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Combined Gas Law Calculator

To use combined gas law calculator, select missing value, fill the required input boxes, and click calculate button

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Combined gas law calculator  

Find any missing parameters from the combined gas law equation with this calculator. This tool is easy to operate because its interface is understandable at one glance. 

What is the combined gas law?

The Combined Gas Law is derived from the three fundamental gas laws and allows us to predict the outcome when multiple gas properties change simultaneously while other factors, such as the amount of gas and the gas constant, remain unchanged.

 

These laws are:

  • Boyle's Law                 P₁V₁ = P₂V₂
  • Charles's Law             V₁/T₁ = V₂/T₂
  • Gay-Lussac's Law      P₁/T₁ = P₂/T₂

combined gas law

When you combine these three laws into the Combined Gas Law, you get the following equation:

                                        (P₁ * V₁) / T₁ = (P₂ * V₂) / T₂

Combined Gas Law applies to ideal gases, which are gases that strictly follow the ideal gas equation PV = nRT

How to derive the combined gas law equation?

We start by considering Boyle's Law, which states that the pressure (P) and volume (V) of a gas are inversely proportional at constant temperature (T). Mathematically, Boyle's Law can be expressed as:

P₁ * V₁ = P₂ * V₂

where P1  and P2 are the initial and final pressure, and V1 and  V2 are the initial and final volume, respectively.

Next, consider Charles's Law, which states that the volume (V) and temperature (T) of a gas are directly proportional at constant pressure (P). Mathematically, Charles's Law is written as:

V₁ / T₁ = V₂ / T₂

where V₁ and T₁ are the initial volume and temperature, and V₂ and T₂ are the final volume and temperature, respectively.

Lastly, take Gay-Lussac's Law, which states that the pressure (P) and temperature (T) of a gas are directly proportional at constant volume (V). Mathematically, Gay-Lussac's Law can be expressed as:

P₁ / T₁ = P₂ / T₂

where P₁ and T₁ are the initial pressure and temperature, and P₂ and T₂ are the final pressure and temperature, respectively.

Now, to combine these three laws into a single equation (Combined Gas Law), we need to find a way to relate all three variables (P, V, and T) together. This can be done by rearranging the equations above to isolate P, V, and T on one side:

From Boyle's Law:

P₁ = (P₂ * V₂) / V₁

From Charles's Law:

V₁ = (V₂ * T₁) / T₂

From Gay-Lussac's Law:

P₁ = (P₂ * T₁) / T₂

Combining all the laws:

P₁ * V₁ / T₁ = (P₂ * V₂) / V₁ * (V₂ * T₁) / T₂ * (P₂ * T₁) / T₂

Cancel out some terms:

P₁ * V₁ / T₁ = P₂ * V₂

This is the final equation for the Combined Gas Law:

(P₁ * V₁) / T₁ = (P₂ * V₂) / T₂

How to find the unknown term in combined gas law?

To find an unknown term in the Combined Gas Law, rearrange the equation based on the given data and solve for the desired variable. Let's go through some examples to illustrate the process:

Example 1:

A sample of gas has an initial pressure of 2 atm, an initial volume of 4 liters, and an initial temperature of 300 K. If the final pressure is 3 atm, and the final volume is 6 liters, what is the final temperature?

Given:

P₁ = 2 atm (initial pressure)

V₁ = 4 liters (initial volume)

T₁ = 300 K (initial temperature)

P₂ = 3 atm (final pressure)

V₂ = 6 liters (final volume)

T₂ = ? (final temperature, to be determined)

Using the Combined Gas Law equation:

(P₁ * V₁) / T₁ = (P₂ * V₂) / T₂

Step 1: Substitute the known values:

(2 atm * 4 liters) / 300 K = (3 atm * 6 liters) / T₂

Step 2: Solve for T₂:

(8 atm * liters) / 300 K = (18 atm * liters) / T₂

Step 3: Cross-multiply to isolate T₂.

8 atm * liters * T₂ = 18 atm * liters * 300 K

Step 4: Divide both sides by 8 atm * liters:

T₂ = (18 atm * liters * 300 K) / (8 atm * liters)

T₂ ≈ 675 K

So, the final temperature (T₂) is approximately 675 Kelvin.

Example 2:

A gas at 1 atm pressure and 25°C occupies a volume of 10 liters. If the final pressure is 2 atm, and the final temperature is 50°C, what will be the final volume?

Given:

P₁ = 1 atm (initial pressure)

V₁ = 10 liters (initial volume)

T₁ = 25°C (initial temperature)

P₂ = 2 atm (final pressure)

V₂ = ? (final volume, to be determined)

T₂ = 50°C (final temperature)

Use the temperature calculator for Celsius to Kelvin conversion.

T₁ = 25°C = 25 + 273.15 = 298.15 K (initial temperature)

T₂ = 50°C = 50 + 273.15 = 323.15 K (final temperature)

Using the Combined Gas Law equation:

(P₁ * V₁) / T₁ = (P₂ * V₂) / T₂

Step 1: Substitute the known values:

(1 atm * 10 liters) / 298.15 K = (2 atm * V₂) / 323.15 K

Step 2: Solve for V₂:

(10 liters * 323.15 K) / (298.15 K) = 2 atm * V₂

Step 3: Divide both sides by 2 atm to isolate V₂.

V₂ = (10 liters * 323.15 K) / (2 atm * 298.15 K)

V₂ ≈ 5.45 liters

So, the final volume (V₂) is approximately 5.45 liters.

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