satisfies all the fundamental assumptions of the kinetic theory of gases,such as negligible molecular volume and elastic collisions.

(D) For an ideal gas,the molecules are assumed to be point masses with no intermolecular forces of attraction or repulsion.
Since there are no intermolecular forces,there is no potential energy associated with the configuration of the molecules.
Therefore,the total internal energy of an ideal gas consists solely of the kinetic energy of its molecules due to their random motion.
Thus,the internal energy is only kinetic energy.

(D) According to the kinetic theory of gases,the average kinetic energy of gas molecules is given by $K = \frac{3}{2} RT$.
This implies $K \propto T$.
Since kinetic energy cannot be negative,its minimum value is zero.
Therefore,the minimum possible temperature is $0 \ K$,at which molecular motion ceases.
Converting this to Celsius: $T(^{\circ} C) = T(K) - 273 = 0 - 273 = -273^{\circ} C$.