Ideal Gas Equation — Revision Notes

⚡ 30-Second Revision

Ideal Gas Equation: — $PV = nRT$
Combined Gas Law: — $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ (for constant $n$)
Density Form: — $\rho = \frac{PM}{RT}$ or $M = \frac{\rho RT}{P}$
Temperature: — Always in Kelvin ($T(\text{K}) = T(^\circ\text{C}) + 273.15$)
**Gas Constant ($R$):**

* $0.0821\,\text{L atm mol}^{-1}\text{K}^{-1}$ (for P in atm, V in L) * $8.314\,\text{J mol}^{-1}\text{K}^{-1}$ (for P in Pa, V in $\text{m}^3$)

STP: — $0^\circ\text{C}$ (273.15 K) and $1\,\text{atm}$. Molar volume at STP = $22.4\,\text{L}$.
Ideal Gas Assumptions: — Negligible volume of particles, no intermolecular forces, elastic collisions, random motion, KE $\propto T$.
Real Gas Deviation: — Most at high P, low T (due to molecular volume and intermolecular forces).

2-Minute Revision

The Ideal Gas Equation, $PV=nRT$, is the core formula for understanding ideal gas behavior. Remember, $P$ is pressure, $V$ is volume, $n$ is moles, $T$ is absolute temperature (always in Kelvin!), and $R$ is the ideal gas constant.

The value of $R$ depends on the units of $P$ and $V$; commonly $0.0821\,\text{L atm mol}^{-1}\text{K}^{-1}$ or $8.314\,\text{J mol}^{-1}\text{K}^{-1}$. This equation is a combination of Boyle's, Charles's, and Avogadro's laws.

It's used to calculate any one variable if the others are known. Crucially, it can be rearranged to find gas density ($\rho = \frac{PM}{RT}$) or molar mass ($M = \frac{mRT}{PV}$). For problems involving changes in conditions, use the Combined Gas Law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$.

Always convert Celsius to Kelvin and ensure unit consistency. Real gases deviate from ideal behavior at high pressures and low temperatures because their molecular volume and intermolecular forces become significant.

5-Minute Revision

Mastering the Ideal Gas Equation, $PV=nRT$, is essential. An ideal gas is a theoretical model where particles have no volume and no interactions. Real gases approximate this behavior at high temperatures and low pressures.

The key variables are pressure ($P$), volume ($V$), number of moles ($n$), and absolute temperature ($T$, always in Kelvin: $T(\text{K}) = T(^\circ\text{C}) + 273.15$). The ideal gas constant ($R$) links these, with values like $0.

0821\,\text{L atm mol}^{-1}\text{K}^{-1}$(for P in atm, V in L) or$8.314\,\text{J mol}^{-1}\text{K}^{-1}$(for P in Pa, V in$\text{m}^3$).

Key Applications:

1
Direct Calculation: — Find any one variable if the other three are known. Example: Calculate $P$ if $n, V, T$ are given.
2
Combined Gas Law: — For a fixed amount of gas changing conditions: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$. This is very common. Example: A gas at $1\,\text{atm}, 10\,\text{L}, 300\,\text{K}$ changes to $2\,\text{atm}, 400\,\text{K}$. Find $V_2$. $V_2 = \frac{P_1V_1T_2}{P_2T_1} = \frac{1 \times 10 \times 400}{2 \times 300} = 6.67\,\text{L}$.
3
Density and Molar Mass: — Use $n = m/M$ to derive $\rho = \frac{PM}{RT}$ or $M = \frac{mRT}{PV}$. Example: Find density of $\text{O}_2$ at $2\,\text{atm}, 300\,\text{K}$. $M(\text{O}_2) = 32\,\text{g/mol}$. $\rho = \frac{2 \times 32}{0.0821 \times 300} \approx 2.60\,\text{g/L}$.
4
STP/NTP: — Remember $1\,\text{mole}$ of ideal gas occupies $22.4\,\text{L}$ at STP ($0^\circ\text{C}$, $1\,\text{atm}$).

Common Mistakes to Avoid: Incorrect unit conversions (especially T to Kelvin), using the wrong $R$ value, and arithmetic errors. Always check units for consistency.

Prelims Revision Notes

The Ideal Gas Equation, $PV=nRT$, is central to the Gaseous State chapter. For NEET, focus on its direct application and derived forms.

1. The Equation and Variables:

* $P$: Pressure (atm, Pa, kPa, mmHg, bar) * $V$: Volume (L, $\text{m}^3$, $\text{cm}^3$, $\text{mL}$) * $n$: Number of moles * $R$: Ideal Gas Constant (choose based on P, V units) * $0.0821\,\text{L atm mol}^{-1}\text{K}^{-1}$ * $8.314\,\text{J mol}^{-1}\text{K}^{-1}$ (or $\text{Pa m}^3 \text{mol}^{-1}\text{K}^{-1}$) * $1.987\,\text{cal mol}^{-1}\text{K}^{-1}$ * $T$: Absolute Temperature (ALWAYS Kelvin: $T(\text{K}) = T(^\circ\text{C}) + 273.15$)

2. Derived Forms:

* Combined Gas Law: For fixed $n$: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$. Use when conditions change. * **Density ($\rho$) and Molar Mass ($M$):** Since $n = m/M$, substitute into $PV=nRT$ to get $PV = \frac{m}{M}RT$. Rearrange to $\rho = \frac{PM}{RT}$ or $M = \frac{mRT}{PV}$. These are vital for calculating gas density or molar mass.

3. Standard Conditions:

* STP (Standard Temperature and Pressure): $0^\circ\text{C}$ (273.15 K) and $1\,\text{atm}$ pressure. At STP, $1\,\text{mole}$ of any ideal gas occupies $22.4\,\text{L}$. * NTP (Normal Temperature and Pressure): $20^\circ\text{C}$ (293.15 K) and $1\,\text{atm}$ pressure. Molar volume at NTP is approximately $24.04\,\text{L}$.

4. Ideal vs. Real Gases:

* Ideal gases follow $PV=nRT$ perfectly. Real gases deviate. * Deviation Conditions: Real gases deviate most at high pressure and low temperature. High P: Molecular volume becomes significant. Low T: Intermolecular forces become significant. * Compressibility Factor (Z): $Z = \frac{PV}{nRT}$. For ideal gas, $Z=1$. For real gases, $Z \neq 1$. $Z<1$ (attractive forces dominate), $Z>1$ (repulsive forces/molecular volume dominate).

5. Problem-Solving Tips:

* Always ensure unit consistency with the chosen $R$ value. Convert all temperatures to Kelvin first. Practice problems involving stoichiometry, gas mixtures (Dalton's Law), and changes in state.

Vyyuha Quick Recall

To remember the conditions for ideal gas behavior, think of 'HILP': High Ideal Low Pressure. This means a gas behaves most ideally at High temperatures and Low Pressures. For the equation itself, just remember 'Pee-Vee equals N-R-T', $PV=nRT$.