Question

A certain mass of a gas occupies a volume of 10 litres at a pressure of one atmosphere at \(3000 \mathrm{k}\). Which of the following changes does not produce any change in the volume of a gas ? (1) Pressure doubled and absolute temperature halved (2) Doubling both pressure and absolute temperature (3) Pressure halved and absolute temperature doubled (4) Pressure unchanged and absolute temperature halved

Short answer

Case 2: Doubling both pressure and absolute temperature leaves the volume unchanged.

Step-by-step solution

Understanding the Ideal Gas Law

The Ideal Gas Law is given by the equation \[ PV = nRT \], where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles of gas, \( R \) is the gas constant, and \( T \) is the absolute temperature.

Analyzing the Changes in Pressure and Temperature

For changes in pressure and temperature, the volume effect can be analyzed using \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]. When considering volume changes: \[ V_2 = V_1 \times \frac{P_1}{P_2} \times \frac{T_2}{T_1} \].

Case 1: Pressure doubled and absolute temperature halved

\( P_2 = 2P_1 \) and \( T_2 = \frac{T_1}{2} \). Plug these values into the equation, \[ V_2 = V_1 \times \frac{P_1}{2P_1} \times \frac{\frac{T_1}{2}}{T_1} = V_1 \times \frac{1}{2} \times \frac{1}{2} = \frac{V_1}{4} \]. The volume decreases to one-fourth.

Case 2: Doubling both pressure and absolute temperature

\( P_2 = 2P_1 \) and \( T_2 = 2T_1 \). Plug these values into the equation, \[ V_2 = V_1 \times \frac{P_1}{2P_1} \times \frac{2T_1}{T_1} = V_1 \times \frac{1}{2} \times 2 = V_1 \]. The volume remains unchanged.

Case 3: Pressure halved and absolute temperature doubled

\( P_2 = \frac{P_1}{2} \) and \( T_2 = 2T_1 \). Plug these values into the equation, \[ V_2 = V_1 \times \frac{P_1}{\frac{P_1}{2}} \times \frac{2T_1}{T_1} = V_1 \times 2 \times 2 = 4V_1 \]. The volume increases to four times.

Case 4: Pressure unchanged and absolute temperature halved

\( P_2 = P_1 \) and \( T_2 = \frac{T_1}{2} \). Plug these values into the equation, \[ V_2 = V_1 \times \frac{P_1}{P_1} \times \frac{\frac{T_1}{2}}{T_1} = V_1 \times 1 \times \frac{1}{2} = \frac{V_1}{2} \]. The volume decreases to half.

Key concepts

Understanding Gas Volume

Gas volume is a measurement of how much space a gas occupies. When we talk about the volume of a gas, we are referring to the space that the gas molecules fill. This can change based on different conditions.
The three main factors that influence gas volume are:

• Pressure
• Temperature
• Amount of gas

This concept is crucial because it helps us understand how gases will behave, especially under varying conditions.

Pressure and Temperature Relationship

Pressure and temperature have a direct and vital relationship when it comes to the behavior of gases. This is especially important when considering gas laws like the Ideal Gas Law.
When we change the pressure or temperature of a gas, we can predict how the gas volume will change using these relationships:

• Increasing the temperature of a gas while keeping pressure constant causes the volume to increase.
• Decreasing the temperature while keeping pressure constant causes the volume to decrease.
• Increasing the pressure while keeping temperature constant causes the volume to decrease.
• Decreasing the pressure while keeping temperature constant causes the volume to increase.

Understanding these principles can help you solve many problems involving the behavior of gases.

The Ideal Gas Equation

The Ideal Gas Law gives us a way to describe how gases behave under different conditions, using the equation \[ PV = nRT \]
Here's what each term represents:

• \(P\) stands for pressure
• \(V\) stands for volume
• \(n\) stands for the number of moles of gas
• \(R\) is the gas constant
• \(T\) is the absolute temperature

By rearranging this equation, we can solve for any of the variables, and understand how changing one variable affects the others. For example:

• If the pressure is doubled and the temperature is also doubled, the volume of the gas stays the same.
• If the pressure is doubled but the temperature is halved, the volume of the gas is reduced to a quarter.
• If the pressure is halved and the temperature is doubled, the volume of the gas increases four times.
• If the pressure is kept constant and the temperature is halved, the volume of the gas is halved too.

This mathematical relationship provides a powerful tool for understanding and predicting the behavior of gases in different scenarios.