# The Gas Laws The **gas laws** are a set of fundamental principles that describe the relationship between the macroscopic properties of a gas: **pressure ($P$)**, **volume ($V$)**, and **temperature ($T$)**, for a given amount of gas ($n$). These laws were developed through experimental observations and form the basis of our understanding of gas behavior. They are combined in the **Ideal Gas Law**. Pressure is a measured in Pascals (Pa). ## Boyle's Law (Constant Temperature) Boyle's Law describes the relationship between pressure and volume when the temperature and the amount of gas are held constant. **Statement:** *For a fixed mass of gas at constant temperature, the volume is inversely proportional to the pressure.* This means that if you increase the pressure on a gas, its volume will decrease proportionally, and vice versa. **Mathematical Formulation:** $$ V \propto \frac{1}{P} \quad \text{or} \quad PV = \text{Constant} $$ For a gas changing from an initial state (1) to a final state (2): $$ P_1V_1 = P_2V_2 $$ **Graphical Representation:** The graph of pressure versus volume for an isothermal (constant temperature) process is a hyperbola. ## Charles's Law (Constant Pressure) Charles's Law describes the relationship between volume and temperature when the pressure and the amount of gas are held constant. **Statement:** *For a fixed mass of gas at constant pressure, the volume is directly proportional to its absolute temperature (in Kelvin).* This means that heating a gas will cause it to expand, and cooling it will cause it to contract. **Mathematical Formulation:** $$ V \propto T \quad \text{or} \quad \frac{V}{T} = \text{Constant} $$ For a gas changing from an initial state (1) to a final state (2): $$ \frac{V_1}{T_1} = \frac{V_2}{T_2} $$ **Graphical Representation:** The graph of volume versus absolute temperature is a straight line passing through the origin. ## Gay-Lussac's Law (Constant Volume) Gay-Lussac's Law (also known as Amontons's Law) describes the relationship between pressure and temperature when the volume and the amount of gas are held constant. **Statement:** *For a fixed mass of gas at constant volume, the pressure is directly proportional to its absolute temperature (in Kelvin).* This explains why the pressure inside a sealed container of gas increases when it is heated. **Mathematical Formulation:** $$ P \propto T \quad \text{or} \quad \frac{P}{T} = \text{Constant} $$ For a gas changing from an initial state (1) to a final state (2): $$ \frac{P_1}{T_1} = \frac{P_2}{T_2} $$ ## Ideal Gas Equation By combining Boyle's Law, Charles's Law, and Avogadro's Law, we derive the General Gas Equation: $$PV = nRT$$ Where $R$ is the Universal Gas Constant. For an ideal gas, the internal energy depends solely on temperature. ## Possible Questions/Answer - **Q:** What is Boyle’s Law? **A:** Boyle’s Law states that for a fixed amount of gas at constant temperature, the volume of the gas is inversely proportional to its pressure. Mathematically, $PV = \text{Constant}$. - **Q:** How does Charles's Law describe the relationship between temperature and volume? **A:** Charles's Law indicates that for a fixed amount of gas at constant pressure, the volume is directly proportional to its absolute temperature (in Kelvin). This means that as the temperature increases, the volume of the gas also increases. | Law | Constant Variable | Relationship | Formula | | :--- | :--- | :--- | :--- | | **Boyle's Law**| Temperature | $P \propto 1/V$ | $P_1V_1 = P_2V_2$ | | **Charles's Law**| Pressure | $V \propto T$ | $V_1/T_1 = V_2/T_2$ | | **Gay-Lussac's Law**| Volume | $P \propto T$ | $P_1/T_1 = P_2/T_2$ |