Question

Given a fixed quantity of a gas at constant temperature, calculate the new volume the gas would occupy if the pressure were changed as shown in the following table. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Initial } \\ \text { Volume } \end{array} & \begin{array}{c} \text { Initial } \\ \text { Pressure } \end{array} & \begin{array}{c} \text { Final } \\ \text { Pressure } \end{array} & \begin{array}{c} \text { Final } \\ \text { Volume } \end{array} \\ \hline 2.50 \mathrm{~L} & 2.50 \mathrm{~atm} & 725 \mathrm{~atm} & ? \\ \hline 6.25 \mathrm{~mL} & 825 \text { torr } & 456 \text { torr } & ? \\ \hline 450 \mathrm{~mL} & 50.0 \text { torr } & 30.0 \text { torr } & ? \\ \hline \end{array} $$

Short answer

The final volumes for the first, second, and third sets of data are approximately 0.0086 L, 14.3 mL, and 750 mL, respectively.

Step-by-step solution

Apply Boyle's law for first set of values

The initial conditions are 2.5 L and 2.5 atm and the final pressure is 725 atm. Substituting these into Boyle's law gives us \(2.5 \times 2.5 = 725 \times V_2\). Solve this equation for \(V_2\) to find the final volume.

Apply Boyle's law for second set of values

The initial conditions are 6.25 mL and 825 torr and the final pressure is 456 torr. Again, substituting these into Boyle's law \(6.25 \times 825 = 456 \times V_2\), we can solve for \(V_2\) to find the final volume.

Apply Boyle's law for third set of values

The initial conditions are 450 mL and 50.0 torr, with final pressure 30.0 torr. Substituting these into Boyle's law \(450 \times 50 = 30 \times V_2\), we solve for \(V_2\) to find the final volume.

Key concepts

Gas Laws

Gas laws are fundamental principles governing the behavior of gases under various conditions of pressure, temperature, and volume. One of the most commonly studied laws is Boyle's Law, which describes the pressure-volume relationship for a gas held at a constant temperature. Boyle's Law states that for a given mass of gas, at constant temperature, the pressure of the gas is inversely proportional to its volume. This means that if the volume of the gas is increased, the pressure will decrease, and vice versa, assuming the temperature and the amount of gas are held constant.

Understanding Boyle's Law is crucial for anyone working in chemistry or physics because it provides a predictable understanding of how gases will react to changes in volume and pressure. This law has various applications, from calculating how the volume of a gas changes as a diver ascends or descends to designing syringes and other equipment where gas compression is a factor.

Pressure-Volume Relationship

The pressure-volume relationship expressed by Boyle's Law can be written mathematically as: \( P_1V_1 = P_2V_2 \), where \( P_1 \) and \( P_2 \) represent the initial and final pressures of the gas, and \( V_1 \) and \( V_2 \) represent the initial and final volumes of the gas, respectively. When tackling a chemistry problem involving this concept, it's vital to ensure that the pressure and volume values are in consistent units. Common units for pressure include atmospheres (atm) and torr (mmHg), while volume is typically measured in liters (L) or milliliters (mL).

For example, if a sample of gas has an initial volume of \( 2.50 \mathrm{L} \) at an initial pressure of \( 2.50 \mathrm{atm} \), and the pressure is changed to \( 725 \mathrm{atm} \), the final volume can be found using Boyle's Law. By rearranging the equation to \( V_2 = \frac{P_1V_1}{P_2} \) and substituting the known values, the new volume can be calculated.

Chemistry Problem Solving

Chemistry problem solving often involves applying laws and principles to find an unknown variable, as demonstrated in the solution of Boyle's Law exercises. The steps to effectively solving these problems include:

• Identifying the known and unknown variables.
• Ensuring that units are consistent across all variables.
• Applying the appropriate chemical law (in this case, Boyle's Law).
• Algebraically rearranging the equation if necessary.
• Performing the calculation to find the unknown variable.

Here, we saw these steps in action by first writing out the formula presented by Boyle's Law, plugging in the known values, and calculating the unknown final volume in each scenario. Problem-solving in chemistry also requires attention to detail and the ability to convert units when necessary, which ensures accurate calculations and understanding of the physical changes occurring within the problem scenario.