A Rigid 2.50 L Bottle Contains 0.458 Mol Of He. The Pressure Of The Gas Inside The Bottle Is 1.83 Atm. If 0.713 Mol Of Ar Is Added To The Bottle And The Pressure Increases To 2.05 Atm, What Is The Change In Temperature Of The Gas Mixture? Use The
Introduction
In this problem, we are given a rigid 2.50 L bottle containing 0.458 mol of helium (He) at a pressure of 1.83 atm. We are also told that 0.713 mol of argon (Ar) is added to the bottle, causing the pressure to increase to 2.05 atm. Our goal is to calculate the change in temperature of the gas mixture.
The Ideal Gas Law
To solve this problem, we will use the ideal gas law, which is given by the equation:
PV = nRT
where:
- P is the pressure of the gas
- V is the volume of the gas
- n is the number of moles of the gas
- R is the gas constant
- T is the temperature of the gas
Step 1: Calculate the Initial Temperature of the Helium Gas
First, we need to calculate the initial temperature of the helium gas. We can do this by rearranging the ideal gas law equation to solve for T:
T = PV / nR
We are given the following values:
- P = 1.83 atm
- V = 2.50 L
- n = 0.458 mol
- R = 0.08206 L atm/mol K (this is the gas constant)
Plugging these values into the equation, we get:
T = (1.83 atm x 2.50 L) / (0.458 mol x 0.08206 L atm/mol K)
T = 3.63 K
Step 2: Calculate the Final Temperature of the Gas Mixture
Next, we need to calculate the final temperature of the gas mixture after the argon gas is added. We can do this by using the same equation, but this time we will use the final values for P, V, and n:
- P = 2.05 atm
- V = 2.50 L
- n = 0.458 mol + 0.713 mol = 1.171 mol (this is the total number of moles of the gas mixture)
Plugging these values into the equation, we get:
T = (2.05 atm x 2.50 L) / (1.171 mol x 0.08206 L atm/mol K)
T = 4.33 K
Step 3: Calculate the Change in Temperature of the Gas Mixture
Finally, we need to calculate the change in temperature of the gas mixture. We can do this by subtracting the initial temperature from the final temperature:
ΔT = T_final - T_initial
ΔT = 4.33 K - 3.63 K
ΔT = 0.70 K
Conclusion
In this problem, we calculated the change in temperature of the gas mixture after 0.713 mol of argon is added to a rigid 2.50 L bottle containing 0.458 mol of helium at a pressure of 1.83 atm. We found that the change in temperature of the gas mixture is 0.70 K.
Discussion
This problem demonstrates the application of the ideal gas law to a real-world scenario. The ideal gas law is a fundamental concept in chemistry that describes the behavior of ideal gases. In this problem, we used the ideal gas law to calculate the initial and final temperatures of the gas mixture, and then calculated the change in temperature.
The addition of argon to the bottle causes the pressure to increase, which in turn causes the temperature to increase. This is because the argon gas molecules are in constant motion, colliding with the walls of the bottle and transferring their kinetic energy to the surrounding gas molecules. As a result, the temperature of the gas mixture increases.
This problem also demonstrates the importance of understanding the behavior of gases in different scenarios. In this case, we used the ideal gas law to calculate the change in temperature of the gas mixture, which is a critical concept in chemistry and physics.
Applications
This problem has several applications in real-world scenarios. For example, in the production of semiconductors, the temperature of the gas mixture must be carefully controlled to ensure that the semiconductor material is produced with the correct properties. In this case, the ideal gas law can be used to calculate the change in temperature of the gas mixture, which is critical for the production of high-quality semiconductors.
In addition, the ideal gas law can be used to calculate the change in temperature of the gas mixture in a variety of other scenarios, such as in the production of fuels, in the operation of engines, and in the design of refrigeration systems.
Limitations
This problem has several limitations. For example, the ideal gas law assumes that the gas molecules are point particles, which is not always the case in real-world scenarios. In addition, the ideal gas law assumes that the gas molecules are in constant motion, which is not always the case in real-world scenarios.
In addition, the ideal gas law is only applicable to ideal gases, which are gases that obey the ideal gas law. Real-world gases, such as air and water vapor, do not always obey the ideal gas law, and therefore the ideal gas law is not always applicable to these gases.
Future Work
In the future, it would be interesting to explore the behavior of gases in different scenarios, such as in the production of semiconductors, in the operation of engines, and in the design of refrigeration systems. This could involve using the ideal gas law to calculate the change in temperature of the gas mixture, and then using this information to design and optimize the system.
In addition, it would be interesting to explore the behavior of real-world gases, such as air and water vapor, in different scenarios. This could involve using more complex models, such as the van der Waals equation, to describe the behavior of these gases.
References
- Kittel, C. (2005). Introduction to Solid State Physics. John Wiley & Sons.
- Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
- Cengel, Y. A. (2014). Thermodynamics: An Engineering Approach. McGraw-Hill Education.
Introduction
In our previous article, we explored the behavior of a rigid 2.50 L bottle containing 0.458 mol of helium (He) at a pressure of 1.83 atm. We calculated the initial and final temperatures of the gas mixture after 0.713 mol of argon (Ar) is added to the bottle, causing the pressure to increase to 2.05 atm. In this article, we will answer some frequently asked questions (FAQs) related to this problem.
Q: What is the ideal gas law?
A: The ideal gas law is a fundamental concept in chemistry that describes the behavior of ideal gases. It is given by the equation:
PV = nRT
where:
- P is the pressure of the gas
- V is the volume of the gas
- n is the number of moles of the gas
- R is the gas constant
- T is the temperature of the gas
Q: What is the significance of the ideal gas law?
A: The ideal gas law is significant because it allows us to calculate the temperature of a gas mixture given its pressure, volume, and number of moles. This is a critical concept in chemistry and physics, as it is used to design and optimize a wide range of systems, including engines, refrigeration systems, and semiconductor production.
Q: What is the difference between an ideal gas and a real gas?
A: An ideal gas is a gas that obeys the ideal gas law, meaning that it has no intermolecular forces and its molecules are point particles. A real gas, on the other hand, is a gas that does not obey the ideal gas law, meaning that it has intermolecular forces and its molecules are not point particles.
Q: What are some examples of real gases?
A: Some examples of real gases include air, water vapor, and carbon dioxide. These gases do not obey the ideal gas law because they have intermolecular forces and their molecules are not point particles.
Q: How does the addition of argon to the bottle affect the temperature of the gas mixture?
A: The addition of argon to the bottle causes the pressure to increase, which in turn causes the temperature to increase. This is because the argon gas molecules are in constant motion, colliding with the walls of the bottle and transferring their kinetic energy to the surrounding gas molecules.
Q: What is the change in temperature of the gas mixture?
A: The change in temperature of the gas mixture is 0.70 K.
Q: What are some applications of the ideal gas law?
A: Some applications of the ideal gas law include the production of semiconductors, the operation of engines, and the design of refrigeration systems. The ideal gas law is also used to calculate the temperature of a gas mixture given its pressure, volume, and number of moles.
Q: What are some limitations of the ideal gas law?
A: Some limitations of the ideal gas law include the fact that it assumes that the gas molecules are point particles and that they have no intermolecular forces. In addition, the ideal gas law is only applicable to ideal gases, which are gases that obey the ideal gas law.
Q: What are some future directions for research in this area?
A: Some future directions for research in this area include exploring the behavior of real gases in different scenarios, such as in the production of semiconductors, in the operation of engines, and in the design of refrigeration systems. This could involve using more complex models, such as the van der Waals equation, to describe the behavior of these gases.
Q: What are some references for further reading?
A: Some references for further reading include:
- Kittel, C. (2005). Introduction to Solid State Physics. John Wiley & Sons.
- Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
- Cengel, Y. A. (2014). Thermodynamics: An Engineering Approach. McGraw-Hill Education.
Conclusion
In this article, we have answered some frequently asked questions (FAQs) related to the behavior of a rigid 2.50 L bottle containing 0.458 mol of helium (He) at a pressure of 1.83 atm. We have also discussed the significance of the ideal gas law, the difference between an ideal gas and a real gas, and some applications and limitations of the ideal gas law. We hope that this article has been helpful in providing a better understanding of this topic.