Question

The product of the pressure and volume of an ideal gas \(1 \mathrm{~s}\) (A) A constant (B) Directly proportional to its temperature. (C) Inversely proportional to its temperature. (D) Approx. equal to the universal gas constant.

Short answer

The correct relationship between the product of the pressure and volume of an ideal gas and its temperature is (B) Directly proportional to its temperature. This is based on the Ideal Gas Law formula \(PV = nRT\), where if n and R remain constant, then PV is directly proportional to T.

Step-by-step solution

Analyze the given options in terms of the Ideal Gas Law

We can compare the given options with the Ideal Gas Law formula to see which one is true: (A) A constant (B) Directly proportional to its temperature (C) Inversely proportional to its temperature (D) Approx. equal to the universal gas constant

Assess option (A) - A Constant

From the Ideal Gas Law, we know that PV = nRT. In option (A), we are checking if PV is a constant value. If we assume the number of moles (n) and the gas constant (R) remain constant, then PV should be directly proportional to T. Therefore, option (A) is incorrect.

Assess option (B) - Directly proportional to its temperature

In this option, we want to see if the product of pressure and volume (PV) is directly proportional to the temperature (T) of the gas. From the Ideal Gas Law formula, we can see that if n and R remain constant, then indeed, PV is directly proportional to T. Therefore, option (B) is correct.

Assess option (C) - Inversely proportional to its temperature

In option (C), we are checking if the product of pressure and volume (PV) is inversely proportional to the temperature (T). According to the Ideal Gas Law, PV = nRT. If PV were inversely proportional to T, the equation would be of the form PV = nR/T. This contradicts the Ideal Gas Law, so option (C) is incorrect.

Assess option (D) - Approx. equal to the universal gas constant

According to the Ideal Gas Law, PV = nRT. If PV were approximately equal to the universal gas constant (R), then the equation would be of the form PV ≈ R. This would mean that the product of P and V is independent of n and T, which contradicts the Ideal Gas Law. Therefore, option (D) is incorrect. Since we have analyzed all the options and found that option (B) - "Directly proportional to its temperature" - is the correct answer, we have successfully solved the exercise.

Key concepts

Pressure-Volume Relationship

In the context of the Ideal Gas Law, understanding the pressure-volume relationship of a gas is crucial. The Ideal Gas Law is typically expressed as \( PV = nRT \), where \( P \) represents the pressure, \( V \) is the volume, \( n \) is the number of moles of the gas, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.
The pressure-volume relationship essentially tells us how the pressure of a gas responds to volume changes under constant temperature and mole number conditions. According to Boyle's Law, a part of the Ideal Gas Law, at a constant temperature, the pressure of a gas varies inversely with its volume. This means if you increase the volume, the pressure decreases, and vice versa, provided the temperature does not change.

This relationship can be illustrated with:

• Blowing up a balloon: As you inflate a balloon, you increase its volume, which decreases the pressure inside (if the temperature remains constant).
• Compressing a syringe: Pressing down on the plunger decreases the volume, increasing the pressure.

These examples demonstrate an inverse pressure-volume relationship under constant temperature, which is a key aspect of gas behavior described by the Ideal Gas Law.

Temperature Proportionality

Temperature plays a significant role in the behavior of gases, especially when examining its direct proportionality to the pressure and volume product in the Ideal Gas Law. In the formula \( PV = nRT \), if the number of moles \( n \) and the universal gas constant \( R \) are constant, then \( PV \) is directly proportional to \( T \).
This means, as the temperature of the gas increases, the product of pressure and volume also increases, assuming no change in the number of moles of the gas. This direct proportionality is critical for understanding how gases behave with temperature changes.

Consider these scenarios:

• Heating a sealed container: Doing so increases the gas temperature, which results in increased pressure and possibly volume, depending on the container's flexibility.
• A car tire on a hot day: The air inside heats up, increasing the pressure due to the increased temperature.

In these contexts, the temperature proportionality of the Ideal Gas Law explains how gases expand when heated and compress when cooled, provided the amount of gas remains constant.

Universal Gas Constant

The universal gas constant \( R \) is a fundamental component in the Ideal Gas Law, binding together the relationships involving pressure, volume, and temperature. Its standard value is approximately \( 8.314 \) J/(mol·K), and it serves as a conversion factor that ensures the properties of a gas relate consistently in the equation \( PV = nRT \).
Understanding the role of \( R \) allows us to predict and quantify the behavior of gases under a variety of conditions, making it indispensable in calculations involving the Ideal Gas Law.

The universal gas constant helps in:

• Converting energy units: It ensures the energy equivalence between gas components like pressure and volume.
• Balancing equations: Enables consistent and accurate representations of gas behavior across different chemical and physical processes.

\( R \) remains constant across varied states and compositions of gases, embodying the uniformity of natural laws in chemical processes across different environments.