Explain the combined gas law.

Short Answer:

The combined gas law is a single gas equation that combines Boyle’s law, Charles’s law, and Gay-Lussac’s law. It relates pressure (P), volume (V), and temperature (T) of a fixed amount of gas, allowing us to calculate how one property changes when the other two change, as long as the number of moles remains constant.

The combined gas law formula is:
(P₁ × V₁) / T₁ = (P₂ × V₂) / T₂
This law is helpful in solving real-world gas problems where pressure, volume, and temperature all change at the same time.

Detailed Explanation:

Combined gas law

In thermodynamics, gases often undergo changes where pressure, volume, and temperature all vary. Instead of using Boyle’s law, Charles’s law, or Gay-Lussac’s law separately, we can use the combined gas law to analyze such situations. It is a convenient and powerful formula that helps solve problems involving gases when the amount of gas remains constant.

This law is derived by combining the three individual gas laws:

  1. Boyle’s law (P ∝ 1/V at constant T)
  2. Charles’s law (V ∝ T at constant P)
  3. Gay-Lussac’s law (P ∝ T at constant V)

Combining these gives:

(P × V) / T = constant

If the gas goes from an initial state to a final state:

(P₁ × V₁) / T₁ = (P₂ × V₂) / T₂

Where:

  • P₁, V₁, T₁ = initial pressure, volume, and temperature
  • P₂, V₂, T₂ = final pressure, volume, and temperature
  • Temperature must always be in Kelvin

Understanding Each Term

  • Pressure (P): Force the gas exerts on the container walls.
  • Volume (V): The space the gas occupies.
  • Temperature (T): Must be in Kelvin. Add 273.15 to Celsius to convert.

This formula tells us that if one property changes, the other two must also adjust to maintain balance.

Real-Life Example

Example:

A gas occupies 4 liters at 300 K and 2 atm pressure. What will be its volume at 400 K and 3 atm?

Using the formula:
(P₁ × V₁) / T₁ = (P₂ × V₂) / T₂

Substitute values:
(2 × 4) / 300 = (3 × V₂) / 400
8 / 300 = 3V₂ / 400
Cross-multiplied:
8 × 400 = 3V₂ × 300
3200 = 900V₂
V₂ = 3200 / 900 = 3.56 liters

So, the new volume is 3.56 liters.

Applications of Combined Gas Law

  1. Weather prediction – Atmospheric gases follow this law as pressure and temperature change.
  2. Aviation – Cabin pressure and temperature are managed using gas behavior.
  3. Diving equipment – Gas cylinders must be calculated correctly for safety.
  4. Industrial gas storage – Combined law helps in adjusting volume with pressure and temperature changes.
  5. Refrigeration systems – Understanding gas expansion and compression.

Importance in Engineering

  • Helps solve real situations where more than one gas property changes.
  • Essential for engine design, HVAC systems, and pressure vessel calculations.
  • Simplifies thermodynamic analysis when ideal gas behavior can be assumed.

Limitations of Combined Gas Law

  • Assumes the gas behaves ideally.
  • Not accurate at very high pressures or very low temperatures.
  • Does not account for chemical reactions or gas leaks.
  • Only valid when the amount of gas remains constant.

For more complex systems, the ideal gas law (PV = nRT) or real gas equations are used.

Conclusion

The combined gas law is a practical and useful equation that links pressure, volume, and temperature for a fixed amount of gas. It simplifies the calculation when all three properties change at once. It brings together three simple gas laws and is widely used in engineering, aviation, refrigeration, and daily life applications involving gas behavior. It helps predict how gases respond to varying environmental conditions.