Question

If the absolute temperature of an ideal gas having volume \(V \mathrm{~cm}^{3}\) is doubled and the pressure is reduced to half, the final volume of gas will be (a) \(0.25 \mathrm{~V}\) (b) \(0.50 \mathrm{~V}\) (c) \(2 V^{2}\) (d) \(4 V\)

Short answer

The final volume of the gas will be (d) \(4 V\).

Step-by-step solution

Identify the Given Information

In the exercise, we're given that an ideal gas at an initial volume of V has its absolute temperature doubled and pressure reduced to half. We need to determine the final volume of the gas.

Recall the Ideal Gas Law

The relation between the pressure (P), volume (V), temperature (T), and amount of substance (n) of an ideal gas is given by the ideal gas equation, PV = nRT, where R is the ideal gas constant. Because n and R are constants for a given sample of gas, we can focus on the relationship P1V1/T1 = P2V2/T2 for the initial (P1, V1, T1) and final states (P2, V2, T2) of the gas.

Apply the Conditions from the Exercise

From the conditions given, we have T2 = 2T1 (temperature is doubled) and P2 = 0.5P1 (pressure is reduced to half). The number of moles remains the same. We can substitute these into the equation for the ideal gas law.

Set Up the Equation

Using the initial and final states of the gas, the condition can be written as (P1)(V1)/(T1) = (0.5P1)(V2)/(2T1) since P2 = 0.5P1 and T2 = 2T1.

Solve for the Final Volume

Simplifying the equation results in V1 = (0.5)(V2). Dividing by 0.5 on both sides gives V2 = 2V1. The final volume, V2, is twice the initial volume, V1.

Key concepts

Chemistry Competitive Exams

Chemistry competitive exams, such as the Chemistry Olympiads, AP Chemistry, and various college entrance exams, test a student's grasp on a wide range of chemical concepts, principles, and problem-solving abilities. Mastering topics in Physical Chemistry, including the ideal gas law, is essential because these concepts frequently appear in different formats of questions. Look for specificity in question phrasing and scenarios, as understanding the context is key in applying the appropriate formula and principles.

• Learn the underlying principles of each topic, not just the formulas.
• Practice with a variety of exercises to navigate through tricker problems.
• Conceptual clarity can simplify problem-solving, improving speed and accuracy.

To excel in these exams, focus on the conceptual understanding that enables you to interpret and solve complex problems under timed conditions.

Physical Chemistry

Physical Chemistry combines principles of physics and chemistry to explain how chemical systems behave. It helps us understand phenomena on a molecular and atomic level, often by applying mathematical models and theories. When considering ideal gas problems, for instance, Physical Chemistry provides the theoretical basis for the Ideal Gas Law, connecting macroscopic measurements of pressure, volume, and temperature with the microscopic behavior of gas molecules.

Key Areas in Physical Chemistry

• Thermodynamics: Understanding heat, energy, and work in chemical systems.
• Kinetics: Examining rates of reactions and factors affecting speed.
• Quantum Chemistry: Analyzing how quantum mechanics applies to chemical systems.

Gaining proficiency in Physical Chemistry helps you to apply such concepts succinctly to solve problems in a logical and systematic manner.

Ideal Gas Equation

The Ideal Gas Equation, PV = nRT, is an invaluable tool in solving problems related to the behavior of gases under various conditions. It correlates the volume (V), pressure (P), and temperature (T) of an ideal gas to the amount of substance (n) and a constant (R, the ideal gas constant). The beauty of this equation lies in its ability to predict the behavior of an ideal gas in response to changes in any of the variables.

Assumptions of the Ideal Gas Law

• Gas molecules do not attract or repel each other.
• The volume of the gas particles is negligible compared to the container.
• Gas particles are in constant, random, straight-line motion.
• All collisions are elastic, meaning there's no net loss of energy.

Although real gases do not perfectly obey the Ideal Gas Law, it serves as an excellent approximation under many conditions. Understanding how to manipulate this equation and its variables is critical when dealing with problems like the one mentioned, where the change in one variable will affect another.