Temperature Required To Double Gas Volume And Pressure A Chemistry Problem

When dealing with gases, understanding the relationships between pressure, volume, and temperature is crucial. The ideal gas law, expressed as PV = nRT, provides a fundamental framework for analyzing these relationships. In this equation, P represents the pressure of the gas, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. This law highlights that pressure and volume are directly proportional to temperature when the number of moles is constant. This means that if you increase the temperature of a gas, either its pressure or its volume (or both) will increase proportionally, assuming the amount of gas remains the same.

To delve deeper into this concept, let's consider a scenario where we want to change both the volume and pressure of a gas simultaneously. Suppose we start with a gas at a specific temperature and we want to know what new temperature is required to double both the pressure and the volume. This is a classic problem in chemistry and physics that requires a clear understanding of how these variables interact. The ideal gas law provides the perfect tool to solve such problems. By manipulating the equation and considering the changes in pressure and volume, we can derive a relationship that directly links the initial and final temperatures. This involves setting up ratios and comparing the initial state of the gas with its final state. Understanding this relationship is not just academically important but also has practical applications in various fields, including engineering, meteorology, and even cooking, where understanding gas behavior is essential. The ideal gas law, therefore, serves as a cornerstone in understanding and predicting the behavior of gases under different conditions.

Applying the Ideal Gas Law to Solve the Problem

In this specific problem, we are given an initial temperature of 273 K and asked to find the final temperature when both the volume and pressure are doubled. To solve this, we can set up a ratio using the ideal gas law, comparing the initial and final states of the gas. Let's denote the initial pressure and volume as P1 and V1, respectively, and the initial temperature as T1. Similarly, the final pressure and volume are P2 and V2, and the final temperature is T2. Since the number of moles (n) and the ideal gas constant (R) remain constant, we can write the ideal gas law for both the initial and final states:

P1V1 = nRT1 P2V2 = nRT2

Now, we are told that the final pressure (P2) is twice the initial pressure (P1), so P2 = 2P1. Similarly, the final volume (V2) is twice the initial volume (V1), so V2 = 2V1. We can substitute these relationships into the second equation:

(2P1)(2V1) = nRT2 4P1V1 = nRT2

Next, we can divide the second equation by the first equation to eliminate n and R:

(4P1V1) / (P1V1) = (nRT2) / (nRT1) 4 = T2 / T1

This simplified equation shows that the final temperature (T2) is four times the initial temperature (T1). Given that the initial temperature (T1) is 273 K, we can now calculate the final temperature:

T2 = 4 * T1 T2 = 4 * 273 K T2 = 1092 K

Therefore, the gas must be raised to a temperature of 1092 K to double both its volume and pressure. This calculation demonstrates the power of the ideal gas law in predicting how gases behave under changing conditions. By understanding and applying these principles, we can solve a wide range of problems related to gases in various scientific and practical contexts. The step-by-step approach used here, from setting up the initial equations to the final calculation, provides a clear method for tackling similar problems in the future.

Analyzing the Answer Choices

After calculating the required temperature, it's crucial to compare our result with the provided answer choices to identify the correct option. The answer choices are:

A. 298 K B. 300 K C. 546 K D. 819 K E. 1092 K

Our calculated temperature of 1092 K matches option E. Therefore, option E is the correct answer. The other options can be eliminated as they do not satisfy the condition of doubling both the volume and pressure. Options A and B are close to the initial temperature and would not result in significant changes in volume and pressure. Options C and D represent intermediate temperatures that do not align with the quadrupling effect we determined through the ideal gas law. This process of comparing the calculated result with the provided options is a critical step in problem-solving, ensuring that the final answer is both mathematically correct and logically consistent with the given context.

This comparison also reinforces the importance of careful calculation and attention to detail. A small error in the calculation could lead to selecting the wrong answer choice. By systematically working through the problem and verifying the result against the options, we can confidently arrive at the correct solution. This methodical approach not only helps in answering the specific question but also enhances problem-solving skills that are valuable in various areas of science and engineering. The ability to accurately apply scientific principles and interpret results is a key skill for anyone working with quantitative data and complex systems.

Additional Insights into Gas Behavior

Beyond the ideal gas law, several other factors influence the behavior of gases. Understanding these factors provides a more comprehensive view of gas dynamics and their applications in various fields. One important concept is the kinetic molecular theory of gases, which describes gases as a collection of particles in constant, random motion. This theory helps explain why gases exert pressure, why they can be compressed, and how their temperature relates to the average kinetic energy of their molecules. The higher the temperature, the faster the molecules move, leading to increased pressure and volume.

Another key aspect of gas behavior is deviations from the ideal gas law. While the ideal gas law provides a good approximation for many gases under normal conditions, it doesn't perfectly describe all gases, especially at high pressures or low temperatures. Real gases exhibit intermolecular forces and have finite molecular volumes, which are not accounted for in the ideal gas law. These deviations are captured by more complex equations of state, such as the van der Waals equation, which includes correction terms for intermolecular attractions and molecular size. These corrections provide a more accurate representation of gas behavior under extreme conditions.

Furthermore, the concept of partial pressures is crucial when dealing with gas mixtures. Dalton's law of partial pressures states that the total pressure exerted by a mixture of gases is the sum of the pressures that each gas would exert if it occupied the same volume alone. This principle is particularly important in fields like respiratory physiology and atmospheric science, where understanding the composition and behavior of gas mixtures is essential. For example, in the study of air pollution, partial pressures help in quantifying the concentrations of different pollutants and their impact on the environment and human health.

In summary, while the ideal gas law provides a fundamental understanding of gas behavior, a broader perspective incorporating the kinetic molecular theory, deviations from ideality, and the concept of partial pressures offers a more complete picture. This holistic understanding is invaluable in various scientific and engineering applications, enabling us to predict and control the behavior of gases in a wide range of scenarios.

Conclusion: Mastering Gas Law Problems

In conclusion, solving problems involving gas laws, such as the one presented here, requires a solid understanding of the ideal gas law and its applications. By systematically applying the equation PV = nRT and understanding the relationships between pressure, volume, and temperature, we can accurately predict how gases will behave under different conditions. The key to success lies in setting up the problem correctly, identifying the knowns and unknowns, and using the appropriate formulas to calculate the desired quantity. In this specific case, we determined that to double both the volume and pressure of a gas initially at 273 K, the gas must be heated to 1092 K.

Moreover, this problem highlights the importance of careful analysis and attention to detail. Each step in the calculation process must be executed accurately to arrive at the correct answer. The ability to manipulate equations, substitute values, and solve for unknowns is a fundamental skill in chemistry and physics. Furthermore, understanding the underlying principles behind the gas laws, such as the kinetic molecular theory, provides a deeper appreciation of gas behavior and enhances problem-solving capabilities. This comprehensive understanding not only helps in academic settings but also has practical applications in various fields, including engineering, environmental science, and medicine.

Finally, mastering gas law problems involves not just knowing the formulas but also being able to apply them in different contexts. This requires practice and exposure to a variety of problems. By working through different scenarios and challenging problems, students can develop the confidence and skills needed to tackle any gas law problem effectively. The ability to analyze, interpret, and solve these problems is a valuable asset in any scientific or technical field, demonstrating a strong foundation in fundamental scientific principles. The ideal gas law, therefore, serves as a cornerstone in understanding the physical world and solving practical problems related to gases.