Question

If \(456 \mathrm{dm}^{3}\) of krypton at \(101 \mathrm{kPa}\) and \(21^{\circ} \mathrm{C}\) is compressed into a \(30.1-\mathrm{dm}^{3}\) tank at the same temperature, what is the pressure of krypton in the tank?

Short answer

The final pressure of krypton is approximately 1529.57 kPa.

Step-by-step solution

Identify Given Properties

We have the initial volume \( V_1 = 456 \, \mathrm{dm}^3 \), the initial pressure \( P_1 = 101 \, \mathrm{kPa} \), and the final volume \( V_2 = 30.1 \, \mathrm{dm}^3 \). The temperature remains constant, so we can use Boyle's Law for this problem.

Use Boyle's Law

Boyle's Law states that for a given amount of gas at constant temperature, the pressure and volume are inversely related: \( P_1 \cdot V_1 = P_2 \cdot V_2 \). We need to solve for the final pressure \( P_2 \).

Rearrange the Formula

Rearrange Boyle’s Law to solve for \( P_2 \):\[ P_2 = \frac{P_1 \cdot V_1}{V_2} \]

Substitute Known Values and Solve

Substitute the known values into the rearranged equation:\[ P_2 = \frac{101 \, \mathrm{kPa} \times 456 \, \mathrm{dm}^3}{30.1 \, \mathrm{dm}^3} \]Calculate the value for \( P_2 \).

Final Calculation

The computation gives:\[ P_2 = \frac{101 \, \times 456}{30.1} = 1529.57 \, \mathrm{kPa} \]So, the final pressure of krypton in the tank is approximately \( 1529.57 \, \mathrm{kPa} \).

Key concepts

Gas Laws

The gas laws are fundamental principles in chemistry and physics that describe how gases behave under different conditions. These laws provide a framework to understand the relationship between variables such as pressure, volume, and temperature.

Among the core gas laws, we find Boyle's Law, Charles's Law, and Avogadro's Law. Each of these examines the interaction of two variables while keeping the others constant. For this exercise, Boyle's Law is in focus. It describes the relationship between the pressure and volume of a gas when temperature is held constant.

Understanding these laws helps in predicting how a gas will react when exposed to changes in conditions. By knowing just a few properties, like the initial volume and pressure, one can calculate unknowns such as final pressure or volume with ease using the appropriate gas law.

Pressure Calculations

Pressure calculations are an essential part of understanding how gases behave, especially when they are compressed or decompressed. In our exercise, we need to determine the final pressure of krypton gas when its volume is reduced.

This can be done using Boyle’s Law, which relates pressure and volume directly. We start by identifying all known values: the initial pressure, initial volume, and final volume. Then, we rearrange the equation to solve for the unknown, which in this instance is the final pressure.

The formula used here is particularly handy:

• For constant temperature, Boyle's Law is used: \( P_1 \cdot V_1 = P_2 \cdot V_2 \).
• Rearranging gives \( P_2 = \frac{P_1 \cdot V_1}{V_2} \).

With this approach, you can efficiently calculate the desired pressure value by plugging in the known quantities.

Volume and Pressure Relationship

The relationship between volume and pressure is a core concept in understanding how gases work, especially under conditions where the temperature remains constant. According to this principle, known as Boyle's Law, the pressure of a gas is inversely proportional to its volume. This means when one increases, the other decreases if all other factors stay the same.

For example, if a gas is compressed to half its volume, its pressure will double given that the temperature does not change. This is represented by the equation:\[ P_1 \cdot V_1 = P_2 \cdot V_2 \]
In practical terms, this law helps solve problems like the one in the exercise, where you start with an initial state of a gas and change one of its properties.

By observing how volume reduction affects pressure, one can design systems such as cylinders and pipes to safely handle gas pressure changes, critical for industries dealing with gases.