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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

## 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

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 **P**** _{1} ** and

**P**

**are the initial and final pressure, and**

_{2}**V**

**and**

_{1}**V**

**are the initial and final volume, respectively.**

_{2}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:** **S**olve 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.