Q.Isothermal equation of an ideal gas is

The Combined Gas Law: One Rule to Rule Them All

You already know that gases are sensitive. Squeeze them, they get smaller. Heat them, they expand. But what happens when you do both at once? That’s where the Combined Gas Law comes in — it’s the single equation that handles pressure, volume, and temperature changing together.

The Intuition: A Balloon in Two Hands

Imagine a balloon filled with air. Now picture two things happening at the same time:

• You push on the balloon (increase pressure). The balloon shrinks.
• You also put the balloon in the sun (increase temperature). The balloon expands.

Which wins? The Combined Gas Law tells you the net result. It’s like having two forces pulling in opposite directions — the law gives you the final size and pressure after both changes.

The Three Individual Laws It Combines

Before the combined version, scientists discovered three separate relationships:

1. Boyle’s Law (constant temperature): $P_1 V_1 = P_2 V_2$ — pressure and volume are inversely related.
2. Charles’s Law (constant pressure): $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ — volume and temperature are directly related.
3. Gay-Lussac’s Law (constant volume): $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ — pressure and temperature are directly related.

Each law holds one variable fixed. But in real life, nothing stays fixed. The Combined Gas Law is the merger of all three.

The Precise Statement

Where:

• $P$ = pressure (any unit, as long as it’s the same on both sides)
• $V$ = volume (any unit, consistent)
• $T$ = absolute temperature (must be in Kelvin, never Celsius or Fahrenheit)

The subscripts 1 and 2 refer to “before” and “after” the change.

Why Temperature Must Be in Kelvin

This is the most common mistake. Celsius and Fahrenheit have negative numbers. If you plug in $0^\circ\text{C}$, you get division by zero — nonsense. Kelvin starts at absolute zero ($-273.15^\circ\text{C}$), so all temperatures are positive and proportional to actual molecular motion.

How to Use It: A Simple Strategy

When you see a problem where pressure, volume, and temperature all change, follow these steps:

1. Identify what’s given and what’s asked. Write down $P_1$, $V_1$, $T_1$, $P_2$, $V_2$, $T_2$ — some will be unknown.
2. Convert all temperatures to Kelvin.
3. Plug into $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$.
4. Solve for the unknown. If you need $V_2$, rearrange: $V_2 = \frac{P_1 V_1 T_2}{T_1 P_2}$.

Worked Example

A gas occupies 5.0 L at 2.0 atm and 300 K. What volume will it occupy at 1.0 atm and 400 K?

Step 1: $P_1 = 2.0$, $V_1 = 5.0$, $T_1 = 300$, $P_2 = 1.0$, $T_2 = 400$, $V_2 = ?$ …