Question

A gas starts with initial pressure of 7.11 atm, initial temperature of \(66^{\circ} \mathrm{C}\), and initial volume of \(90.7 \mathrm{~mL}\). If its conditions change to \(33^{\circ} \mathrm{C}\) and \(14.33 \mathrm{~atm}\), what is its final volume?

Short answer

The final volume is approximately 42.52 mL.

Step-by-step solution

Identify Known Values and Convert Units

First, list the initial conditions (P₁, T₁, V₁) and final conditions (P₂, T₂) provided in the exercise. Convert all temperatures from Celsius to Kelvin by adding 273.15. The initial pressure P₁ is 7.11 atm, initial temperature T₁ is 66 + 273.15 = 339.15 K, and initial volume V₁ is 90.7 mL. The final pressure P₂ is 14.33 atm and final temperature T₂ is 33 + 273.15 = 306.15 K.

Apply the Combined Gas Law

The Combined Gas Law is \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). Substitute the known values: \( \frac{7.11 \times 90.7}{339.15} = \frac{14.33 \times V_2}{306.15} \).

Solve for the Final Volume (V₂)

To solve for V₂, rearrange the equation: \( V_2 = \frac{7.11 \times 90.7 \times 306.15}{339.15 \times 14.33} \). Calculate each component and simplify. The calculation results in: \( V_2 \approx 42.516 \, \mathrm{mL} \).

Key concepts

Gas Laws

Gas laws describe how gases behave and interact under different conditions. These laws include Boyle's Law, Charles's Law, and Avogadro's Law. The Combined Gas Law is a unifying principle that comes into play when more than one condition changes, such as pressure, volume, or temperature. It combines Boyle's Law, which relates pressure and volume, Charles's Law, which links volume and temperature, and Gay-Lussac's Law, which addresses pressure and temperature. The Combined Gas Law formula is:

• \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \)

Here, \(P_1\) and \(P_2\) are initial and final pressures, \(V_1\) and \(V_2\) are initial and final volumes, and \(T_1\) and \(T_2\) are initial and final temperatures in Kelvin. When using this law, ensure that the units are consistent, typically using atmospheres for pressure, liters for volume, and Kelvin for temperature. Knowing how to manipulate this equation allows us to determine an unknown variable when the state of a gas changes.

Temperature Conversion

When dealing with gas laws, proper unit conversion is essential, especially when it comes to temperature. Gas law calculations require temperature to be in the Kelvin scale. This is because Kelvin is an absolute scale where 0 K is absolute zero, a theoretical point at which particles have minimal vibrational motion. Celsius is more commonly used in everyday applications, but for scientific calculations involving gases, it's crucial to convert these temperatures to Kelvin. To convert Celsius to Kelvin, use the simple formula:

• \( K = ^\circ C + 273.15 \)

For instance, the initial temperature of \(66^{\circ} \mathrm{C}\) becomes \(339.15 \mathrm{K}\), and the final temperature of \(33^{\circ} \mathrm{C}\) becomes \(306.15 \mathrm{K}\). Remember, this conversion ensures that mathematical operations involving temperature, pressure, and volume in the gas laws produce accurate and meaningful results.

Chemical Calculations

Chemical calculations are a key part of understanding complex scientific processes, including those involving gases. They involve using mathematical methods to calculate quantities like volume, pressure, and temperature changes. When using the Combined Gas Law, a thorough understanding of solving equations and isolating variables is necessary. To solve for a missing quantity like final volume \(V_2\), rearrange the equation:

• \( V_2 = \frac{P_1 V_1 T_2}{T_1 P_2} \)

Substitute the known values: initial pressure (\(7.11 \mathrm{~atm}\)), initial volume (\(90.7 \mathrm{~mL}\)), initial temperature (\(339.15 \mathrm{K}\)), final pressure (\(14.33 \mathrm{~atm}\)), and final temperature (\(306.15 \mathrm{K}\)). The computation gives us the final volume \(V_2\) which is approximately \(42.516 \mathrm{~mL}\). Understanding these calculations helps to predict the behavior of a gas when subjected to changes, crucial for chemistry and physics problems alike.