Behaviour of Perfect Gas and Kinetic Theory — Revision Notes

⚡ 30-Second Revision

Ideal Gas Law: — $PV = nRT$ or $PV = N k_B T$
Combined Gas Law (n=const): — $rac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$
Kinetic Energy per molecule: — $langle E_k \rangle = \frac{3}{2} k_B T$
RMS Speed: — $v_{rms} = sqrt{\frac{3RT}{M}} = sqrt{\frac{3 k_B T}{m}}$
Degrees of Freedom ($f$): — Monoatomic $f=3$, Diatomic $f=5$, Polyatomic $f=6$
Internal Energy (per mole): — $U = \frac{f}{2} RT$
Molar Specific Heat at constant volume: — $C_V = \frac{f}{2} R$
Molar Specific Heat at constant pressure: — $C_P = (\frac{f}{2} + 1) R$
Mayer's Relation: — $C_P - C_V = R$
Ratio of Specific Heats: — $gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f}$
Mean Free Path: — $lambda = \frac{k_B T}{sqrt{2} pi d^2 P}$
Real Gas Equation (van der Waals): — $(P + \frac{an^2}{V^2})(V - nb) = nRT$

2-Minute Revision

Quickly revise the core concepts of perfect gas behavior and Kinetic Theory. Start with the Ideal Gas Law, $PV=nRT$, remembering to always use absolute temperature (Kelvin). This equation encapsulates Boyle's, Charles's, and Gay-Lussac's laws.

The Kinetic Theory of Gases (KTG) provides the microscopic view: gas pressure comes from molecular collisions, and temperature is a direct measure of the average translational kinetic energy of molecules ($langle E_k \rangle = \frac{3}{2} k_B T$).

Remember the RMS speed formula, $v_{rms} = sqrt{3RT/M}$, and its dependence on temperature and molar mass. Crucially, recall degrees of freedom ($f=3$ for monoatomic, $f=5$ for diatomic at moderate T) and their role in specific heats.

The Law of Equipartition of Energy states each degree gets $rac{1}{2} k_B T$. This leads to $C_V = \frac{f}{2}R$ and $C_P = C_V + R$, and the ratio $gamma = 1 + \frac{2}{f}$. Finally, briefly recall the concept of mean free path and the qualitative differences between ideal and real gases, especially the conditions for deviation (high P, low T).

Focus on the formulas and their direct applications.

5-Minute Revision

For a more comprehensive revision, let's break down the key areas. First, ensure a solid understanding of the **Ideal Gas Law ($PV=nRT$)**. Practice problems where you need to calculate an unknown variable given the others, and especially those involving changes in state ($rac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$).

Always convert temperatures to Kelvin. Remember the values of $R$ in different units. Next, delve into the Kinetic Theory of Gases (KTG). Recall its postulates: point masses, no intermolecular forces, random elastic collisions.

The most important outcome is the kinetic interpretation of temperature: $langle E_k \rangle = \frac{3}{2} k_B T$. This means temperature is a direct measure of the average translational kinetic energy.

Understand the different molecular speeds: $v_{mp} < v_{avg} < v_{rms}$. The RMS speed, $v_{rms} = sqrt{\frac{3RT}{M}}$, is frequently tested. Practice comparing RMS speeds for different gases or at different temperatures.

For example, if temperature doubles, $v_{rms}$ increases by $sqrt{2}$.

Crucially, master **Degrees of Freedom ($f$) and Specific Heats**. For monoatomic gases ($He, Ne$), $f=3$. For diatomic gases ($O_2, N_2$), $f=5$ (at moderate temperatures, ignoring vibration). For non-linear polyatomic gases ($H_2O, NH_3$), $f=6$.

The Law of Equipartition of Energy states each degree of freedom contributes $rac{1}{2} k_B T$ to the internal energy. This leads to the molar internal energy $U = \frac{f}{2}RT$. From this, $C_V = \frac{dU}{dT} = \frac{f}{2}R$.

Then, use Mayer's Relation, $C_P - C_V = R$, to find $C_P = (\frac{f}{2}+1)R$. Finally, the ratio of specific heats is $gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f}$. Be ready to calculate these values or identify the gas type from given $gamma$.

Lastly, briefly review the mean free path ($lambda propto T/P$) and the reasons for real gas deviation from ideal behavior (finite molecular volume, intermolecular forces) and the conditions (high P, low T) where these deviations are significant.

The van der Waals equation qualitatively explains these deviations.

Prelims Revision Notes

Behaviour of Perfect Gas and Kinetic Theory - NEET Revision Notes

1. Ideal Gas Equation:

$PV = nRT$ (for $n$ moles)
$PV = N k_B T$ (for $N$ molecules, $k_B = R/N_A$ is Boltzmann constant)
$P$: Pressure (Pa, atm), $V$: Volume (m$^3$, L), $n$: moles, $R$: Universal Gas Constant ($8.314,\text{J mol}^{-1}\text{K}^{-1}$ or $0.0821,\text{L atm mol}^{-1}\text{K}^{-1}$), $T$: Absolute Temperature (Kelvin).
Conversion: — $T_K = T_C + 273.15$.
Combined Gas Law (for fixed mass): — $rac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$.

2. Kinetic Theory of Gases (KTG) Postulates:

Gas consists of a large number of identical, point-like molecules.
Molecules are in continuous, random motion.
Collisions are perfectly elastic.
No intermolecular forces (except during collisions).
Volume of molecules is negligible compared to container volume.

3. Kinetic Interpretation of Temperature:

Average translational kinetic energy per molecule: $langle E_k \rangle = \frac{1}{2} m langle v^2 \rangle = \frac{3}{2} k_B T$.
Total translational kinetic energy for $n$ moles: $U_{trans} = n \frac{3}{2} RT$.

4. Molecular Speeds:

RMS Speed: — $v_{rms} = sqrt{langle v^2 \rangle} = sqrt{\frac{3RT}{M}} = sqrt{\frac{3 k_B T}{m}}$ ($M$ = molar mass, $m$ = molecular mass).
Average Speed: — $v_{avg} = sqrt{\frac{8RT}{pi M}}$.
Most Probable Speed: — $v_{mp} = sqrt{\frac{2RT}{M}}$.
Order: — $v_{mp} < v_{avg} < v_{rms}$.

5. Degrees of Freedom ($f$) and Law of Equipartition of Energy:

Degrees of Freedom: — Independent ways a molecule can possess energy.

* Monoatomic (He, Ne): $f=3$ (3 translational) * Diatomic (O$_2$, N$_2$): $f=5$ (3 translational + 2 rotational, at moderate T) * Polyatomic (non-linear, H$_2$O): $f=6$ (3 translational + 3 rotational)

Equipartition Law: — Each degree of freedom contributes $rac{1}{2} k_B T$ per molecule (or $rac{1}{2} RT$ per mole) to the internal energy.
Total Internal Energy (per mole): — $U = \frac{f}{2} RT$.

6. Molar Specific Heats:

At constant volume ($C_V$): — $C_V = \frac{f}{2} R$.
At constant pressure ($C_P$): — $C_P = C_V + R$ (Mayer's Relation).
Ratio of Specific Heats ($gamma$): — $gamma = \frac{C_P}{C_V} = 1 + \frac{R}{C_V} = 1 + \frac{2}{f}$.

* Monoatomic ($f=3$): $gamma = 5/3 approx 1.67$ * Diatomic ($f=5$): $gamma = 7/5 = 1.40$ * Polyatomic ($f=6$): $gamma = 4/3 approx 1.33$

7. Mean Free Path ($lambda$):

Average distance a molecule travels between collisions.
$lambda = \frac{1}{sqrt{2} pi d^2 n} = \frac{k_B T}{sqrt{2} pi d^2 P}$ ($d$=molecular diameter, $n$=number density).
$lambda propto T$, $lambda propto 1/P$.

8. Real Gases:

Deviate from ideal behavior at high pressure and low temperature.
Reasons: Finite molecular volume, intermolecular forces.
Van der Waals Equation: — $(P + \frac{an^2}{V^2})(V - nb) = nRT$. ('a' accounts for forces, 'b' for volume).

Vyyuha Quick Recall

For specific heats and degrees of freedom: Monoatomic has 3 degrees, so 5/3 gamma. Diatomic has 5 degrees, so 7/5 gamma. Polyatomic has 6 degrees, so 4/3 gamma. (Remember $C_V = fR/2$, $C_P = C_V + R$, $gamma = 1 + 2/f$)