Step-By-Step Guide To Deriving The Combined Gas Law Formula
Understanding the Combined Gas Law is essential for anyone diving into the fascinating world of chemistry and physics. This law elegantly combines the principles of Boyle's Law, Charles's Law, and Gay-Lussac's Law into one comprehensive formula. But let’s be real—deriving it can seem like a daunting task at first glance. Don’t worry! We’re here to break it down step by step, making it as easy as pie (or should we say, as easy as gas laws?).
By the end of this guide, you'll not only be able to derive the Combined Gas Law but also understand its applications and significance in real-world scenarios. Let’s dive in, shall we?
What is the Combined Gas Law?
Before we jump into the derivation, let’s quickly clarify what the Combined Gas Law actually is. In simple terms, it's a formula that relates the pressure, volume, and temperature of a gas when the amount of gas is held constant. The law can be mathematically expressed as:
[
\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}
]
Where:
- (P) = Pressure
- (V) = Volume
- (T) = Temperature (in Kelvin)
This equation essentially states that for a given amount of gas, the ratio of pressure times volume to temperature remains constant.
Step-by-Step Derivation of the Combined Gas Law
Step 1: Start with Individual Gas Laws
To derive the Combined Gas Law, we begin with three fundamental gas laws:
-
Boyle’s Law: At constant temperature, the pressure and volume of a gas are inversely proportional.
[
P_1 V_1 = P_2 V_2
] -
Charles’s Law: At constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin.
[
\frac{V_1}{T_1} = \frac{V_2}{T_2}
] -
Gay-Lussac’s Law: At constant volume, the pressure of a gas is directly proportional to its temperature in Kelvin.
[
\frac{P_1}{T_1} = \frac{P_2}{T_2}
]
Step 2: Rearranging the Laws
Now, let’s rearrange each of these laws to set them up for combination:
-
From Boyle’s Law:
[
P_1 V_1 = P_2 V_2 \Rightarrow \frac{P_1}{P_2} = \frac{V_2}{V_1}
] -
From Charles’s Law:
[
\frac{V_1}{T_1} = \frac{V_2}{T_2} \Rightarrow V_1 T_2 = V_2 T_1
] -
From Gay-Lussac’s Law:
[
\frac{P_1}{T_1} = \frac{P_2}{T_2} \Rightarrow P_1 T_2 = P_2 T_1
]
Step 3: Combining the Rearranged Laws
Now, let’s combine these equations. If we multiply the rearranged forms of Boyle’s and Charles’s Law together, we get:
[
\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}
]
This elegantly leads us to the Combined Gas Law.
Understanding the Derivation of the Combined Gas Law
Now that we have derived the formula, let’s break down what each part means:
- Pressure ((P)): The force exerted by gas particles colliding with the walls of its container.
- Volume ((V)): The space occupied by the gas.
- Temperature ((T)): A measure of the kinetic energy of the gas particles, expressed in Kelvin.
This relationship shows how changes in one of these variables will affect the others, as long as the amount of gas remains constant.
Combined Gas Law Derivation Examples
To truly grasp the Combined Gas Law, let’s look at some practical examples:
Example 1: Compressing a Gas
Imagine you have a gas at a pressure of 1 atm, a volume of 10 L, and a temperature of 300 K. If you compress the gas to a volume of 5 L at a temperature of 600 K, what is the new pressure?
Using the Combined Gas Law:
[
\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}
]
Plugging in the values:
[
\frac{1 , \text{atm} \times 10 , \text{L}}{300 , \text{K}} = \frac{P_2 \times 5 , \text{L}}{600 , \text{K}}
]
Solving for (P_2) gives:
[
P_2 = \frac{(1 , \text{atm} \times 10 , \text{L} \times 600 , \text{K})}{(300 , \text{K} \times 5 , \text{L})} = 4 , \text{atm}
]
Example 2: Heating a Gas
Suppose you have a gas with a pressure of 2 atm and a volume of 8 L at 200 K. What happens to the volume if the temperature is increased to 400 K?
Using the Combined Gas Law:
[
\frac{2 , \text{atm} \times 8 , \text{L}}{200 , \text{K}} = \frac{P_2 \times V_2}{400 , \text{K}}
]
Assuming the pressure remains constant:
[
\frac{2 \times 8}{200} = \frac{2 \times V_2}{400}
]
Solving gives:
[
V_2 = 16 , \text{L}
]
Conclusion
And there you have it—a comprehensive breakdown of the Combined Gas Law and its derivation! By following our step-by-step derivation of combined gas law, you now have a solid understanding of how it works and why it’s important. Remember, the beauty of this law lies in its ability to simplify complex gas behavior into a single equation.
So, next time someone asks about the mathematical derivation of the combined gas law, you can confidently explain it with style! Want to dive deeper into gas laws? Keep experimenting—science is all about curiosity and discovery!
