# Ideal Gas Equation ## What is an Ideal Gas? - An ideal gas is a theoretical model of a gas with specific properties - It behaves in a way that can be perfectly described by mathematical equations - Real gases approximate ideal gas behaviour under certain conditions ### Properties of an Ideal Gas An ideal gas has the following characteristics: - **Negligible molecular volume** - the actual volume occupied by gas molecules themselves is insignificant compared to the total volume of the container - **Elastic collisions** - when molecules collide with each other or the container walls, no kinetic energy is lost - **Cannot be liquefied** - it remains in the gaseous state at all temperatures and pressures - **No intermolecular forces** - molecules don't attract or repel each other, except during the instant of collision - **Obeys the gas laws** - follows Boyle's law, Charles' law and the Pressure law perfectly

An ideal gas obeys the equation of state pV = nRT at all pressures, volumes and temperatures

- All these conditions hold at any temperature or pressure - Real gases (like oxygen, nitrogen, helium) behave approximately like ideal gases at high temperatures and low pressures ## The Ideal Gas Equation (Molar Form) - The ideal gas equation relates pressure, volume, temperature and the amount of gas in moles - The equation is: $$pV = nRT$$ ### Variables in the Equation | Variable | Symbol | Unit | Description | |----------|--------|------|-------------| | Pressure | $p$ | Pa (pascals) | Force per unit area exerted by gas | | Volume | $V$ | m³ | Space occupied by the gas | | Number of moles | $n$ | mol | Amount of substance | | Temperature | $T$ | K (kelvin) | Absolute temperature | | Molar gas constant | $R$ | J mol⁻¹ K⁻¹ | $8.31$ J mol⁻¹ K⁻¹ |

The values of R and k are always given in the data sheet during exams

## The Ideal Gas Equation (Molecular Form) - The ideal gas equation can also be written in terms of the number of molecules - This form uses the Boltzmann constant instead of the molar gas constant - The equation is: $$pV = NkT$$ ### Variables in the Molecular Form | Variable | Symbol | Unit | Description | |----------|--------|------|-------------| | Pressure | $p$ | Pa | Force per unit area exerted by gas | | Volume | $V$ | m³ | Space occupied by the gas | | Number of molecules | $N$ | - | Total number of particles | | Boltzmann constant | $k$ | J K⁻¹ | $1.38 \times 10^{-23}$ J K⁻¹ | | Temperature | $T$ | K | Absolute temperature | ### Relationship Between the Two Forms - The Boltzmann constant $k$ is related to the molar gas constant $R$ and Avogadro's constant $N_A$ by: $$k = \frac{R}{N_A}$$ - This means: $k = \frac{8.31}{6.02 \times 10^{23}} = 1.38 \times 10^{-23}$ J K⁻¹ - Both forms describe the same physical behaviour, just using different units for amount ## Using the Ideal Gas Equation ### Key Points for Calculations

Temperature must always be in kelvin (K). Convert from Celsius: T(K) = T(°C) + 273

- Volume must be in m³ (convert cm³ by dividing by $10^6$) - Pressure must be in Pa (pascals) - Choose the appropriate form: $pV = nRT$ for moles, $pV = NkT$ for molecules ### Worked Example A sealed container holds an ideal gas with a volume of $2.5 \times 10^4$ cm³. The gas is at a temperature of 27 °C and a pressure of $3.0 \times 10^6$ Pa. Calculate the number of moles of gas in the container. **Answer:** **Step 1: State the known quantities** - Pressure, $p = 3.0 \times 10^6$ Pa - Volume, $V = 2.5 \times 10^4$ cm³ $= 2.5 \times 10^4 \times 10^{-6} = 2.5 \times 10^{-2}$ m³ - Temperature, $T = 27$ °C $= 27 + 273 = 300$ K **Step 2: Write down the ideal gas equation** $$pV = nRT$$ **Step 3: Rearrange for the number of moles $n$** $$n = \frac{pV}{RT}$$ **Step 4: Substitute values and calculate** $$n = \frac{3.0 \times 10^6 \times 2.5 \times 10^{-2}}{8.31 \times 300} = \frac{7.5 \times 10^4}{2493} = 30 \text{ mol}$$ ## Choosing Between the Two Equations - Use $pV = nRT$ when: - The question gives or asks for mass or moles - You need to convert between mass and number of particles - Use $pV = NkT$ when: - The question gives or asks for number of molecules or atoms - Working with microscopic particle behaviour - Both equations are equivalent and describe the same physical system - The choice depends on what information you have and what you're solving for