Overview
This lecture explains how to rearrange the combined gas law equation to solve for any variable using both a shortcut cross-multiplication method and a traditional algebraic method.
Shortcut (Cross Multiplication) Method
- To solve for a variable, cross multiply both sides to eliminate denominators.
- Get the desired variable alone and ensure it appears in the numerator (top) of the fraction.
- Example: To solve for T₂: P₁ × V₁ × T₂ = T₁ × P₂ × V₂ → T₂ = (T₁ × P₂ × V₂) / (P₁ × V₁)
- Example: To solve for V₂: P₁ × V₁ × T₂ = T₁ × P₂ × V₂ → V₂ = (P₁ × V₁ × T₂) / (T₁ × P₂)
- You can always flip the equation so the solved variable is on the left side.
Traditional Algebraic Method
- Isolate the target variable by stepwise multiplication or division to move terms across the equation.
- To solve for P₁: Multiply both sides by T₁, then divide both sides by V₁.
- Example: P₁ = (P₂ × V₂ × T₁) / (T₂ × V₁)
- To solve for T₁: Multiply both sides by T₁, then T₂, then divide by P₂ and V₂.
- Example: T₁ = (T₂ × P₁ × V₁) / (P₂ × V₂)
- Always ensure the variable you solve for is alone and in the numerator.
- Variable placement (left or right of equals sign) is interchangeable.
Key Terms & Definitions
- Combined Gas Law — An equation relating pressure (P), volume (V), and temperature (T) of a gas: (P₁ × V₁) / T₁ = (P₂ × V₂) / T₂
- Cross Multiplication — Method where both sides of an equation are multiplied across to remove denominators.
- Numerator — The top part of a fraction.
- Denominator — The bottom part of a fraction.
Action Items / Next Steps
- Practice rearranging the combined gas law for other variables not covered in the examples.
- Use both shortcut and traditional algebra methods for upcoming homework.