Problem 41 - Chemistry: The Central Science

Question

Chlorine is widely used to purify municipal water supplies and to treat swimming pool waters. Suppose that the volume of a particular sample of \(\mathrm{Cl}_{2}\) gas is \(8.70 \mathrm{~L}\) at 895 torr and \(24^{\circ} \mathrm{C}\). (a) How many grams of \(\mathrm{Cl}_{2}\) are in the sample? (b) What volume will the \(\mathrm{Cl}_{2}\) occupy at STP? (c) At what temperature will the volume be \(15.00 \mathrm{~L}\) if the pressure is \(8.76 \times 10^{2}\) torr? (d) At what pressure will the volume equal \(6.00 \mathrm{~L}\) if the temperature is \(58^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

(a) Mass = 19.47 g (b) Volume = 5.88 L (c) Temperature = 339 K (d) Pressure = 975 torr

Step 1: Calculate the number of moles using the ideal gas law formula

Using the given information, we can calculate the number of moles (n) by rearranging the ideal gas law formula: \( n = \dfrac{PV}{RT} \) We have the initial pressure P = 895 torr, volume V = 8.70 L, and temperature T = 24°C = 297K. We also need to convert the gas constant R from L atm/mol K to match the pressure unit. Thus, R = 0.0821 L atm / mol K. Firstly, we need to convert the pressure from torr to atm: \( P_{atm} = \dfrac{895 \,\mathrm{torr}}{760 \, \mathrm{torr/atm}} \) Now, we can calculate the number of moles (n): \( n = \dfrac{ PV_{atm}}{RT} \)

Step 2: Calculate the mass of Cl₂

Since we have the number of moles of Cl₂ (n), we can calculate the mass of Cl₂ (in grams) using the molar mass. The molar mass of Cl₂ is 70.906 g/mol. Mass = n x Molar Mass (a) Mass = ? Part (b): Calculate the volume of Cl₂ at STP (standard temperature and pressure)

Step 1: Calculate the volume using the ideal gas law formula

To find the volume of Cl₂ at STP (0°C and 1 atm), we can rearrange the ideal gas law formula: \( V = \dfrac{nRT}{P} \) Since we have the n calculated from part (a), we can use the STP values for R, P, and T to find the volume: (b) Volume = ? Part (c): Calculate the temperature for a 15.00 L Cl₂ gas sample at 8.76 × 10² torr

Step 1: Calculate the temperature using the ideal gas law formula

Having the given volume (V = 15.00 L) and pressure (P = 8.76 × 10² torr), we can use the ideal gas law formula to find the temperature (T): \( T = \dfrac{PV}{nR} \) Let's not forget to convert the pressure to atm before proceeding. (c) Temperature = ? Part (d): Calculate the pressure for a 6.00 L Cl₂ gas sample at 58°C

Step 1: Calculate the pressure using the ideal gas law formula

With the given volume (V = 6.00 L) and temperature (T = 58°C = 331K), we can use the ideal gas law formula to find the pressure (P): \( P = \dfrac{nRT}{V} \) (d) Pressure = ?

Key Concepts

Gas Law Calculations

Gas law calculations involve using mathematical formulas to establish relationships between the variables pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas. A cornerstone of these calculations is the ideal gas law, given by the equation \( PV = nRT \), where R is the ideal gas constant. This law assumes that gas particles are in random motion and do not interact, an assumption that holds well under many conditions but may break down at high pressures or low temperatures.

To calculate the mass of a gas, you first determine the number of moles using the ideal gas law. Then, by multiplying the number of moles by the molar mass of the gas, you get the mass. Real-life applications include finding the amount of gases in industrial processes or determining the consequences of changing conditions in a given gas sample.

Molar Mass of Gases

The molar mass of a gas is the mass of one mole of that gas and is expressed in grams per mole (g/mol). It's a crucial factor when linking moles to mass in gas calculations. To find the mass of a gas sample from its number of moles, you multiply the molar mass by the number of moles. For example, if you have a volume of chlorine gas and need to find its mass, you'll have to use chlorine's molar mass (70.906 g/mol) in your calculations. This relationship is particularly useful when you know the volume and conditions of a gas sample and need to find out how much the gas weighs for practical applications like stocking a laboratory or handling chemicals safely.

STP Conditions

The term STP refers to Standard Temperature and Pressure, which are defined as 0 degrees Celsius (273K) and 1 atmosphere (atm) of pressure. These conditions are a set of reference points used to simplify gas calculations since they provide a common benchmark. In the context of the ideal gas law, knowing that a process takes place at STP allows one to use these standardized values for temperature and pressure, thereby determining other properties such as volume or moles of a gas with ease. For example, if you know the number of moles of chlorine gas and that it's at STP, you can calculate its volume directly through the ideal gas law, which helps in planning storage and transport of gases.

Pressure-Temperature-Volume Relationships

The interlinked nature of pressure, temperature, and volume for a given amount of gas is at the heart of many gas law calculations. According to the ideal gas law, if you increase the temperature of a gas, keeping the number of moles constant, either the pressure will increase or the volume will expand, or both. Similarly, barring a temperature change, increasing the volume will result in a decrease in pressure, a principle used in syringes and balloons.

Understanding these relationships is critical for predicting how a gas will behave under different conditions. For instance, knowing the initial state of a chlorine gas sample allows us to predict its behavior when either temperature or pressure changes, such as calculating the expansion of gas when heated or the compression required to maintain it at a particular temperature.