Question

An open flask containing air is heated from \(300 \mathrm{~K}\) to \(500 \mathrm{~K}\). What percentage of air will be escaped to the atmosphere, if pressure is keeping constant? (a) 80 (b) 40 (c) 60 (d) 20

Short answer

40% of the air will escape.

Step-by-step solution

Identify the known variables

Before solving the problem, identify the key variables given. We have the initial temperature of air in the flask, which is 300 K, and the final temperature, which is 500 K. Pressure is constant.

Apply Charles's Law

Since the pressure is constant, we can apply Charles's Law, which states that the volume of a gas is directly proportional to the absolute temperature when the pressure is held constant. The formula based on Charles's Law is given by: \( V_1/T_1 = V_2/T_2 \) where \( V_1 \) and \( V_2 \) are the initial and final volumes, and \( T_1 \) and \( T_2 \) are the initial and final temperatures, respectively.

Calculate the final volume

Rearrange the equation to solve for \( V_2 \): \( V_2 = V_1 \cdot (T_2/T_1) \). We can substitute the known temperatures in Kelvin to find the ratio of the volumes.

Calculate the percentage of air escaped

Since the volume after heating will be greater than the initial volume, calculate the difference to find the amount of air that will escape: \( V_{escaped} = V_2 - V_1 \). Then, calculate the percentage of air that escaped: \( Percentage = (V_{escaped} / V_1) \times 100% \).

Substitute the temperatures and perform calculations

With the given temperature values: \( T_1 = 300 K \) and \( T_2 = 500 K \), plugging these into the equation for \( V_2 \/ V_1 \), we get: \( V_2 \/ V_1 = \frac{T_2}{T_1} = \frac{500 K}{300 K} = \frac{5}{3} \). The volume increases by a factor of \( \frac{5}{3} \), meaning that for every 3 volumes of air, an additional 2 volumes of air will occupy the space when heated. Therefore, the escaped air is \( 2/5 \) of the total new volume.

Arrive at the final answer

The percentage of air that has escaped is \( 2/5 \) of the air which is \( (2/5) \times 100% = 40% \).

Key concepts

Gas Laws Chemistry

The study of gas laws is essential if we want to understand the behavior of gases under various conditions. Gas laws describe how the measurable properties of gases—pressure, volume, temperature, and amount of gas—interact with one another. For students aiming to excel in JEE physical chemistry, mastering these laws is a must.

One fundamental aspect of gas laws is the relationship between temperature and volume, known as Charles's Law. Charles's Law asserts that at a constant pressure, the volume of a gas changes in direct proportion to its temperature in Kelvin. This means that if a gas is heated, its volume will increase, while cooling the gas will result in a decrease in volume. This is precisely the concept tested by the exercise presented.

Understanding gas laws, including Charles's Law, is vital for interpreting phenomena in real-world situations, such as understanding how hot air balloons work or calculating changes in tire pressure during temperature fluctuations. For students aspiring to crack competitive exams like JEE, being able to solve gas law problems with precision will go a long way.

Temperature-Volume Relationship

When diving into the temperature-volume relationship, we're looking at Charles's Law in action. This law highlights the direct proportionality between the volume of a gas and its absolute temperature, as long as the pressure remains constant. To visualize this relationship, imagine a fixed amount of gas inside a balloon. As the temperature increases, the kinetic energy of the gas particles also rises, causing them to move more vigorously and push against the balloon's walls, which results in an expansion of the balloon.

The relationship is beautifully simplified in the equation \((V_1/T_1 = V_2/T_2)\), linking initial and final states of a gas undergoing a temperature change. By manipulating the equation, we can solve various problems, like determining the new volume after a temperature change or finding the percentage of gas escaped as in our exercise. This relationship is witnessed often in everyday life, from the functioning of refrigerators to the behavior of car engines.

JEE Physical Chemistry

JEE, or Joint Entrance Examination, is a highly competitive exam in India, consisting of rigorous testing in various subjects, including physical chemistry. This domain involves understanding the physical properties of chemical substances and the principles that govern their behavior, including gas laws like Charles's Law.

For JEE physical chemistry, mastering the conceptual underpinnings of gas laws and being able to apply them to solve complex problems is required. Being clear on how gases behave at different temperatures and pressures isn't just academic; it's also practical, as these principles apply to a multitude of real-world applications. Proficiency in problems like the textbook exercise shown here demonstrates a student's ability to handle real-life scenarios that involve changes in gas properties due to temperature fluctuations. At its core, JEE physical chemistry encourages a deep understanding of phenomena to foster innovation and problem-solving skills in the field of chemistry.