Ideal Gas Equation — Revision Notes

⚡ 30-Second Revision

Ideal Gas Equation: — $PV = nRT$
Combined Gas Law: — $rac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ (for constant $n$)
Density Form: — $ho = \frac{PM}{RT}$ 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 $ext{m}^3$)

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 ($ho = \frac{PM}{RT}$) or molar mass ($M = \frac{mRT}{PV}$). For problems involving changes in conditions, use the Combined Gas Law: $rac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$.

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, ext{L atm mol}^{-1} ext{K}^{-1}$(for P in atm, V in L) or$8.314, ext{J mol}^{-1} ext{K}^{-1}$(for P in Pa, V in$ ext{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: $rac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$. 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 $ho = \frac{PM}{RT}$ or $M = \frac{mRT}{PV}$. Example: Find density of $ext{O}_2$ at $2,\text{atm}, 300,\text{K}$. $M(\text{O}_2) = 32,\text{g/mol}$. $ho = \frac{2 \times 32}{0.0821 \times 300} approx 2.60,\text{g/L}$.
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, $ext{m}^3$, $ext{cm}^3$, $ext{mL}$) * $n$: 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 $ext{Pa m}^3 \text{mol}^{-1}\text{K}^{-1}$) * $1.987,\text{cal mol}^{-1}\text{K}^{-1}$ * $T$: Absolute Temperature (ALWAYS Kelvin: $T(\text{K}) = T(^circ\text{C}) + 273.15$)

2. Derived Forms:

* Combined Gas Law: For fixed $n$: $rac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$. Use when conditions change. * **Density ($ho$) and Molar Mass ($M$):** Since $n = m/M$, substitute into $PV=nRT$ to get $PV = \frac{m}{M}RT$. Rearrange to $ho = \frac{PM}{RT}$ 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, $Z eq 1$. $Z<1$ (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', $PV=nRT$.