Question

If a gas expands at constant temperature: (1) The pressure decreases (2) The kinetic energy of the molecules remains the same (3) The kinetic energy of the molecules decreases (4) The number of molecules of the gas increase (a) 1,2 (b) \(1,2,3\) (c) \(1,2,4\) (d) 2,3

Short answer

(a) 1,2

Step-by-step solution

Understand the Concepts Involved

This problem deals with a gas expanding at a constant temperature, which is an example of an isothermal process. In an isothermal process, the temperature of the gas remains constant, which implies that the average kinetic energy of the gas molecules does not change because kinetic energy is directly related to temperature.

Analyze Each Statement

1. The pressure decreases: According to Boyle's Law for isothermal processes, \( PV = \text{constant} \), so if the gas expands (volume increases), then the pressure must decrease.2. The kinetic energy of molecules remains the same: Since the temperature is constant, the kinetic energy, which is related to temperature, remains unchanged.3. The kinetic energy of the molecules decreases: This contradicts the situation since the temperature is constant, so this statement is false.4. The number of molecules of the gas increases: For an ideal gas in a closed system, the number of molecules remains constant.

Determine Correct Combination of Statements

From the analysis, statements 1 and 2 are correct. The pressure decreases due to the expansion at constant temperature, and the kinetic energy remains the same. Statement 3 is incorrect as the kinetic energy does not decrease, and statement 4 is incorrect because the number of molecules doesn't change.

Choose the Correct Answer

The correct combination of true statements given the scenario is 1 and 2. Therefore, the correct answer is option (a) 1,2.

Key concepts

Boyle's Law

Boyle's Law is a crucial principle in understanding the behavior of gases under constant temperature conditions. It describes how the pressure of a gas tends to decrease as the volume of the gas increases, provided the temperature remains constant. The law is expressed mathematically as \[ P \times V = \text{constant} \]- **Pressure (\( P \))** is the force exerted by the gas molecules on the walls of its container.- **Volume (\( V \))** is the amount of space the gas occupies.
In an isothermal process, the temperature is kept constant, which means that any change in volume directly affects the pressure. If a gas expands, meaning its volume increases, the pressure must decrease in order for the product of pressure and volume to remain constant (as dictated by Boyle's Law). Understanding these relationships helps predict how gases will react under different conditions, such as expansion or compression.

Kinetic Energy

Kinetic energy in the context of gases refers to the energy that gas molecules have due to their motion. The temperature of the gas is directly related to this kinetic energy, as increased molecular motion at higher temperatures results in increased kinetic energy. However, during an isothermal process: - **Temperature remains constant:** Since temperature does not change, the average kinetic energy of gas molecules remains unchanged as well. - **Molecular movement:** The speed and movement of the molecules stay the same as long as the temperature stays consistent. This means that even if the gas expands or contracts, the kinetic energy of the individual gas molecules remains the same during an isothermal process. It's interesting to note that since kinetic energy does not change, the internal energy of an ideal gas is only a function of temperature.

Ideal Gas Law

The Ideal Gas Law is another fundamental concept that describes the behavior of a gas in relation to its pressure, volume, and temperature. The equation is given as \[ PV = nRT \]where:- **\( P \)** is pressure,- **\( V \)** is volume,- **\( n \)** is the number of moles,- **\( R \)** is the ideal gas constant,- **\( T \)** is the temperature in Kelvin.
The Ideal Gas Law combines several simpler gas laws, including Boyle's Law and Charles's Law, into one comprehensive equation. In an isothermal process:- **Temperature (\( T \))** is constant,- The product of pressure and volume (\( PV \)) will change proportionately.Although pressure and volume change, the number of gas molecules (\( n \)) in a closed system remains constant. This law is instrumental for making predictions about how a gas will behave under different conditions of pressure, volume, and temperature, assuming the gas behaves ideally.