Question

If increase in temperature and volume of an ideal gas is two times, then the initial pressure \(\mathrm{P}\) changes to (a) \(4 \mathrm{P}\) (b) \(2 \mathrm{P}\) (c) \(\mathrm{P}\) (d) \(3 \mathrm{P}\)

Short answer

(c) \( \mathrm{P} \) - the pressure remains the same.

Step-by-step solution

Understand the Problem

We have an ideal gas where the temperature and volume are both increased by two times. We are asked how the initial pressure \( P \) changes given these conditions.

Recall the Equation of State

The ideal gas law is given by \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature.

Express Initial and Final Conditions

Initially, the pressure, volume, and temperature are \( P, V, \) and \( T \) respectively. After the increase, these become \( P', 2V, \) and \( 2T \). The number of moles and gas constant remain the same.

Apply Ideal Gas Law to Initial State

For the initial state, the ideal gas equation is \( PV = nRT \).

Apply Ideal Gas Law to Final State

For the final state, the equation becomes \( P'(2V) = nR(2T) \), because the volume and temperature are both doubled.

Simplify the Final State Equation

The equation from the final state becomes \( 2P'V = 2nRT \). Dividing both sides by 2, we get \( P'V = nRT \).

Compare Initial and Final Equations

Notice that \( P'V = nRT \) is identical to the initial equation \( PV = nRT \). This means \( P = P' \).

Conclude the Change in Pressure

Since \( P' = P \), the pressure remains unchanged even after doubling the temperature and volume.

Key concepts

Pressure Change

Pressure in an ideal gas context is quite an interesting subject. It's affected by changes in volume and temperature, yet our problem considers a situation where both factors double. Initially, you'd think an increase in temperature might elevate the pressure, right?
Surprisingly, when both temperature and volume increase by the same factor, the pressure doesn't change.
With the ideal gas law, you can see that pressure only alters if the ratio of temperature to volume changes. When these changes are proportionate, overall pressure remains constant.

Temperature and Volume Increase

Increasing the temperature and volume of a gas entails a fascinating interplay of properties.
When the temperature of a gas is doubled, its molecules move faster and possess more kinetic energy.
This usually raises pressure if the volume is constant. However, when volume also doubles, the increased space allows gas molecules to spread out, offsetting the acceleration in molecular movement.
In a scenario where temperature and volume double precisely, these effects balance each other out, thus maintaining the original pressure.

Ideal Gas Equation

The ideal gas equation, represented as \(PV = nRT\), is crucial in solving many gas-related problems. It perfectly depicts the relationship among pressure \(P\), volume \(V\), number of moles \(n\), the ideal gas constant \(R\), and temperature \(T\).
This formula helps us see how a change in one variable, like temperature, impacts others.
In our problem, incorporating changes into this equation unveils that when both temperature and volume change equally, the terms cancel each other out, keeping pressure constant.

Gas Properties

Understanding the properties of gases provides insight into their behavior under various conditions.
Gases are composed of molecules in constant motion, spread across any space available to them.
Factors like temperature and pressure influence these movements. The concept of an ideal gas simplifies real gas behavior by assuming no intermolecular forces and perfectly elastic collisions.
This assumption helps us efficiently apply equations like the ideal gas law, even if gases do not always behave ideally.

• Gases expand to fill their containers.
• They exert pressure due to collisions with container walls.
• Their behavior can be predicted through mathematical expressions like \(PV = nRT\).

These characteristics offer a powerful framework for understanding how gases behave when subjected to conditions like those in our exercise.