Heat and Thermodynamics

Let's analyze the given statements one by one in detail: Statement (I): The mean free path of gas molecules is inversely proportional to the square of molecular diameter.

The mean free path ($\lambda$) of gas molecules is the average distance a molecule travels before colliding with another molecule.

The formula for the mean free path in terms of molecular diameter ($d$) is given by:

Here, $k_B$ is the Boltzmann constant, $T$ is the temperature, and $P$ is the pressure.

From this equation, it is evident that the mean free path ($\lambda$) is inversely proportional to the square of the molecular diameter ($d^2$).

Hence, Statement I is true.

Statement (II): Average kinetic energy of gas molecules is directly proportional to absolute temperature of gas.

The average kinetic energy ($E_{\text{avg}}$) of gas molecules is given by:

where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature.

This shows that the average kinetic energy is directly proportional to the absolute temperature of the gas.

Hence, Statement II is true.

Considering the analysis above, the correct answer is: Option B Both Statement I and Statement II are true

To find the ratio of specific heats at constant volume ($C_V$) of the gases, we need to understand the degrees of freedom each type of gas molecule has, as this determines their specific heat capacity at constant volume.

Degrees of freedom refer to the number of independent ways in which a molecule can store energy.

A monoatomic gas molecule has 3 translational degrees of freedom.

This is because it can move in three dimensions: x, y, and z.

A diatomic gas, if we assume it to be rigid for this context, has 5 degrees of freedom: 3 translational like the monoatomic gas and 2 rotational since it can rotate around two axes perpendicular to the bond axis connecting the two atoms.

Vibrational modes are not considered at room temperature for a rigid diatomic gas, as these require higher energy to become accessible.

The specific heat capacity at constant volume for a monoatomic gas is given by:

And for a diatomic gas, it's:

Where $R$ is the ideal gas constant.

To find the ratio of their specific heats at constant volume, we divide the specific heat of the monoatomic gas by that of the diatomic gas:

Thus, the correct ratio of the specific heats at constant volume of the monoatomic to diatomic (rigid) gases is given by Option B: $\dfrac{3}{5}$.

The collision frequency ($Z$) of gas molecules is proportional to the square root of the absolute temperature ($T$) of the system.

Mathematically, it can be represented as $Z \propto \sqrt{T}$.

This implies that when the temperature changes, the collision frequency changes as well according to the formula:

where: $Z_1$ is the collision frequency at temperature $T_1$, $Z_0$ is the initial collision frequency at temperature $T_0$, $T_1$ is the final absolute temperature, and $T_0$ is the initial absolute temperature.

To solve the given problem, we first need to convert the provided temperatures from Celsius to Kelvin (since absolute temperature in Kelvin should be used): $T_0 = 27^{\circ}C = 27 + 273 = 300$ K $T_1 = 127^{\circ}C = 127 + 273 = 400$ K Using the formula for collision frequency change and substituting the given values:

This can be simplified to:

Thus, the collision frequency of the system at $127^{\circ}C$ is $\dfrac{2}{\sqrt{3}}$ times the collision frequency at $27^{\circ}C$, making Option B the correct answer:

To begin with, we're told that during an adiabatic process, the pressure $P$ of a gas is directly proportional to the cube of its absolute temperature $T$, that is, $P \propto T^3$.

From this, we can express the relation as $P = kT^3$, where $k$ is a constant.

In an adiabatic process, $PV^\gamma = \text{constant}$, where $\gamma = \dfrac{C_P}{C_V}$, $P$ is the pressure, $V$ is the volume, $C_P$ is the heat capacity at constant pressure, and $C_V$ is the heat capacity at constant volume.

Also, the ideal gas equation is $PV = nRT$, where $n$ is the amount of substance, $R$ is the ideal gas constant, and $T$ is the absolute temperature.

This can be rearranged to express $P$ in terms of $T$ and $V$, resulting in $P = \dfrac{nRT}{V}$.

Given that $P = kT^3$, we can equate this to the expression we got from the ideal gas law: $\dfrac{nRT}{V} = kT^3$.

Simplifying, we get $V = \dfrac{nR}{kT^2}$, which shows that $V$ is inversely proportional to $T^2$.

Now, let's recall the adiabatic condition $PV^\gamma = \text{constant}$.

Substituting the proportionalities $P \propto T^3$ and $V \propto T^{-2}$ into this equation, it becomes $T^3(T^{-2})^\gamma = \text{constant}$, simplifying to $T^{3-2\gamma} = \text{constant}$.

For the expression $T^{3-2\gamma} = \text{constant}$ to hold true in an adiabatic process, where the only variable is temperature $T$, the exponent must equal to zero (as the quantity of $T$ to some power equals to a constant suggests that the change in $T$ does not alter the value of the expression).

This implies $3 - 2\gamma = 0$, solving for $\gamma$: $3 - 2\gamma = 0 \implies 2\gamma = 3 \implies \gamma = \dfrac{3}{2}$ Therefore, the ratio of $\dfrac{C_P}{C_V}$ for the gas is $\dfrac{3}{2}$, which matches with Option B $\dfrac{3}{2}$.

The mean free path $\lambda$ is the average distance covered by a molecule between two successive collisions.

It is given by the formula:

Where:

is the number density of the molecules, i.e., the number of molecules per unit volume,

is the diameter of the molecules, and

arises from considering the relative motion of a pair of particles in a gas, taking into account that either particle can be moving towards the other, which effectively doubles the cross-sectional area through which they can collide.

This formula shows that the mean free path is inversely proportional to the number density

of the molecules and the square of the diameter

of the molecules.

It also takes into account that collisions are more likely as the cross-sectional area of the molecules increases, or as the density of the molecules increases.

Therefore, the correct answer is: Option A:

To evaluate the veracity of the given statements, we need to understand the physical quantities involved and their dimensional formulas.

Specifically, we're looking at the dimensions of specific heat and gas constant.

Specific Heat: Specific heat (c) is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin, since the increment is the same in both scales).

Its formula is

, where

is the heat added,

is the mass,

is the specific heat, and

is the change in temperature. From the formula, we can deduce the dimensions of specific heat as follows:

Knowing that the dimension of heat (q) is equivalent to energy, which is

, the mass (m) is

, and temperature ($\Delta T$) is

, we can solve for

:

So,

This reveals that Statement (I) is correct.

Gas Constant: The gas constant (R) appears in the ideal gas law, represented as

, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature.

The dimensions of the gas constant can be derived from this relation.

Pressure (P) has dimensions

, volume (V) has dimensions

, and temperature (T) has dimensions

. Therefore,

Considering that the mole (n) is a dimensionless quantity, we can deduce the dimensions of R as:

This indicates that Statement (II) has stated the dimensions incorrectly, presenting them as

when it should be

, making Statement (II) incorrect.

Based on the analysis: Option A, "Statement (I) is incorrect but statement (II) is correct," is wrong because Statement (I) is correct.

Option B, "Both statement (I) and statement (II) are incorrect," is wrong because Statement (I) is correct.

Option C, "Both statement (I) and statement (II) are correct," is wrong because Statement (II) is incorrect.

Option D, "Statement (I) is correct but statement (II) is incorrect," is the correct choice, reflecting the true nature of the statements provided.

The energy of a diatomic molecule depends on the degrees of freedom it has.

For a non-rigid diatomic molecule, there are more degrees of freedom compared to a rigid diatomic molecule.

Specifically, a non-rigid diatomic molecule has translational, rotational, and vibrational degrees of freedom.

The translational and rotational degrees of freedom are the same for both rigid and non-rigid diatomic molecules, which include: 3 translational degrees of freedom, 2 rotational degrees of freedom (since linear molecules do not have the third rotational freedom because it requires rotating around the bond axis, which doesn't change the energy for linear molecules).

Additionally, non-rigid diatomic molecules have vibrational degrees of freedom.

For a simple diatomic molecule, there is 1 vibrational degree of freedom (since vibration along the bond axis is possible).

However, considering the energy distribution across these vibrations requires accounting for both the potential and kinetic energy associated with vibrations, effectively doubling the vibrational degrees of freedom for energy calculations to 2 (one for kinetic and one for potential energy).

Thus, the total degrees of freedom for a non-rigid diatomic molecule are: 3 (translational) + 2 (rotational) + 2 (vibrational) = 7 degrees of freedom.

The formula for the average energy of a molecule in terms of degrees of freedom at temperature T is given by: $E = \dfrac{f}{2}k_\mathrm{B}T$ where $f$ is the total number of degrees of freedom, $k_\mathrm{B}$ is the Boltzmann constant, and $T$ is the temperature.

Plugging in the values for a non-rigid diatomic molecule: $E = \dfrac{7}{2}k_\mathrm{B}T$ Therefore, the total energy for 10 such molecules would be: $10 \times \dfrac{7}{2}k_\mathrm{B}T = 35k_\mathrm{B}T$ So, the correct answer is: Option D: 35 $k_\mathrm{B}T$.

To solve this problem, we can use the first law of thermodynamics, which states:

where:

is the heat transferred to the system, in this case,

.

is the change in internal energy of the system.

is the work done by the system.

For one mole of an ideal gas, the change in internal energy can be calculated using the equation:

where:

is the number of moles, which is

in this case.

is the molar specific heat capacity at constant volume.

is the change in temperature, in Kelvins. Given that we are dealing with helium, a monatomic ideal gas, we know that:

Substituting the given values, we get:

Since a change in temperature of

is equivalent to a change in temperature of

, we can directly substitute

as a

change without needing to convert Celsius to Kelvin as both scales have the same magnitude of degree. Therefore,

Now, substituting

and

into the first law of thermodynamics:

Solving for

we find:

Thus, the work done by the gas is

, which corresponds to Option A.

To convert a temperature difference into electrical energy effectively, the material in question should have low thermal conductivity and high electrical conductivity.

Such materials are classified as thermoelectric materials.

A low thermal conductivity ensures that the material does not easily transfer heat, which is crucial for maintaining a significant temperature difference between its hot and cold sides.

This helps in generating a greater voltage.

Meanwhile, high electrical conductivity is essential for facilitating the efficient movement of electrons across the material when there is a temperature difference, leading to maximal energy conversion.

Therefore, a material that optimally harvests heat energy into electrical energy should embody these characteristics.