Question

The volume of a gas at 99.0 kPa is 300.0 mL. If the pressure is increased to 188 kPa, what will be the new volume?

Short answer

The new volume of the gas under the increased pressure of \(188\,\text{kPa}\) is approximately \(157.77\,\text{mL}\).

Step-by-step solution

Write Down Given Variables

Write down the given variables using Boyle's Law: \[P_1V_1 = P_2V_2\] \(P_1 = 99.0\,\text{kPa}\) \(V_1 = 300.0\,\text{mL}\) \(P_2 = 188\,\text{kPa}\) We need to find \(V_2\).

Rearrange the Equation to Solve for V2

Rearrange Boyle's Law equation to isolate the final volume (\(V_2\)): \[V_2 = \frac{P_1V_1}{P_2}\]

Plug in Given Values and Calculate V2

Now that we've rearranged the equation, we can plug in the given values and calculate the new volume of the gas, \(V_2\): \[V_2 = \frac{(99.0\,\text{kPa})(300.0\,\text{mL})}{188\,\text{kPa}}\]

Calculate and Express the Final Answer

Perform the calculation: \[V_2 = \frac{(99.0\,\text{kPa})(300.0\,\text{mL})}{188\,\text{kPa}} = 157.77\,\text{mL}\] The new volume under the increased pressure condition is approximately \(157.77\,\text{mL}\).

Key concepts

Gas laws

Understanding the behavior of gases is critical in the field of chemistry, and the gas laws provide the foundational knowledge required to predict how gases will respond to changes in pressure, volume, and temperature. Among these is Boyle's Law, which is often introduced as the simplest of the gas laws because it focuses on the relationship between pressure and volume when the temperature of the gas is held constant.

Boyle's Law states that for a given mass of gas at constant temperature, the volume of the gas is inversely proportional to its pressure. This means that if you increase the pressure exerted on a gas, its volume will decrease, and vice versa, assuming the temperature doesn't change. The historical experiments by Robert Boyle in the 17th century demonstrated this behavior, leading to the formulation of the law that now bears his name.

This law is particularly useful when dealing with closed systems where no gas molecules are added or removed. Boyle's Law has practical applications in various scientific fields such as chemistry, physics, and engineering, as well as in everyday life, including the workings of syringes, human lungs, and even scuba diving where divers experience changes in pressure as they descend or ascend.

Pressure-volume relationship

The pressure-volume relationship, also known as Boyle's Law, is one of the most fundamental concepts in gas laws. It quantifies how pressure (\(P\)) and volume (\(V\)) of a gas are related. Specifically, the law is given by the equation \[P_1V_1 = P_2V_2\] where \(P_1\) and \(V_1\) represent the initial pressure and volume, and \(P_2\) and \(V_2\) represent the final pressure and volume, respectively.

In simple terms, when the volume of the gas decreases, the particles are compressed into a smaller space, resulting in more frequent collisions with the container walls and thus a higher pressure. Conversely, if the volume increases, the particles have more space to move around, leading to less frequent collisions and a lower pressure.

It's essential to note that Boyle's Law assumes the gas behaves ideally, meaning that the particles do not attract or repel each other and occupy negligible space. Real gases may exhibit slight deviations from this law under certain conditions, such as high pressures or low temperatures, where the ideal gas assumption becomes less accurate.

Chemistry calculations

Chemistry calculations are a vital part of understanding chemical processes and reactions. When working with gases, precise calculations can predict how changes in conditions, like pressure or volume, affect the gas. Boyle's Law provides a straightforward formula that can be applied to solve for unknown variables. In the example provided, the goal is to find the new volume (\(V_2\)) after an increase in pressure.

The step-by-step solution demonstrates a typical chemistry calculation process:

• Identify the known variables (\(P_1, V_1, P_2\)).
• Rearrange the equation to solve for the unknown (\(V_2\)).
• Insert the known values into the arranged equation.
• Perform the arithmetic computation.

Ensuring units are consistent throughout the calculation is pivotal for accuracy. In the end, it’s important to reflect on whether the result is logical based on the principles of Boyle's Law. If the pressure of a gas increases and the temperature remains the same, we should anticipate a decrease in volume, which is confirmed by the calculated value of \(V_2 = 157.77\text{ mL} \). This helps students not only learn how to perform the calculations but also understand the underlying principles that govern chemical behavior.