Question

An unknown diatomic gas has a density of 3.164 g/L at STP. What is the identity of the gas?

Short answer

The unknown diatomic gas can be identified by calculating its molar mass using the given density at STP. The molar mass is calculated using the formula: \(Molar\; mass = Density \times Molar\; volume\) which comes out to be \(70.88 g/mol\). Comparing this value to the molar masses of known diatomic gases, the unknown diatomic gas is Chlorine (Cl2), having a molar mass of \(70.9 g/mol\).

Step-by-step solution

Write down the given information

We are given the following information: - Density of the unknown gas: \(3.164 \frac{g}{L}\) - Conditions: STP, which means a temperature of \(273.15 K\) and a pressure of \(1 atm\).

Understand the relationship between density, molar mass, and molar volume

At STP, one mole of any ideal gas occupies a volume of \(22.4 L\). This is known as the molar volume of an ideal gas at STP. The density of a gas is defined as its mass per unit volume, so we can find its molar mass by using the following equation: \[Density = \frac{molar\; mass}{molar\; volume}\]

Calculate the molar mass of the gas

Rearrange the equation in step 2 and solve for the molar mass: \[Molar\; mass = Density \times Molar\; volume\] \[Molar\; mass = 3.164 \frac{g}{L} \times 22.4 L\] \[Molar\; mass = 70.88 g/mol\] The calculated molar mass of the unknown gas is approximately \(70.88 g/mol\).

Identify the diatomic gas

Now compare the calculated molar mass with the molar mass of known diatomic gases: - Hydrogen (H2): \(2 g/mol\) - Nitrogen (N2): \(28 g/mol\) - Oxygen (O2): \(32 g/mol\) - Fluorine (F2): \(38 g/mol\) - Chlorine (Cl2): \(70.9 g/mol\) - Bromine (Br2): \(160 g/mol\) - Iodine (I2): \(254 g/mol\) Our calculated molar mass (\(70.88 g/mol\)) is closest to the molar mass of Chlorine (Cl2), which is \(70.9 g/mol\). Therefore, we can conclude that the unknown diatomic gas is Chlorine (Cl2).

Key concepts

Diatomic Gas

Diatomic gases are molecules composed of two atoms. They are one of the simplest and most common types of molecular structures. Some examples include the gases we breathe every day, like oxygen (\[O_2\]) and nitrogen (\[N_2\]).
These molecules can be made from the same element, such as hydrogen (\[H_2\]), or from different elements. In our exercise, the focus is on identifying a diatomic gas based on its properties. Recognizing these gases involves understanding their simple structures and common presence in nature.

• Diatomic molecules include essential atmospheric gases.
• They have straightforward molecular structures, easier to analyze for identification.

Appreciating these gases helps in many scientific fields, including chemistry and environmental science. Their behavior and properties make them crucial in understanding basic and advanced gas laws.

Molar Volume

The concept of molar volume is essential in understanding gas behavior, especially under standard conditions. At Standard Temperature and Pressure (STP), one mole of a gas occupies a fixed and known volume. This volume is approximately 22.4 liters for any ideal gas.
A fixed molar volume helps chemists and physicists use the ideal gas laws to predict and compute values related to gases, linking the number of moles directly to volume. This makes calculations feasible and simplifies many problem-solving processes.

• One mole of an ideal gas at STP occupies 22.4 liters.
• Molar volume simplifies gas calculations by establishing a standard reference.

By knowing the molar volume, we can deduce other properties, like molar mass, through density measurements. This becomes instrumental in identifying unknown gases, as seen in our exercise.

Standard Temperature and Pressure (STP)

When dealing with gases, conditions can significantly affect their volume, pressure, and temperature. STP is a common reference point used to compare gas behavior consistently. It is defined as a temperature of 273.15 Kelvin and a pressure of 1 atmosphere (atm).
These conditions allow scientists and students to relate and predict gas behavior in a controlled manner. Without STP, comparing gases would be challenging, given that their properties change under different temperatures and pressures.

• STP is defined as 273.15 Kelvin and 1 atmosphere pressure.
• It provides a baseline for scientific comparisons and calculations.

Understanding STP is crucial because it standardizes measurements, making calculations, like determining molar mass from gas density, accessible and reliable. It becomes the backbone of many calculations, ensuring robust and exact results.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in g/mol. For gases, it becomes a crucial feature in determining their identity. By utilizing the density and molar volume concepts, we can calculate a gas's molar mass.
In our example, the molar mass was calculated using the density (\[3.164 \frac{g}{L}\]) and molar volume (\[22.4 L\]) of the gas. By multiplying these two values, the molar mass turns out to be around 70.88 g/mol. This value helps identify which gas we are dealing with.

• Molar mass determines the mass of a mole of particles.
• It is essential for identifying gases by comparing with known values.

With the molar mass in hand, comparing against known diatomic gases helps pinpoint our unknown gas with confidence. Thus, understanding molar mass opens doors to identifying unknown substances.

Gas Identification

Once the molar mass is known, identifying a gas becomes straightforward. For diatomic gases, we compare the calculated molar mass with the known molar masses of standard diatomic gases.In our scenario, we calculated the molar mass to be 70.88 g/mol. By checking against common diatomic gases like oxygen, nitrogen, and chlorine, we determined that the unknown gas resembles chlorine (\[Cl_2\]). This method effectively narrows down possibilities.

• Compare calculated molar mass with known gases.
• Utilize known densities and molar masses for deduction.

With practice, identifying gases through calculations becomes second nature. Logical reasoning and mathematical calculations together make the identification process rewarding and precise.