The Equation of Gay-Lussac's Law

18th Dec 2019 @ 3 min read

The Equation of Gay-Lussac's law is P = kT. Here, P and T are the pressure and temperature of an enclosed ideal gas and k is a constant of proportionality.

Gay-Lussac's law says the pressure of an enclosed ideal gas is directly proportional to its temperature at the constant volume of the gas. The law is sometimes called pressure law because it relates the pressure of a gas to the temperature of the gas.

For a fixed amount of gas at the constant volume, the pressure P is directly proportional to the temperature T.

Removing the proportionality,

Here, k is a constant of proportionality.

Rearranging the equation,

From the above expression, the ratio of pressure to temperature (^P⁄_V) remains constant when the volume (V) and the amount (n) of gas are constant.

Note: In the above expression, the temperature is in an absolute scale (kelvin).

The equation of Gay-Lussac's law in the celsius scale

Let T and t be the temperature in the kelvin and the degree celsius.

Substituting T,

Here, k₁ = k × 273.15. The above equation is a straight similar to y = mx + c.

Gay-Lussac's law under different conditions

Let P₁ and P₂ be pressures at temperatures T₁ and T₂ respectively. We can establish the relation between the two conditions.

Pressure and temperature at two different conditions

For an enclosed ideal gas at the constant volume,

Combining both equations,

From the above formula, we can calculate the pressure or the temperature at any unknown condition if the pressure and the temperature are known at any one condition.

Let take an example to better understand it.

Example 1

Statement: A cruise ship voyages from the port of San Diego to the port of Seattle. The ship is equipped with carbon dioxide (CO₂) cylinders for fire safety. The temperature and pressure of the cylinder gas at San Diego is 12 °C and 50.1 bar. What will the pressure of the gas when the ship arrives at Seattle? The temperature at Seattle will be 5 °C when the ship arrives.

Solution: Let T₁ and T₂ be the temperatures at San Diego and Seattle.

Let P₁ and P₂ be the pressures at San Diego and Seattle. P₁ = 50.1 bar.

From Gay-Lussac's law,

The ratio of pressure to temperature at both conditions

Thus, the pressure at Seattle is 48.9 bar.

The pressure at San Diego and Seattle — The temperature and pressure at San Diego and Seattle

When the temperature of the gas decreases from 12 °C to 5 °C, the pressure also decreases from 50.1 bar to 48.9 bar. This explains Gay-Lussac's law.

We can generalize the equation to any number of conditions. Consider a fixed amount of gas of the volume V. Let (P₁, T₁), (P₂, T₂), (P₃, T₃)… (P_i, T_i) be pressure and temperature at different conditions.

Pressure and temperature at i conditions — Pressure and temperature at i different conditions

According to the law, the ratio of pressure to temperature remains constant.

Example 2

Statement: Let us continue with the above example. The ship after reaching the port of Seattle is scheduled to travel towards the port of Anchorage, Alaska. Find the pressure of the gas in Anchorage if the temperature is −7 °C?