Question

If the pressure on a 1.00 -L sample of gas is increased from \(760 .\) mm Hg to 2.00 atm at constant temperature, what will the volume of the gas sample become?

Short answer

The final volume of the gas sample after increasing the pressure from 760 mm Hg to 2.00 atm at constant temperature is \(0.50 \, L\).

Step-by-step solution

List down given information

Here the given information: Initial Pressure, \(P_1 = 760\) mm Hg Initial Volume, \(V_1 = 1.00\) L Final Pressure, \(P_2 = 2.00\) atm Our first step is to convert the pressures to the same unit. Let's convert it to atm.

Convert pressures to the same unit

To convert the pressure from mm Hg to atm, we can use the conversion factor: 1 atm = 760 mm Hg. So, \(P_1 = 760 \, mm \, Hg * \frac{1 \, atm}{760 \, mm \, Hg}\) \(P_1 = 1 \, atm\) Now, we have the initial pressure in atm: \(P_1 = 1\) atm Since the final pressure is already given in atm: \(P_2 = 2\) atm

Use Boyle's Law and substitute values to find the final volume

Boyle's Law states that \(P_1V_1 = P_2V_2\). Now we can substitute our given values to find the final volume, \(V_2\). \(P_1V_1 = P_2V_2\) \((1 \, atm)(1.00 \, L) = (2.00 \, atm)V_2\)

Solve for the final volume, V_2

We have this equation: \(1.00 \, L = 2.00 \, atm * V_2\) To find \(V_2\), we will divide both sides by \(2.00 \, atm\): \(V_2 = \frac{1.00 \, L}{2.00 \, atm}\) \(V_2 = 0.50 \, L\) The final volume of the gas sample is \(0.50 \, L\).

Key concepts

Gas Laws

Understanding gas laws is essential for students studying chemistry and physics as they describe how various properties of gases—pressure, volume, temperature, and amount—are interconnected. In particular, gas laws help to predict one property if the others are known, making them invaluable to scientists and engineers in practical applications, such as inflating balloons, designing engines, and even predicting weather patterns.

One of the fundamental gas laws is Boyle's Law, which specifically addresses the pressure-volume relationship for a fixed amount of gas at constant temperature. This law posits that, for a given mass of an enclosed gas, its pressure and volume are inversely proportional. Other gas laws include Charles's Law, which describes the volume-temperature relationship at constant pressure, and the Combined Gas Law, which combines Boyle's and Charles's laws to relate pressure, volume, and temperature simultaneously.

Pressure-Volume Relationship

The pressure-volume relationship is a key aspect of Boyle's Law, a cornerstone in the study of gas behaviors. Boyle's Law states that if the temperature of the gas remains unchanged (isothermal conditions), the volume of a gas is inversely proportional to the pressure applied to it. This means that as the pressure on a gas increases, its volume decreases, as long as the amount of gas and the temperature remain constant.

To explore this concept further, consider a syringe filled with air. If you plug the opening and press the plunger, you're increasing the pressure on the gas inside. According to Boyle's Law, the volume of the air should decrease as a result, and indeed, the plunger moves inward, reducing the space available for the air. This relationship can be mathematically expressed as \(P_1V_1 = P_2V_2\), where \(P_1\) and \(V_1\) are the original pressure and volume, and \(P_2\) and \(V_2\) are the new conditions after a change in pressure.

Chemical Education

Chemical education aims to simplify complex scientific concepts to enhance understanding and interest among students. The use of practical examples, like Boyle's Law, allows educators to demonstrate how the theoretical principles apply to everyday life. Approaches might include hands-on experiments, simulations, and problem-solving activities that encourage active learning and help students to internalize the concepts.

By guiding students through exercises such as the pressure-volume relationship problem given in the original exercise, educators can foster critical thinking and practical skills. Breaking down the steps to reach a solution, they should not only offer the mathematical procedure but also connect it to real-world examples and emphasize the applicability of these laws beyond academic exercises. This contextualization helps students appreciate the relevance of science in their lives and may inspire future investigations.