Problem 82 - Chemistry Matter and Change

Question

If the volume of a sample of chlorine gas is 8.2 \(\mathrm{L}\) at 1.2 atm and 306 \(\mathrm{K}\) , what volume will the gas occupy at STP? (Chapter 13\()\)

Step-by-Step Solution

Answer

The volume of the chlorine gas at STP will be approximately 9.69 L, using the combined gas law equation: \( V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \) and plugging in the given values.

Step 1: Write down the given information

We know the initial conditions of the gas: - Initial Volume(V1) = 8.2 L - Initial Pressure(P1) = 1.2 atm - Initial Temperature(T1) = 306 K And the Standard Conditions: - Standard Temperature(T2) = 273 K - Standard Pressure(P2) = 1 atm We want to find the final volume(V2) at STP.

Step 2: Set up the combined gas law equation

The combined gas law is: \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \) We will plug in the known values into this equation and solve for the final volume(V2).

Step 3: Rearrange the equation to solve for V2

To find V2, we will rearrange the combined gas law as follows: \( V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \)

Step 4: Plug in the values and solve for V2

Now, plug in the given values and solve for V2: \( V_2 = \frac{(1.2 \ \mathrm{atm}) (8.2 \ \mathrm{L}) (273 \ \mathrm{K})}{(1 \ \mathrm{atm}) (306 \ \mathrm{K})} \) When we plug in the values and perform the calculations, we get: \( V_2 = \frac{ (1.2)(8.2)(273)}{(1)(306)} \) \( V_2 \approx 9.69 \ \mathrm{L} \) The volume of the chlorine gas at STP will be approximately 9.69 L.

Key Concepts

Chlorine Gas

Chlorine gas is a significant element often represented by the chemical symbol \( \text{Cl}_2 \). It is a member of the halogen group in the periodic table and has a variety of uses in everyday applications. Chlorine gas is a greenish-yellow diatomic molecule that exists naturally as a gas under standard conditions.

Chlorine is commonly associated with sanitation processes and is widely used to disinfect drinking water and in the production of numerous consumer goods. It is important to handle chlorine with care due to its reactive nature and potential health hazards if inhaled.

In gas law calculations, such as the one provided, chlorine is often evaluated under changing conditions of temperature and pressure. Understanding these changes can help predict the behavior of chlorine gas in laboratory and industrial settings.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, abbreviated as STP, is a set of conditions for the measurement of gases. When scientists reference gases at STP, they mean a temperature of 273 K (0°C) and a pressure of 1 atm. These conditions are used as a baseline for comparing the behavior of different gases.

STP is particularly useful in gas law calculations, such as when converting or predicting volumes under different conditions. In our given exercise, we need to determine how chlorine gas behaves when it is brought to STP from another set of conditions. This standardized setting allows for consistency and comparability in experiments and calculations across different environments and studies.

By knowing values at STP, we can apply the combined gas law or other gas equations to better predict how a gas's volume, pressure, and temperature will interact and change. This foundational knowledge underpins many applications and allows scientists and engineers to work effectively with gases.

Gas Volume Calculation

Calculating the volume of a gas under different conditions is essential for many scientific and engineering tasks. In our problem, we're using the Combined Gas Law, which is derived from Boyle's, Charles's, and Gay-Lussac's laws. The formula \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \) combines these principles and illustrates how pressure, volume, and temperature interdependently affect a gas.

To solve for the volume of chlorine gas at STP, we rearranged the combined gas law to isolate \( V_2 \), the variable for the final volume: \( V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \). Substituting the given values into this equation helps us determine the new volume under standard conditions.

By performing the calculations, we find that the volume of chlorine gas increases to approximately 9.69 liters at STP. This reflects how the gas expands because of decreased pressure and temperature relative to its initial state. Understanding these calculations can be crucial in fields ranging from chemical manufacturing to atmospheric science.