Question

A sample of nitrogen gas has a pressure of 1.22 atm. If the amount of gas and the temperature are kept constant, what is the pressure if (a) its volume is decreased by \(38 \%\) ? (b) its volume is decreased to \(38 \%\) of its original volume?

Short answer

Answer: The pressure of the nitrogen gas when the volume is decreased by 38% is approximately 1.97 atm. The pressure of the nitrogen gas when the volume is decreased to 38% of its original volume is approximately 3.21 atm.

Step-by-step solution

Write the Boyle's Law formula

Boyle's Law states that, for a given mass of gas at constant temperature, the pressure is inversely proportional to its volume. Mathematically, we can represent this as: \(P_1V_1=P_2V_2\) where: - \(P_1\) is the initial pressure - \(V_1\) is the initial volume - \(P_2\) is the final pressure - \(V_2\) is the final volume

Find the pressure when volume is decreased by 38%

For part (a), we need to find the new pressure when the volume is decreased by 38%. Let's calculate the final volume first: \(V_2=V_1*(1- 0.38)\) Now, we can use Boyle's Law to find the new pressure: \(P_1V_1=P_2V_2\) => \(P_2=\dfrac{P_1V_1}{V_2}\) => \(P_2=\dfrac{P_1V_1}{V_1*(1- 0.38)} = \dfrac{P_1}{1-0.38}\) Substitute the initial pressure, \(P_1=1.22\) atm: \(P_2=\dfrac{1.22}{1-0.38}=\dfrac{1.22}{0.62}=1.97\text{ atm}\) So, the pressure of the nitrogen gas when the volume is decreased by 38% is approximately 1.97 atm.

Find the pressure when volume is decreased to 38% of its original volume

For part (b), we need to find the new pressure when the volume is decreased to 38% of its original volume. Let's calculate the final volume first: \(V_2=V_1*0.38\) Now, we can use Boyle's Law to find the new pressure: \(P_1V_1=P_2V_2\) => \(P_2=\dfrac{P_1V_1}{V_2}\) => \(P_2=\dfrac{P_1V_1}{V_1*0.38} = \dfrac{P_1}{0.38}\) Substitute the initial pressure, \(P_1=1.22\) atm: \(P_2=\dfrac{1.22}{0.38}=3.21\text{ atm}\) So, the pressure of the nitrogen gas when the volume is decreased to 38% of its original volume is approximately 3.21 atm.

Key concepts

Basics of Gas Laws

Gas laws are crucial in understanding and predicting the behavior of gases under different conditions. These laws help us relate key properties of gases: pressure, volume, temperature, and amount of gas. One fundamental gas law is Boyle's Law, which is particularly relevant when we consider situations where the temperature and the number of gas molecules are constant.

Boyle's Law specifically describes the pressure-volume relationship for a gas. Its discovery was among the earliest descriptions of how gases behave and is widely relevant, from chemistry labs to respiratory physiology. Understanding Boyle's Law allows us to make precise calculations on how gases will respond to changes in volume or pressure, provided that temperature and number of molecules are kept constant.

Pressure-Volume Relationship

The pressure-volume relationship, central to Boyle's Law, is an inverse relationship meaning that as the volume of a gas decreases, its pressure increases, and vice versa, as long as temperature and quantity of gas remain unchanged. This can be visualized when imagining squeezing a balloon; the more you squeeze and reduce its volume, the harder the balloon pushes back, indicating an increase in pressure.

When we talk about this relationship in quantitative terms, we refer to the Boyle's Law formula: \(P_1V_1 = P_2V_2\). This formula expresses the constancy of the product of pressure and volume for a specific amount of gas at a constant temperature. By accurately manipulating this equation, we can make predictions and solve problems related to the compression and expansion of gases.

Mastering Chemistry Calculations

Understanding the theory behind Boyle's Law is one thing, but applying it to solve chemistry calculations is where its practical value is shown. Chemistry calculations often require careful attention to units and a systematic approach to formulas. For Boyle's Law problems, it involves a series of steps such as identifying initial and final states of pressure and volume, ensuring units match, and applying the law correctly.

In the exercises provided, calculations were made to determine the new pressures after the volume of a gas changes. Part (a) required a calculation after a 38% decrease in volume, while part (b) involved reducing the volume to 38% of the original. These two scenarios might seem similar, but they describe different situations and lead to different outcomes - a critical distinction that can affect the results of chemistry calculations. Clearly identifying the end conditions is vital for accurate results in not only Boyle's Law problems but all chemical computations.