Problem 26 - Chemistry Central Science

Question

A fixed quantity of gas at \(21{ }^{\circ} \mathrm{C}\) exhibits a pressure of 752 torr and occupies a volume of \(4.38 \mathrm{~L}\). (a) Use Boyle's law to calculate the volume the gas will occupy if the pressure is increased to \(1.88 \mathrm{~atm}\) while the temperature is held constant. (b) Use Charles's law to calculate the volume the gas will occupy if the temperature is increased to \(175^{\circ} \mathrm{C}\) while the pressure is held constant.

Step-by-Step Solution

Answer

(a) Using Boyle's Law, the volume of the gas when the pressure is increased to 1.88 atm is \(V_2 = 2.29 \mathrm{~L}\). (b) Using Charles's Law, the volume of the gas when the temperature is increased to 175°C while the pressure is held constant is \(V_2 = 6.58 \mathrm{~L}\).

Step 1: Understand Boyle's Law

Boyle's Law states that for a fixed amount of gas at a constant temperature, the pressure (P) of the gas is inversely proportional to its volume (V). Mathematically, this can be expressed as: \(PV = k\) where 'k' is the constant for the given amount of gas at a constant temperature.

Step 2: Calculate the volume using Boyle's Law

We are given the initial pressure (P1) and volume (V1), and we need to find the volume (V2) when the pressure is increased to 1.88 atm. We just need to convert the given pressure from torr to atm and apply Boyle's Law. 1. Convert 752 torr to atm: \(1 \mathrm{~atm} = 760 \mathrm{~torr}\) \(P_1 = \frac{752 \mathrm{~torr}}{760 \mathrm{~torr/atm}}= 0.989 \mathrm{~atm}\) 2. Use Boyle's Law formula: \(P_1V_1 = P_2V_2 \) We know the values for P1, V1, and P2; we need to find V2. \((0.989 \mathrm {~atm})(4.38 \mathrm{~L}) = (1.88 \mathrm{~atm}) V_2 \) 3. Solve for V2: \(V_2 = \frac{(0.989 \mathrm {~atm})(4.38 \mathrm{~L})}{1.88 \mathrm{~atm}} = 2.29 \mathrm{~L}\) So, the volume of the gas when the pressure is increased to 1.88 atm is 2.29 L.

Step 3: Understand Charles's Law

Charles's Law states that at a constant pressure, the volume (V) of a fixed amount of gas is directly proportional to its absolute temperature (T). Mathematically, this can be expressed as: \(\frac{V}{T} = k\) where 'k' is the constant for the given amount of gas at a constant pressure.

Step 4: Calculate the volume using Charles's Law

We need to find the volume (V2) of the gas when the temperature is increased to 175°C, while the pressure is held constant. First, we should convert the given temperatures to Kelvin and then apply Charles's Law. 1. Convert the temperatures to Kelvin: \(T_1 = 21°C + 273.15 = 294.15 \mathrm{K} \) \(T_2 = 175°C + 273.15 = 448.15 \mathrm{K}\) 2. Use Charles's Law formula: \(\frac{V_1}{T_1} = \frac{V_2}{T_2} \) We know the values for V1, T1, and T2; we need to find V2. \(\frac{4.38 \mathrm{~L}}{294.15 \mathrm{K}} = \frac{V_2}{448.15 \mathrm{K}} \) 3. Solve for V2: \(V_2 = \frac{(4.38 \mathrm{~L})(448.15 \mathrm{K})}{294.15 \mathrm{K}} = 6.58 \mathrm{~L}\) So, the volume of the gas when the temperature is increased to 175°C while the pressure is held constant is 6.58 L.

Key Concepts

Boyle's Law

Boyle's Law is a fundamental principle in the field of gas laws in chemistry. It elaborates on the relationship between pressure and volume for a fixed amount of gas at a constant temperature. According to Boyle's Law, the pressure of a gas is inversely proportional to its volume. This implies that as the volume of the gas decreases, the pressure increases, provided the temperature doesn't change.

The Mathematical Representation of Boyle's Law

Using Boyle's Law in calculations, we typically work with the equation \(PV = k\), where \(P\) stands for pressure, \(V\) denotes volume, and \(k\) is a constant value specific to the amount of gas. If we have two sets of pressure and volume conditions - \(P_1\) and \(V_1\), and \(P_2\) and \(V_2\) - they can be related as \(P_1V_1 = P_2V_2\).

To put this into practice, suppose a gas displays a pressure of 752 torr in a 4.38 L container. If the pressure is increased to 1.88 atm, while maintaining the same temperature, we can find the new volume \(V_2\). However, consistency in units is crucial in calculations. In this case, we must convert 752 torr to atm before applying the formula. Once this is completed, \(V_2\) is found using simple algebra. Through such exercises, the inverse relationship of pressure and volume in Boyle's Law becomes apparent.

Charles's Law

Expanding our understanding of gas behaviors, Charles's Law provides insight into how gas volume is affected by temperature changes, maintaining constant pressure. Charles's Law states the volume of a given amount of gas is directly proportional to its absolute temperature, which must be measured in Kelvin.

Application of Charles's Law

When using Charles's Law in calculations, the formula \(\frac{V}{T} = k\), represents the direct relationship between volume \(V\) and absolute temperature \(T\), with \(k\) being a constant. If the temperature of a gas initially at 294.15 K is increased to 448.15 K, while the pressure remains constant, we can solve for the new volume \(V_2\) using this law. The calculation requires an understanding of how to convert Celsius to Kelvin and the ability to rearrange the equation for the desired variable. By employing Charles's Law, we observe how an increase in temperature (while keeping pressure static) leads to a proportional increase in gas volume.

Gas Law Calculations

In chemistry, the mastery of gas law calculations is essential to explain and predict the behavior of gases under various conditions. Gas laws like Boyle's Law and Charles's Law form the basis of these calculations, but understanding them in isolation is not enough. A comprehensive approach often involves combining these laws to solve for various variables such as pressure, volume, and temperature in different scenarios. For instance, when carrying out calculations, it is vital to ensure consistency in units to achieve accurate results.

To carry out effective gas law calculations:

• Identify the law applicable to the situation (Boyle's for constant temperature or Charles's for constant pressure).
• Ensure all measurements are in the correct units, such as Kelvin for temperature and atmospheres for pressure.
• Rearrange the formulas to solve for the unknown variable.
• Understand the relationships between the variables (direct or inverse proportionality).

By following such organized steps, students can approach complex problems systematically and obtain reliable outcomes.