Gas Volume Calculation At Different Temperatures A Physics Exploration

The question is: 'A gas has a volume of 2.5 L at 25 °C. What will its new volume be if the temperature is lowered to 10 °C?'

In the fascinating world of physics, understanding the behavior of gases is crucial. Gases, unlike solids and liquids, are highly compressible and their volume changes significantly with variations in temperature and pressure. This article delves into a specific scenario: A gas initially occupies a volume of 2.5 liters at a temperature of 25°C. Our goal is to determine the new volume of this gas when the temperature is lowered to 10°C. This exploration will involve applying fundamental gas laws, particularly Charles's Law, which describes the relationship between volume and temperature at constant pressure. By understanding these principles, we can predict how gases will behave under different conditions, a critical skill in various scientific and engineering applications. Understanding gas behavior is not just an academic exercise; it has practical implications in fields ranging from meteorology to industrial processes. For instance, knowing how gases expand and contract with temperature changes is essential in designing pipelines, storing compressed gases, and even understanding weather patterns. Before diving into the calculations, it's important to grasp the underlying concepts and assumptions that govern gas behavior. The ideal gas law, for example, provides a simplified model that works well under certain conditions, while real gases may deviate from this behavior under extreme pressures or temperatures. As we progress through this analysis, we will highlight these considerations and their impact on the final result. The question we are addressing is a classic example of a gas law problem, which involves manipulating variables such as volume, temperature, and pressure. By carefully applying the appropriate laws and principles, we can arrive at a solution that accurately predicts the new volume of the gas. This process not only reinforces our understanding of gas behavior but also demonstrates the power of physics in solving real-world problems.

Problem Statement: Volume Change with Temperature

Our primary objective is to calculate the new volume of a gas when its temperature changes, keeping other factors constant. Specifically, we have a gas that initially occupies a volume (V₁) of 2.5 liters at a temperature (T₁) of 25°C. The temperature is then decreased to T₂ = 10°C, and we need to find the new volume (V₂). This type of problem is a common application of gas laws, which are fundamental principles in thermodynamics and physical chemistry. To solve this problem accurately, we must first identify the relevant gas law that applies to the given conditions. In this case, we are dealing with a situation where the amount of gas (number of moles) and the pressure are kept constant while the temperature and volume change. This scenario aligns perfectly with Charles's Law. Charles's Law states that the volume of a gas is directly proportional to its absolute temperature when the pressure and the amount of gas are held constant. Mathematically, this relationship is expressed as V₁/T₁ = V₂/T₂. This law is a cornerstone in understanding gas behavior and is widely used in various scientific and engineering applications. Before we can apply Charles's Law, however, we need to ensure that our temperature values are in the correct units. Gas laws require the use of absolute temperature, which is measured in Kelvin (K). The conversion from Celsius (°C) to Kelvin is done by adding 273.15 to the Celsius temperature. Therefore, we will need to convert both the initial and final temperatures from Celsius to Kelvin before plugging them into Charles's Law. This conversion is crucial because gas laws are based on the absolute scale of temperature, where zero Kelvin represents absolute zero, the theoretical point at which all molecular motion ceases. By using Kelvin, we ensure that our calculations are consistent with the fundamental principles of thermodynamics. Once we have the temperatures in Kelvin, we can proceed with applying Charles's Law to calculate the new volume. The process involves rearranging the equation to solve for V₂ and then substituting the known values for V₁, T₁, and T₂. This step-by-step approach will allow us to determine the new volume of the gas accurately and gain a deeper understanding of how temperature affects gas volume.

Applying Charles's Law: A Step-by-Step Solution

To accurately determine the new volume of the gas, we will apply Charles's Law, which states that the volume of a gas is directly proportional to its absolute temperature when the pressure and the amount of gas are held constant. The formula for Charles's Law is V₁/T₁ = V₂/T₂. Before we can use this formula, we need to convert the temperatures from Celsius to Kelvin. The initial temperature, T₁, is 25°C, and the final temperature, T₂, is 10°C. To convert to Kelvin, we add 273.15 to each temperature: T₁ (K) = 25°C + 273.15 = 298.15 K and T₂ (K) = 10°C + 273.15 = 283.15 K. Now that we have the temperatures in Kelvin, we can proceed with applying Charles's Law. We know the initial volume V₁ = 2.5 L, T₁ = 298.15 K, and T₂ = 283.15 K. Our goal is to find V₂. We can rearrange Charles's Law to solve for V₂: V₂ = V₁ * (T₂/T₁). Substituting the known values into this equation, we get: V₂ = 2.5 L * (283.15 K / 298.15 K). Performing the calculation, we find: V₂ = 2.5 L * 0.95 = 2.375 L. Therefore, the new volume of the gas when the temperature is lowered to 10°C is approximately 2.375 liters. This result indicates that as the temperature decreases, the volume of the gas also decreases, which is consistent with Charles's Law. This step-by-step approach not only provides the numerical answer but also reinforces the importance of unit conversions and the correct application of gas laws. By carefully following each step, we can confidently predict how gases will behave under different temperature conditions. The final answer, 2.375 liters, provides a clear understanding of the change in volume due to the temperature decrease, highlighting the direct relationship between volume and temperature as described by Charles's Law. This understanding is crucial in various practical applications, from designing gas storage systems to understanding atmospheric phenomena.

Detailed Calculation and Result Interpretation

In the previous section, we applied Charles's Law to determine the new volume of the gas. Now, let's delve into a more detailed breakdown of the calculation and interpret the result in the context of gas behavior. We started with the initial conditions: a volume (V₁) of 2.5 liters at a temperature (T₁) of 25°C. The temperature was then reduced to T₂ = 10°C. The key to solving this problem lies in Charles's Law, which mathematically relates volume and temperature at constant pressure and amount of gas. As we established, Charles's Law is expressed as V₁/T₁ = V₂/T₂. The first crucial step was to convert the temperatures from Celsius to Kelvin. We added 273.15 to both temperatures, resulting in T₁ = 298.15 K and T₂ = 283.15 K. Using Kelvin is essential because gas laws are based on absolute temperature scales, where zero Kelvin represents the theoretical absence of all thermal energy. Next, we rearranged Charles's Law to solve for the final volume, V₂. The rearranged equation is V₂ = V₁ * (T₂/T₁). Substituting the known values, we have: V₂ = 2.5 L * (283.15 K / 298.15 K). Performing the division inside the parentheses, we get approximately 0.95. Multiplying this by 2.5 L gives us V₂ = 2.375 L. This result indicates that the new volume of the gas is 2.375 liters. The interpretation of this result is straightforward: As the temperature decreased from 25°C to 10°C, the volume of the gas also decreased from 2.5 liters to 2.375 liters. This behavior is consistent with Charles's Law, which predicts a direct proportionality between volume and temperature. In other words, when the temperature of a gas decreases, its volume decreases proportionally, assuming the pressure and the amount of gas remain constant. The magnitude of the change is also noteworthy. The temperature decreased by 15°C, which led to a decrease in volume of 0.125 liters. This change highlights the sensitivity of gas volume to temperature variations, especially under constant pressure conditions. This understanding has practical implications in various applications, such as designing containers for gases that may experience temperature fluctuations.

Implications and Real-World Applications

The principles demonstrated in this problem have far-reaching implications and are fundamental to many real-world applications. Understanding how gases behave under varying conditions is crucial in diverse fields, from engineering to environmental science. One significant application is in the design and operation of gas storage systems. For instance, compressed gas cylinders are used to store gases like oxygen, nitrogen, and helium. When designing these cylinders, engineers must account for the changes in temperature that the cylinders may experience. If a cylinder is heated, the pressure inside will increase, potentially leading to a dangerous situation. By applying Charles's Law and other gas laws, engineers can determine the safe operating limits for these cylinders and design appropriate safety mechanisms. Similarly, in the transportation of liquefied natural gas (LNG), understanding gas behavior is critical. LNG is transported in large tankers at very low temperatures to reduce its volume. However, as the temperature of the LNG increases, it will expand, and the tankers must be designed to accommodate this expansion safely. Gas laws also play a vital role in meteorology. The behavior of gases in the atmosphere is governed by these laws, and understanding them is essential for predicting weather patterns. For example, the expansion and contraction of air masses due to temperature changes drive many weather phenomena, such as the formation of clouds and the development of storms. In industrial processes, gas laws are used extensively in chemical reactions and manufacturing processes. Many chemical reactions involve gaseous reactants or products, and controlling the temperature and pressure of these gases is crucial for optimizing the reaction yield. In manufacturing, processes like drying and distillation rely on the principles of gas behavior to separate and purify substances. Furthermore, the study of gas behavior is essential in the field of thermodynamics, which deals with the relationships between heat, work, and energy. Thermodynamics is a cornerstone of many engineering disciplines, including mechanical engineering and chemical engineering. The ideal gas law, which combines Charles's Law, Boyle's Law, and Avogadro's Law, provides a simplified model for understanding the behavior of gases under various conditions. This model is used extensively in the design of engines, turbines, and other thermodynamic systems. In conclusion, the principles governing gas behavior are not just theoretical concepts; they have practical applications that impact our daily lives. From the safe storage of compressed gases to the prediction of weather patterns, understanding gas laws is essential for solving real-world problems and advancing scientific knowledge.

Summary and Key Takeaways

In this comprehensive exploration, we addressed the problem of determining the new volume of a gas when its temperature changes, specifically focusing on a gas that initially occupies 2.5 liters at 25°C and is then cooled to 10°C. Through a step-by-step application of Charles's Law, we successfully calculated the new volume and interpreted the result in the context of gas behavior. Our analysis began by understanding the fundamental principles of gas laws, particularly Charles's Law, which states that the volume of a gas is directly proportional to its absolute temperature when pressure and the amount of gas are held constant. This law is mathematically expressed as V₁/T₁ = V₂/T₂. We emphasized the importance of converting temperatures from Celsius to Kelvin, as gas laws are based on the absolute temperature scale. The conversion formula is T(K) = T(°C) + 273.15. Applying this conversion, we found that the initial temperature of 25°C is equivalent to 298.15 K, and the final temperature of 10°C is equivalent to 283.15 K. Next, we rearranged Charles's Law to solve for the final volume, V₂: V₂ = V₁ * (T₂/T₁). Substituting the known values, we obtained V₂ = 2.5 L * (283.15 K / 298.15 K). Performing the calculation, we determined that the new volume of the gas is approximately 2.375 liters. This result demonstrates that as the temperature of the gas decreases, its volume also decreases, which is consistent with Charles's Law. The interpretation of this result highlights the direct proportionality between volume and temperature, a key concept in understanding gas behavior. We also discussed the practical implications and real-world applications of gas laws, emphasizing their importance in various fields such as engineering, meteorology, and industrial processes. Understanding gas behavior is crucial for designing safe and efficient gas storage systems, predicting weather patterns, and optimizing chemical reactions. In summary, this exploration not only provided a solution to the specific problem but also reinforced the fundamental principles of gas laws and their significance in various scientific and engineering contexts. The key takeaways from this analysis include the importance of using absolute temperature in gas law calculations, the direct relationship between volume and temperature as described by Charles's Law, and the wide range of practical applications of these principles in real-world scenarios.