In Problems $67,68,69,$ and $70$ you are given the equation(s) used to solve a problem. For each of these, you are to write a realistic problem for which this is the correct equation(s).
$p_2=\frac{300{\text{ cm}}^3}{100{\text{ cm}}^3}\times 1\times 2\text{ atm}$

Verified step by step guidance

1. Step 1: Recognize that the given equation resembles the application of Boyle's Law, which states that for a fixed amount of gas at constant temperature, the product of pressure and volume remains constant: \( P_1 V_1 = P_2 V_2 \).
2. Step 2: Identify the variables in the equation. Here, \( P_2 \) is the final pressure, \( V_1 = 300 \; \text{cm}^3 \) is the initial volume, \( V_2 = 100 \; \text{cm}^3 \) is the final volume, and \( P_1 = 2 \; \text{atm} \) is the initial pressure.
3. Step 3: Write a realistic problem based on the context of Boyle's Law. For example: 'A gas is initially at a pressure of 2 atm and occupies a volume of 300 cm³. If the gas is compressed to a volume of 100 cm³ while keeping the temperature constant, what is the final pressure of the gas?'
4. Step 4: Rearrange Boyle's Law equation to solve for \( P_2 \): \( P_2 = \frac{P_1 V_1}{V_2} \). Substitute the given values into the equation: \( P_2 = \frac{2 \; \text{atm} \times 300 \; \text{cm}^3}{100 \; \text{cm}^3} \).
5. Step 5: Simplify the expression to find \( P_2 \). This step involves basic arithmetic, but the final numerical result is not calculated here as per the instructions.

Key Concepts

Gas Laws

Gas laws describe the behavior of gases in relation to pressure, volume, and temperature. The ideal gas law, for example, relates these variables through the equation PV=nRT. Understanding these laws is crucial for solving problems involving gas behavior, as they provide the foundational relationships that govern how gases respond to changes in their environment.

Pressure and Volume Relationship

The relationship between pressure and volume of a gas is often described by Boyle's Law, which states that at constant temperature, the pressure of a gas is inversely proportional to its volume. This means that if the volume of a gas decreases, its pressure increases, and vice versa. This concept is essential for understanding how changes in one variable affect the other in gas-related problems.

Units of Measurement

In physics, using consistent units of measurement is vital for accurate calculations and problem-solving. In the context of gas laws, pressure is often measured in atmospheres (atm), volume in cubic centimeters (cm³), and temperature in Kelvin (K). Understanding how to convert between different units and apply them correctly in equations is crucial for solving real-world problems involving gases.