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Ideal Gas Law

CSCA Ideal Gas Law study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Ideal Gas Law

1. The Ideal Gas Model

An Ideal Gas is a theoretical model established to simplify calculations. It ignores minor factors to capture the main characteristics. **In the CSCA exam, gases are generally treated as ideal gases.**

* **Three Core Assumptions**:

1. **Negligible Volume**: Molecules are treated as point masses.

2. **No Intermolecular Forces**: No attraction or repulsion (except during collisions). This means Internal Energy depends *only* on Temperature.

3. **Perfectly Elastic Collisions**: No loss of kinetic energy during collisions.

2. Core Formulas

#### A. Clapeyron Equation

Describes the relationship between state parameters in a **single equilibrium state**:

$$ pV = \nu R T $$

* **$p$ (Pressure)**: Unit **Pa** (Pascal).

* **$V$ (Volume)**: Unit **m³** (Cubic meter).

* **$T$ (Temperature)**: **Thermodynamic Temperature**, Unit **K** (Kelvin). **Crucial:** $T = t(^{\circ}\text{C}) + 273.15$.

* **$\nu$ (Amount of substance)**: Unit **mol**. *(Note: Sometimes denoted as $n$, but $\nu$ is often used in physics to distinguish from number density)*.

* **$R$**: Universal Gas Constant, $R \approx 8.31 \, \text{J/(mol\cdot K)}$.

#### B. Experimental Laws & Graphs (Special Processes)

When the mass ($\nu$) of the gas is constant, we use the **Combined Gas Law**:

$$ \frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2} = C $$

You must master the **$p-V$ graph** characteristics for three special processes:

1. **Isothermal (Constant $T$)**: $pV = C$. The graph is a **hyperbola** curve.

2. **Isobaric (Constant $p$)**: $V/T = C$. The graph is a **horizontal line**.

3. **Isochoric (Constant $V$)**: $p/T = C$. The graph is a **vertical line**.

3. Useful Derivations

* **Density Form**: $pM = \rho RT$ (where $\rho$ is density, $M$ is molar mass). Useful for finding molar mass from density.

* **Microscopic Form**: $pV = NkT$ ($N$ is total number of molecules, $k$ is Boltzmann constant).

4. Typical Examples

**Example 1: Process Analysis & Graphs**

An ideal gas of fixed mass starts at State A, expands isobarically to State B, and then cools isochorically to State C. Describe the path on a $p-V$ diagram.

**Analysis**:

* A $\rightarrow$ B (Isobaric Expansion): $p$ constant, $V$ increases. Horizontal line to the right.

* B $\rightarrow$ C (Isochoric Cooling): $V$ constant, $T$ decreases. Since $p/T=C$, $p$ must decrease. Vertical line downwards.

**Example 2: Calculation**

An oxygen tank has volume $V=32\text{L}$, pressure $p=1.3 \times 10^7 \text{Pa}$, and temperature $27^{\circ}\text{C}$. Calculate the mass $m$ of the oxygen. ($M_{O_2} = 32 \text{g/mol}$)

**Solution**:

1. **Unit Conversion**:

* $V = 0.032 \, \text{m}^3$

* $T = 300 \, \text{K}$ (Don't forget to convert to Kelvin!)

* $M = 0.032 \, \text{kg/mol}$

2. **Substitute into** $pV = \frac{m}{M}RT$:

$$ m = \frac{pVM}{RT} = \frac{1.3 \times 10^7 \times 0.032 \times 0.032}{8.31 \times 300} \approx 5.34 \, \text{kg} $$