Combined gas law equation; what is it, where to use P1V1/T1 = P2V2/T2

P₁V₁/T₁ = P₂V₂/T₂ represents the combined gas law, a mathematical relationship used to study the behavior of gases under changing temperature, pressure, and volume conditions.

It is referred to as the combined gas law because it is derived by the combination of three different gas laws into a single equation.

In this article, we will teach you what P₁V₁/T₁ = P₂V₂/T₂ represents and how to use it to solve numerous physics, chemistry, and mathematical problems.

But first, let us discuss some background related to the combined gas law.

Which three laws form the basis of the combined gas law?

The combined gas law is a combination of:

• Charles law: It says that the volume of a gas is directly proportional to its temperature if the pressure is kept constant.

V α T

V = kT…..Equation (i)

• Where V = volume, T= temperature, k= proportionality constant

• Boyle’s law: It states that the pressure exerted by a gas is inversely proportional to its volume.

P α 1/V

P = k/V….Equation (ii)

• Where P = pressure, V=volume, k= constant

• Gay-Lussac’s law: It says that the pressure exerted by a gas is directly proportional to the temperature provided to it.

P = kT……Equation (iii)

• Where P = pressure, T = temperature, k= constant

Combining equations (i), (ii), and (iii) gives us PV/T = k, which implies that pressure and volume are inversely related to each other while both are in direct relationship with the temperature of the gas.

Increasing the temperature of a gas in a container increases its volume, which in turn decreases the pressure so that the value of k stays unchanged.

Therefore, at a time, the effect of only two variables can be investigated, keeping the third variable constant.

What does P₁V₁/T₁ = P₂V₂/T₂ represent?

For a gaseous system undergoing some change, P₁V₁/T₁ = P₂V₂/T₂ represents:

• P₁ = Initial pressure of a gaseous system (Units = Pa, atm or mm Hg)
• V₁ = Initial volume of gas (Units = L or m³)
• T₁ = Initial temperature (Units = K)
• P₂ = Final pressure of the gaseous system (Units = Pa, atm or mm Hg)
• V₂ = Final volume of the gas (Units = L or m³)
• T₂ = Final temperature (Units = K)

Now let us practice some examples using P₁V₁/T₁ = P₁V₁/T₂.

Where and how to use P₁V₁/T₁ = P₂V₂/T₂? – Examples

You must note that to use this formula, five variables should ideally be known so that we can find the unknown variable, whether be it pressure, temperature, or volume.

Hence, the formula P₁V₁/T₁ = P₂V₂/T₂ can be rearranged in the following six ways, depending upon which variable is unknown.

Another important point is that consistency in units is very important. The pressure, volume, and temperature must be in the same units on either side of P₁V₁/T₁ = P₂V₂/T₂.

For example, A gas sample in a container exerts an initial pressure of 2.0 atm; the volume of the gas in the container is 3.0 L at 300 K. What is the final volume of the gas if the pressure is increased to 3.5 atm on increasing the temperature to 350 K?

Solution

As per the question statement;

P₁ = 2.0 atm

V₁ = 3.0 L

T₁ = 300 K

P₂ =3.5 atm

V₂ = ?

T₂ = 350 K

We need to insert the above data in the combined gas equation and solve for V₂ by cross multiplication, as shown below:

6(350) = 1050 V₂

2100 = 1050 V₂

∴ V₂ = 2100/1050 = 2

Result: The final volume of gas in the container is 2.0 L.

Another example is- A gas at 110 kPa and 30°C fills a flexible container with an initial volume of 2.00 L. If the temperature is raised by 50°C, the volume decreases to 0.58 L. Calculate the final pressure in the gas container.

Solution

As per the question statement;

P₁ = 110 kPa

V₁ = 2.0 L

T₁ = 30°C

P₂ = ?

V₂ = 0.58 L

T₂ = 30 + 50 = 80 °C

Remember to convert both T₁ and T₂ from °C to Kelvin (K).

1 K = 1°C + 273

T₁ = 30 + 273 = 303 K

T₂ = 80 + 273 = 353 K

Now let’s plug in all the above data into the equation and find P₂:

Cross multiply and make P₂ the subject of the formula:

220(353) = P₂ (175.74)

77660 = 175.74 P₂

∴ P₂ = 77660/175.74 = 441.9 kPa

Note: As P₁ is given in kilopascals (kPa), therefore P₂ is also obtained in kPa, respectively.

Result: The final pressure in the gas container is 441.9 kPa.

More examples on P₁V₁/T₁ = P₂V₂/T₂

If you collect a gas at 620 mmHg and 13°C. At the time of collection, it takes up a volume of 1.3 L. What will be the temperature of the gas in Kelvin if the pressure is increased to 1.2 atm and the volume is decreased to 0.85 L?

As per the question statement; P1 = 620 mm Hg V1 = 1.3 L T1 = 13° C (Converting the temperature into Kelvin gives us 13 + 273) = 286 K P2 = 1.2 atm V2 = 0.85 L T2 =? To bring consistency in units, P2 is also converted from atm into mm Hg as follows: 1 atm = 760 mm Hg 1.2 atm = 1.2 x 760 = 912 mm Hg Now substitute all the above data into the combined gas law equation to find the unknown variable, i.e., T2. Via cross multiplication: 806 (T2) = 286(912 x 0.85) 806 (T2) = 221707.2 ∴ T2 = 221707.2/806 = 275.1 K Result: The final temperature of the gas is 275.1 K

A gas initially occupies a volume of 2.5 L when its temperature is raised twice from 25°C while the pressure changes from 1.75 atm to 3 atm. Find the initial volume of the gas by applying the formula P1V1/T1 = P2V2/T2.

As per the question statement; P1 = 1.75 atm V1 = ? T1 = 25° C (Converting the temperature into Kelvin gives us 25 + 273) = 298 K P2 = 3.0 atm V2 = 2.5 L T2 = 323 K (as the temperature is raised twice, so 2(25) = 50°C = 50 + 273 = 323 K). Now substitute all the above data into the combined gas law to find the unknown variable, i.e., V1. Via cross multiplication: 565.25 (V1) = 298(3.0)(2.5) 565.25 (V1) = 2235 ∴ V1 = 2235/565.25= 3.95 L Result: The initial volume occupied by the gas is 3.95 L.

 FAQ 

How is the combined gas law P1V1/T1 = P2V2/T2 derived?

The combined gas law is derived from the combination of three different gas laws, i.e., Charles’ law, Boyles’s law, and Gay-Lussac’s law.

What do the components of P1V1/T1 = P2V2/T2 represent?

In P1V1/T1 = P2V2/T2, P1 = Initial pressure (in atm, Pa, or mm Hg) V1 = Initial volume (in L or m3) T1 = Initial temperature (in K) P2 = final pressure (in atm, Pa, or mm Hg) V2 = final volume (in L or m3) T2 = final temperature (in K)

In the combined gas law P1V1/T1 = P2V2/T2, whose law is: a) V1/T1 = V2/T2 b) P1V1 = P2V2 c) P1/T1 = P2/T2

a) V1/T1 = V2/T2 represents Charles gas law which says volume is directly proportional to temperature if the pressure is kept constant. b) P1V1 = P2V2 denotes Boyle’s law which says that the pressure and volume of a gas are inversely related to each other at a constant temperature. c) P1/T1 = P2/T2 represents Gay-Lussac’s law, according to which pressure and temperature are inversely proportional to each other at constant volume.

Given the equation P1V1/T1 = P2V2/T2, which of the following is the formula for the final temperature?  A) T2 = P2V2T1/P1V1 B) T2 = P1V2T1/P2V1 C) T2 = P2V1T1/P1V2 D) T2 = P1V1/P2V2T1

Option A is the correct answer. The equation can be rearranged to make T2 the subject of the formula, as shown below:   ⇒  P1V1/T1 = P2V2/T2 Via cross multiplication: P1V1T2 = P2V2T1 ∴ T2 = P2V2T1/P1V1