Calculating Vapor Density Of Gas X₂ A Chemistry Problem Solution
Vapor density is a crucial concept in chemistry, especially when dealing with gases. It provides insights into the molar mass of an unknown gas relative to a known gas, typically hydrogen. This article delves into the principles behind vapor density calculations and demonstrates how to determine the vapor density of a gas using the ideal gas law and stoichiometric relationships. We will explore a specific scenario involving two identical vessels containing ethane and an unknown gas X₂, providing a step-by-step solution to calculate the vapor density of X₂. This exploration will not only enhance your understanding of vapor density but also reinforce fundamental concepts in gas laws and molar mass calculations. Understanding vapor density is paramount in various chemical applications, from identifying unknown gases to predicting gas behavior in reactions. This comprehensive guide aims to equip you with the knowledge and skills to confidently tackle such problems.
Understanding Vapor Density
Vapor density is defined as the ratio of the density of a gas to the density of hydrogen under the same conditions of temperature and pressure. Mathematically, it can be expressed as:
Vapor Density = (Density of gas) / (Density of Hydrogen)
Since density is directly proportional to molar mass, vapor density is also equivalent to half the molar mass of the gas. This relationship is derived from the fact that the molar mass of hydrogen (H₂) is approximately 2 g/mol.
Vapor Density = (Molar mass of gas) / (Molar mass of Hydrogen) Vapor Density = (Molar mass of gas) / 2
This relationship is pivotal in determining the molar mass of an unknown gas if its vapor density is known, or vice versa. The concept of vapor density simplifies gas comparisons, allowing chemists to quickly estimate the molar mass of a gas relative to hydrogen. In practical terms, vapor density helps in identifying gases and understanding their behavior in gaseous mixtures and chemical reactions. By calculating the vapor density, we can infer the molecular weight of a gas, which is a crucial parameter in stoichiometry and chemical kinetics. Furthermore, vapor density plays a significant role in industrial processes involving gas handling and storage, where understanding the relative densities of gases is essential for safety and efficiency.
Problem Statement: Determining the Vapor Density of Gas X₂
We are presented with a scenario involving two identical vessels, A and B. Vessel A contains 15 g of ethane (C₂H₆) at 1 atmosphere (atm) and 25°C. Vessel B contains 75 g of an unknown gas X₂ under the same conditions of temperature and pressure. The objective is to determine the vapor density of gas X₂.
This problem requires a thorough understanding of the ideal gas law and how it relates to molar mass, volume, pressure, and temperature. The identical conditions of temperature and pressure in both vessels allow us to make direct comparisons between the amounts of ethane and gas X₂. We will use the molar mass of ethane to calculate the number of moles in vessel A, which will then help us deduce the molar volume under the given conditions. By comparing the masses and moles of both gases, we can determine the molar mass of gas X₂ and subsequently calculate its vapor density. This problem effectively integrates several fundamental concepts in chemistry, including stoichiometry, gas laws, and the relationship between molar mass and vapor density. The structured approach to solving this problem will not only provide the answer but also enhance your problem-solving skills in chemistry.
Step-by-Step Solution
1. Calculate the Molar Mass of Ethane (C₂H₆)
The molar mass of ethane (C₂H₆) is calculated by summing the atomic masses of its constituent elements:
- Carbon (C): 2 atoms × 12 g/mol = 24 g/mol
- Hydrogen (H): 6 atoms × 1 g/mol = 6 g/mol
Total Molar Mass of Ethane = 24 g/mol + 6 g/mol = 30 g/mol
2. Determine the Number of Moles of Ethane in Vessel A
Using the given mass of ethane (15 g) and its molar mass (30 g/mol), we can calculate the number of moles:
Moles of Ethane = (Mass of Ethane) / (Molar Mass of Ethane) Moles of Ethane = 15 g / 30 g/mol = 0.5 moles
3. Apply the Ideal Gas Law to Ethane
The ideal gas law is given by:
PV = nRT
Where:
- P = Pressure (1 atm)
- V = Volume (unknown)
- n = Number of moles (0.5 moles)
- R = Ideal gas constant (0.0821 L atm / (mol K))
- T = Temperature (25°C = 298 K)
Rearranging the formula to solve for volume (V):
V = (nRT) / P V = (0.5 moles × 0.0821 L atm / (mol K) × 298 K) / 1 atm V ≈ 12.24 L
This calculation gives us the volume of vessel A, which is also the volume of vessel B since they are identical.
4. Calculate the Number of Moles of Gas X₂ in Vessel B
Since vessel B has the same volume, temperature, and pressure as vessel A, the number of moles of gas X₂ can be calculated using the ideal gas law as well, but it’s more straightforward to deduce it based on the given conditions. Both vessels have the same volume (V), pressure (P), and temperature (T). Therefore, for gas X₂:
Moles of X₂ = (P × V) / (R × T)
Since P, V, R, and T are the same for both gases, the ratio of moles is directly proportional to the ratio of masses divided by molar masses. Thus:
Moles of X₂ / Moles of Ethane = Mass of X₂ / Molar mass of X₂ / (Mass of Ethane / Molar mass of Ethane)
We know that moles of ethane = 0.5 moles, Mass of X₂ = 75 g, Mass of Ethane = 15 g, and Molar mass of Ethane = 30 g/mol. Let the molar mass of X₂ be M.
Moles of X₂ / 0.5 = 75 / M / (15 / 30) Moles of X₂ / 0.5 = 75 / M / 0.5 Moles of X₂ = (75 / M) / 1
Using the ideal gas law directly for X₂:
Moles of X₂ = (1 atm × 12.24 L) / (0.0821 L atm / (mol K) × 298 K) Moles of X₂ ≈ 0.5 moles
So, 0. 5 = 75 / M, which implies:
M = 75 / 0.5 = 150 g/mol
5. Calculate the Molar Mass of Gas X₂
Using the number of moles of gas X₂ (0.5 moles) and its mass (75 g), we can calculate its molar mass:
Molar Mass of X₂ = (Mass of X₂) / (Moles of X₂) Molar Mass of X₂ = 75 g / 0.5 moles = 150 g/mol
6. Determine the Vapor Density of Gas X₂
Using the relationship between vapor density and molar mass:
Vapor Density of X₂ = (Molar Mass of X₂) / 2 Vapor Density of X₂ = 150 g/mol / 2 = 75
Therefore, the vapor density of gas X₂ is 75.
Conclusion: The Vapor Density of Gas X₂
In conclusion, by applying the principles of the ideal gas law and the relationship between vapor density and molar mass, we have successfully determined the vapor density of gas X₂. The step-by-step solution involved calculating the moles of ethane, using the ideal gas law to find the volume, determining the moles of gas X₂ under the same conditions, and finally, calculating the molar mass and vapor density of X₂. The vapor density of gas X₂ was found to be 75, which corresponds to option (2) in the given choices.
This exercise underscores the importance of understanding gas laws and molar mass concepts in solving chemical problems. The ability to apply these principles is crucial for various applications in chemistry and related fields. This comprehensive approach to solving the problem not only provides the correct answer but also reinforces the fundamental concepts necessary for tackling similar challenges. Understanding the vapor density of gases is essential in many practical applications, from industrial processes to laboratory research. By mastering these concepts, one can confidently handle a wide range of problems involving gases and their properties.
