# Gay-Lussac's Law

This is part of Year 11 HSC Chemistry course under the topic of Gas Laws.

### HSC Chemistry Syllabus

• Conduct investigations and solve problems to determine the relationship between the Ideal Gas Law and:
• Gay-Lussac's Law (Temperature)

### Gas Laws

Gay-Lussac's law is one of the four individual gas laws which were combined to describe the ideal gas law.

The gas laws describe the behaviour of gases under ideal circumstances and assume their properties that ideal gases:

• Have a low density since the spaces between the particles of gas are large
• Are free flowing/forming meaning that they fill the volumes of their containers and can leak through cracks
• Are compressible and expandable
• Are diffusive meaning that they travel from areas of higher concentration to areas of lower concentration.

Gay-Lussac's law focuses on the direct relationship between pressure and temperature when volume and amount of gas is kept constant.

### Gay-Lussac's Law

Gay-Lussac's Law is named after Joseph Louis Gay-Lussac, an 18th-century chemist and physicist who first observed that volumes of ideal gases react in stoichiometric ratios.

Gay-Lussac's law states that the pressure of a gas is directly proportional to its temperature when volume and amount of gas are kept constant. Mathematically the relationship described by Gay-Lussac's Law can be expressed as

$$P \propto T$$

and thus

$$\frac{P_1}{T_1} = \frac{P_2}{T_2}$$

Where:

P_1 and T_1 represent the initial pressure and temperature

P_2 and T_2 represent the pressure and temperature after a change has been imparted on the gas system.

In the context of Gay-Lussac's Law, pressure refers to the force that is experienced by particles of gas while temperature refers to the average kinetic energy of those gas particles.

Units of Measurement:

• Pressure:
• Pascal (Pa)
• 1 Kilopascal (kPa) = 1000 Pa
• 1 Atmosphere (atm) = 101325 Pa.

• Temperature:
• Kelvin (K)
• Degrees Celsius (℃) = Kelvin - 273.15

### Deriving and understanding Gay-Lussac's Law

The ideal gas law is given by the formula:

$$PV = nRT$$

Where:

• P is pressure (in kPa)
• V is volume (in L)
• n is the number of moles of gas
• R is the universal gas constant = 8.314 J mol–1 K–1
• T is temperature (in Kelvin)

The equation can be rearranged into

$$P = \frac{nRT}{V}$$

When volume and amount are held constant, the expression \frac{nR}{V} is equivalent to a constant k. Thus the following mathematical equation can also be obtained:

$$P = kT$$

For this relationship to be true, an increase in pressure must lead to an equivalent increase in temperature and a decrease in pressure must lead to an equivalent decrease in temperature. Thus when volume and amount are held constant, according to the ideal gas law

$$P \propto T$$

The relationship described by Gay-Lussac's Law is demonstrated in the graph below.

### Real-World Examples & Practical Applications

Understanding Gay-Lussac's law is crucial not just in academic settings but also for everyday applications involving any sealed container of gas, from cooking methods to the use of aerosol products.

1. Pressure Cooker: Initially molecules of gas are moving with some energy inside a pressure cooker. As temperature increases in the pressure cooker and the volume and amount of gas remains the same since the container is sealed by a lid, the average kinetic energy of gas increases causing the internal pressure to rise. This proportional increase in temperature and pressure demonstrates Gay-Lussac's Law.  To release the pressure, a regulator is used but the temperature in the system does not decrease proportionally because after opening the valve, volume and amount are no longer held constant and thus Gay-Lussac's Law no longer applies.

2. Aerosol Cans: Aerosol cans are tightly sealed containers filled with gas and liquid under pressure. When the temperature of an aerosol can increases, the kinetic energy of the gas particles inside also increases, leading to an increase in pressure. Like with the pressure cooker, this demonstrates Gay-Lussac's law as the increase in pressure is directly proportional to the increase in temperature when volume is held constant. As the temperature continues to increase, at a certain point when the increase in pressure is too great, the gas will overcome the structural integrity of the can causing the can to rupture. This is why aerosol cans come with warnings not to expose them to high temperatures or open flames.