Boyle's, Dalton's, and Henry's Laws

Boyle's Law: The Relationship Between Pressure and Volume

• Boyle's Law defines the relationship between gas pressure and volume when temperature is held constant.

• Mathematical Expression: The law is expressed as $P_1V_1 = P_2V_2$.

  • $P_1$: Initial pressure

  • $V_1$: Initial volume

  • $P_2$: Secondary pressure

  • $V_2$: Secondary volume

• Inverse Proportionality: Pressure varies inversely with volume.

  • If the volume of a container decreases, the pressure must increase to maintain the equality.

  • If the volume of a container increases, the pressure must decrease.

• Kinetic Molecular Explanation:

  • At a given temperature, molecules possess a specific amount of kinetic energy and collide with the walls of their container.

  • In a smaller volume with a fixed number of molecules, the molecules collide with the container walls more frequently.

  • More collisions result in higher pressure.

  • Example: A container with three molecules at a constant temperature will exert more pressure if the volume is reduced because the molecules hit the walls more often.

Dalton's Law of Partial Pressures

• Memory Trick: Use the full name "Dalton's Law of Partial Pressures" to distinguish it from other gas laws.

• Principle of Partial Pressure: In a mixture of different gas species (e.g., Nitrogen ($N_2$), Oxygen ($O_2$), and Carbon Dioxide ($CO_2$)), each gas exerts its own pressure against the walls of the container.

• Proportionality: The partial pressure of a specific gas is directly proportional to the percentage of that gas in the total mixture.

• Calculation Example (6 Torr Container):

  • Total Pressure: $6\,Torr$

  • Composition: 3 molecules of $N_2$, 1 molecule of $O_2$, 2 molecules of $CO_2$.

  • Partial Pressure of Nitrogen ($P_{N_2}$): Since $N_2$ makes up half of the molecules, it is responsible for half the pressure: $3\,Torr$.

  • Partial Pressure of Oxygen ($P_{O_2}$): Since there is only one Oxygen molecule out of six, it contributes one-sixth: $1\,Torr$.

  • Partial Pressure of Carbon Dioxide ($P_{CO_2}$): Since there are two molecules of $CO_2$, its contribution is $2\,Torr$.

  • Total Pressure ($P_{total}$): $3\,Torr (N_2) + 1\,Torr (O_2) + 2\,Torr (CO_2) = 6\,Torr$.

Atmospheric Gas Composition and Partial Pressures

• The total atmospheric pressure is defined as $760\,mmHg$.

• Nitrogen ($N_2$):

  • Percentage of atmosphere: $78.6\%$

  • Calculation: $760\,mmHg \times 0.786 = 597\,mmHg$

  • Nitrogen accounts for the vast majority of atmospheric pressure.

• Oxygen ($O_2$):

  • Percentage of atmosphere: approximately $21\%$ (specifically $20.9\%$

  • Calculation: $760\,mmHg \times 0.209 = 159\,mmHg$

• Carbon Dioxide ($CO_2$):

  • Percentage of atmosphere: $0.04\%$

  • Note: CO2 levels are very low, which is why small changes in CO2 emissions can dramatically affect atmospheric dynamics and global warming.

  • Partial Pressure: approximately $0.3\,mmHg$

• Water Vapor ($H_2O$):

  • Percentage of atmosphere: approximately $0.46\%$ (variable based on humidity)

  • Partial Pressure: approximately $3.7\,mmHg$

Mental Calculation Techniques for Exams

• To calculate partial pressures without a calculator based on a total pressure of $760\,mmHg$:

  • 1% Baseline: $1\%$ of $760$ is $7.6$.

  • 0.5% Threshold: Half of $1\%$ ($7.6$) is $3.8$.

  • 0.05% Threshold: One-tenth of $0.5\%$ ($3.8$) is $0.38$.

  • Application to CO2: Since atmospheric $CO_2$ is $0.04\%$, which is slightly less than $0.05\%$, the partial pressure is slightly less than $0.38\,mmHg$ (textbook value is $0.3\,mmHg$).

  • Application to Water Vapor: At $0.46\%$, it is slightly less than the $0.5\%$ value ($3.8\,mmHg$), resulting in the textbook value of $3.7\,mmHg$.

Henry's Law of Solubility

• Henry's Law of Solubility states that a gas will dissolve into a liquid (like blood plasma) in proportion to its partial pressure.

• Mechanism of Movement:

  • Molecules move down their partial pressure gradients (from high partial pressure to low partial pressure).

  • Gases move between the alveoli and the bloodstream (external respiratory membrane) and between the blood and tissues (internal respiratory membrane) based on these gradients.

  • Fluids include both gases and liquids; gases can move into liquids, and molecules can move out of liquids back into the air.

Gas Solubility Rules and Physiological Factors

• While Henry's Law dictates the pressure gradient, the actual amount of gas that dissolves is heavily influenced by specific solubility rules:

  • Nitrogen Solubility: Nitrogen is only half ($1/2$) as soluble as Oxygen in water/blood. Despite its high partial pressure, it is "hardly soluble at all" compared to other gases.

  • Oxygen Solubility: Oxygen is only one-twentieth ($1/20$) as soluble as Carbon Dioxide. Its solubility in blood is significantly enhanced by the protein hemoglobin.

  • Carbon Dioxide Solubility: $CO_2$ is extremely soluble in the blood (forty times ($40\times$) more soluble than Nitrogen). This high solubility allows the body to retain $CO_2$ and maintain necessary acidity levels, rather than losing it all to the atmosphere.

• Nitrogen Absence in Blood: Because nitrogen lacks a specialized carrier like hemoglobin and has very low solubility, it does not dissolve well into the bloodstream despite its massive gradient in atmospheric air.

• Chemical Reactivity: Carbon dioxide's high solubility is further explained by its tendency to react with water, forming a chemical equation that will be explored in later sections.