P1V1/n1T1 = P2V2/n2T2; what is it, and how to use it?

P₁V₁/n₁T₁ = P₂V₂/n₂T₂ is the general gas equation that relates the pressure, volume, temperature, and number of moles of an ideal gas under two different circumstances. It is also known as the combined gas law, as it is derived from the combination of four different gas laws.

In this article, we will discuss these four gas laws as well as we will learn to use P₁V₁/n₁T₁ = P₂V₂/n₂T₂ using plenty of numerical questions.

So come along and continue reading!

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

The formula P₁V₁/n₁T₁ = P₂V₂/n₂T₂ is derived from the 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

PV = k….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 α T

P = kT……Equation (iii)

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

Avogadro’s law: It states that an equal volume of gases at a constant temperature and pressure contain the same number of particles. In other words, the volume occupied by a gas is directly proportional to its number of moles.

V α n

V = kn……Equation (iv)

• Where V= volume, n = number of moles of gas, k= constant

Combining equations (i), (ii), (iii), and (iv) into a single formula gives us:

PV = k (nT) … Equation (v)

As per equation (v), the pressure and volume of a gas are inversely proportional to each other; similarly, temperature and the number of moles of gas are inversely related. In contrast, all the variables lying on opposite sides of the equation are directly related to on another.

However, the combined effect of each of the four variables is such that it leads to a constant value (k).

Under ideal temperature, pressure, and volume conditions, k = R (ideal gas constant) = 0.08206 L.atm/ (mol.K). Thus, the equation (v) turns into the ideal gas equation, i.e., PV = nRT, frequently used in Chemistry. 

Under two different circumstances, the change in either of these variables can be determined using P₁V₁/n₁T₁ = P₂V₂/n₂T₂ if all the other variables are pre-determined.

What does P₁V₁/n₁T₁ = P₂V₂/n₂T₂ represent?

For a gaseous system undergoing some change, P₁V₁/ n₁T₁ = P₂V₂/n₂T₂ 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)
• n₁ = Initial number of moles of gas (Units = mol)
• 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)
• n₂ = Final number of moles of gas (Units = mol)

If the number of moles of gas (n) stays constant, i.e., n₁ = n_2, then the combined gas law is reduced to:

P₁V₁/T₁ = P₂V₂/T₂

Now let us practice some examples using P₁V₁/n₁T₁ = P₁V₁/n₂T₂.

Where and how to use P₁V₁/n₁T₁ = P₂V₂/n₂T₂? – Examples

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

Hence, the formula P₁V₁/n₁T₁ = P₂V₂/n₂T₂ can be rearranged in the following eight 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/n₁T₁ = P₂V₂/n₂T₂.

For example, In a closed, flexible container, 3 moles of a gas have an initial pressure of 2 atm and occupy a volume of 4 L at a temperature of 300 K. If the number of moles of gas is increased to 3.5 while the temperature is raised to 350 K, the pressure inside the container increases to 3 atm. Find the final volume occupied by the gas using P₁V₁/n₁T₁ = P₂V₂/n₂T₂.

Solution

The data obtained from the question statement is as follows:

• P₁ = 2 atm
• V₁ = 4 L
• n₁ = 3 mol
• T₁ = 300 K
• P₂ = 3 atm
• V₂ = ?
• n₂ = 3.5 mol
• T₂ = 350 K

Substitute the above data into the formula:

Via cross-multiplication:

8(3.5) (350) = 9(300) (V₂)

V₂ = 9800/2700

∴ V₂ = 3.63 L

Result: The final volume of gas in the container is 3.63 L.

Another example is- 2 moles of a gas occupies a volume of 2.5 L at 400 K. As the temperature of the gas is increased to 500 K and its number of moles are increased to 3 moles, the pressure of the gas raised to 2 atm while the volume occupied by the gas becomes 7.03 L. Can you find the initial gas pressure by applying the general gas equation, P₁V₁/n₁T₁ = P₂V₂/n₂T₂?

Solution

As per the question statement,

• P₁ = ?
• V₁ = 2.5 L
• n₁ = 2 moles
• T₁ = 400 K
• P₂ = 2 atm
• V₂ = 7.03 L
• n₂ = 3 moles
• T₂ = 500 K

Substitute the above data into the general gas equation and find the initial pressure by making P₁ the subject of the formula:

P₁ (2.5) (3) (500) = (2) (7.03) (2) (400)

P₁ (3750) = 11248

P₁ = 11248/3750

∴ P₁ = 2.999 = 3.0

Result: The initial pressure of the gas is 3.0 atm.

Similarly, you can solve for an unknown temperature or number of moles making the unknown entity the subject of the formula as shown in the above two examples for volume and pressure.

FAQ

Which gas laws are used to derive P1V1/n1T1 = P2V2/n2T2?

The following four gas laws are used to derive P1V1/n1T1 = P2V2/n2T2 : Charles’ law (V α T) Boyles’s law (P α 1/V) Gay-Lussac’s law (P α T) Avogadro’s law  (V α n)

When do we use P1V1 = P2V2 instead of P1V1/n1T1 = P2V2/n2T2?

If the temperature (T) and number of moles of gas (n) are kept constant, then the general gas equation (P1V1/n1T1 = P2V2/n2T2) is reduced to P1V1 = P2V2.

How to find T2 using P1V1/n1T1 = P2V2/n2T2?

The formula P1V1/n1T1 = P2V2/n2T2 can be rearranged to make final temperature (T2) the subject of the formula as follows: ⇒ P1V1/n1T1 = P2V2/n2T2 P1V1 (n2T2) = P2V2 (n1T1) ∴ T2 = P2V2 (n1T1)/P1V1n2