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## Definition of Gay Lussac’s Law

Gay Lussac’s law that the absolute temperature and pressure of an ideal gas are directly proportional under constant mass and volume conditions.

In other words, heating gas in a sealed container causes its pressure to increase, while cooling a gas lowers its pressure.

The reason this happens is that increasing temperature imparts thermal kinetic energy to gas molecules. As the temperature increases, molecules collide more often with the container walls. The increased collisions are seen as increased pressure.

### Other Definitions of Gay Lussac’s Law

- The law is named for French chemist and physicist Joseph Gay-Lussac. Gay-Lussac formulated the law in 1808, but it was a formal statement of the relationship between temperature and pressure described by French physicist Guillaume Amonton in the late 1600s.
- This law, formulated by Gay Lussac, states that “the ratio between the volumes of gaseous reactants and products can be expressed in simple whole numbers.”
**For example**, in the following reaction, the ratio of hydrogen, chlorine, and hydrogen chloride volumes is 1:1:2 (a simple ratio). - Gay-Lussac’s law is a gas law which states that the pressure exerted by a gas (of a given mass and kept at a constant volume) varies directly with the absolute temperature of the gas. In other words, the pressure exerted by a gas is proportional to the temperature of the gas when the mass is fixed, and the volume is constant.

## Formula of Gay Lussac’s Law

The mathematical expression of Gay-Lussac’s law can be written as follows:

P ∝ T

P/T = k

where,

- P is the pressure exerted by the gas.
- T is the absolute temperature of the gas.
- k is a constant.

## Derivation of the Formula

Gay-Lussac’s law implies that the ratio of the initial pressure and temperature is equal to the ratio of the final pressure and temperature for a gas of a fixed mass kept at a constant volume. This formula can be expressed as follows:

(P_{1}/T_{1}) = (P_{2}/T_{2})

where,

P_{1} is the initial pressure

T_{1} is the initial temperature

P_{2} is the final pressure

T_{2} is the final temperature

This expression can be derived from the pressure-temperature proportionality for gas. Since P ∝ T for gases of fixed mass kept at constant volume:

P_{1}/T_{1} = k (initial pressure/ initial temperature = constant)

P_{2}/T_{2} = k (final pressure/ final temperature = constant)

Therefore,

P_{1}/T_{1} = P_{2}/T_{2} = k

Or,

P_{1}T_{2} = P_{2}T_{1}

### Summary:

When the temperature of a sample of gas in a rigid container is increased, the pressure of the gas increases as well. The increase in kinetic energy results in the molecules of gas striking the walls of the container with more force, resulting in greater pressure.

The French chemist Joseph Gay-Lussac (1778-1850) discovered the relationship between the pressure of a gas and its absolute temperature.

Gay-Lussac’s Law states that the pressure of a given mass of gas varies directly with the absolute temperature of the gas when the volume is kept constant.

Gay-Lussac’s Law is very similar to Charles’s Law, with the only difference being the type of container. Whereas the container in a Charles’s Law experiment is flexible, it is rigid in a Gay-Lussac’s Law experiment.

Gay-Lussac is incorrectly recognized for the Pressure Law, which established that the pressure of an enclosed gas is directly proportional to its temperature and which he was the first to formulate (c. 1809).

He is also sometimes credited with being the first to publish convincing evidence that shows the relationship between the pressure and temperature of a fixed mass of gas kept at a constant volume. These laws are also known as the Pressure Law or Amontons’s law and Dalton’s law, respectively.

Gay-Lussac used the formula acquired from ΔV/V = αΔT to define the rate of expansion α for gases.

For air, he found a relative expansion ΔV/V = 37.50% and obtained a value of α = 37.50%/100°C =1/266.66°C which indicated that the value of absolute zero was approximately 266.66°C below 0°C.

The value of the rate of expansion α is approximately the same for all gases, and this is also sometimes referred to as Gay-Lussac’s Law.

## Gay Lussac’s law of Gaseous Volume

Gay-Lussac’s Law shows the relationship between the Temperature and Pressure of a gas, at a fixed volume, the temperature and pressure of a gas are directly proportional to each other.

Pressure∝Temperature

In 1808, the French chemist Joseph Louis Gay-Lussac reported the results of new experiments together with a generalization known today as Gay-Lussac’s law of combining gases.

The volume of gas is directly proportional to the Kelvin temperature if the volume is kept constant.

V_{1} / T_{1}=V_{2} / T_{2}

Where,

V_{1} = original volume

V_{2} = final volume

T_{1} = original temperature (K)

T_{2} = final temperature (K)

Gay-Lussac’s Law is applicable only to gases. The volumes of liquids or solids involved in the reactants or products are not governed by Gay Lussac’s law.

## Real Life Examples of Gay Lussac’s Law

Some real-life examples of Gay-Lussac’s Law are the rupture of a pressure cooker, an aerosol can, and a tire. All these substances explode when exposing to higher temperatures. Gay-Lussac’s Law explains the scientific reason behind the explosion.

Gay-Lussac’s law is the law that says the pressure of gas increases with its temperature, or vice versa.

### Pressure Cooker

The pressure cooker is a sealed utensil for cooking food under steam pressure. It is usually made up of steel or aluminum. When heat is supplied, the water inside the cooker vaporizes, and the steam is produced. The steam is periodically released through a valve to maintain the operating pressure inside the cooker.

If the valve malfunctions and the heat flow is not interrupted, the pressure inside the cooker escalates. The increase in the pressure is due to Gay-Lussac’s law, i.e., the pressure of a fixed amount of gas increases with its temperature at constant volume.

This high pressure may rupture the cooker and may lead to an unfortunate accident.

### Aerosol Can

Aerosol cans or sprays are devices that dispense an aerosol, a suspension of fine solid particles, or liquid droplets in the air. When the valve of a metal can is opened, the gas is driven out to form a mist or an aerosol. One of the components in an aerosol can is a propellant.

The propellant consists of high volatile compounds which are liquefied with high pressure. The propellant gives thrust to other components when the valve is opened.

When an aerosol can is subjected to a hot environment, the propellant gets vaporize. These vaporized gases exert pressure on the wall of the bottle. The pressure on the wall increases with the temperature as per Gay-Lussac’s law.

Finally, the can bursts when the pressure becomes intolerable. This is the reason why it is recommended to keep the aerosol cans away from heat.

### Tires

On hot summer days, the inflated tires of vehicles may burst. Gay-Lussac’s law causes the bursting of tires. The inflated tires are under high pressure. When the temperature of the air rises, the pressure of the gas in the tubes increases. After an unbearable point, the tires fracture.

### Water Heater

The electric water heater is similar to the pressure cooker. The cold water is heated by the heating filaments inside the heater. The hot water generated is released through the outlet nozzle.

Modern electric heaters automatically regulate the temperature of the water. When the system and pressure-relief valve malfunction, the steam is generated by the continuous power supply.

This steam can damage the heater. If the pressure of the steam exceeds the tolerable limit, the heater may burst.

## Limitations of Gay-Lussac’s law

The limitations of Gay Lussac’s law are as follows:

- The law is only applicable to ideal gases.
- Gay-Lussac’s law holds good for real gases at high temperatures and/or low pressure.
- The ratio of the pressure to temperature deviated at high pressures. The ratio decreases with increasing the pressure. This decrease is due to an increase in volume at high pressures which is explained by an increase in repulsive forces among the molecules at high pressures.

## Solved Examples with Gay Lussac’s Law Formula

Some of the solved problems based on the Gay-Lussac Law Formula are given below:

### Example-1

What is the volume of a quantity of gas at 27^{o}C if its volume was 400mL at 0^{o}C? The pressure remains constant.

**Solution:** Since the temperature increased, the volume increased. Here we must use a temperature factor greater than 1, that is 300/273

The temperature must be in degrees kelvin.

20^{o}C + 273 = 300 K

0^{o}C + 273 = 273 K

400mL × 300K / 273K = 439.5 mL

### Example-2

Solve Gay-Lussac’s Law to get an expression for the unknown volume. Substitute the appropriate data into the formula. V_{1} = 400mL, V_{2} = ? mL, T_{1} = 0^{o}C, T_{2} = 27^{o}C

**Solution: ** V_{1} = 400mL, V_{2} = ? mL

T_{1} = 0^{o}C + 273 = 273K

T_{2} = 27^{o}C + 273 = 300K

Substitute all the values in the corresponding formula

V_{1} / T_{1} = V_{2} / T_{2}

V_{2} = (400 mL ×300 )/273 = 439.56 mL

### Example-3

Heating a gas cylinder to 250 K raises its pressure to 2.0 atm. What was its initial temperature, assuming the gas started out at ambient pressure (1.0 atm)?

P_{1} / T_{2} = P_{2} / T_{1}

(1.0 atm) (250 K) = (2.0 atm) (T_{1})

T_{1} = (1.0 atm) (250 K) / (2.0 atm)

T_{1} = 125 K

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