Question

The root mean square velocity of a gas is doubled when the temperature is (1) reduced to half (2) reduced to one-fourth (3) increased four times (4) increased two times

Short answer

Increased four times.

Step-by-step solution

Understand the Relation Between Velocity and Temperature

The root mean square (RMS) velocity of a gas is given by the formula: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \] where \( k \) is Boltzmann's constant, \( T \) is the absolute temperature, and \( m \) is the mass of a gas molecule. Since Boltzmann's constant and mass do not change, we can simplify it to: \[ v_{rms} \propto \sqrt{T} \]

Set Up the Scenario of Doubling the RMS Velocity

If the RMS velocity is doubled, then: \[ 2v_{rms} \propto \sqrt{T'} \] where \( T' \) is the new temperature.

Establish the Proportionality Relationship

Since \( v_{rms} \propto \sqrt{T} \), then \[ \sqrt{T'} = 2 \sqrt{T} \] Squaring both sides gives: \[ T' = 4T \]

Analyze the Options

Compare the new temperature \( T' = 4T \) with the given options: (1) Reduced to half: \( T' = T/2 \) – does not match (2) Reduced to one-fourth: \( T' = T/4 \) – does not match (3) Increased four times: \( T' = 4T \) – matches (4) Increased two times: \( T' = 2T \) – does not match

Key concepts

RMS velocity

Root Mean Square (RMS) velocity is a key concept in understanding the behavior of gas molecules. The RMS velocity is essentially the measure of the average speed of gas particles in a container.
The formula to calculate RMS velocity is given by: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \] Here:

• \(k\) is the Boltzmann's constant.
• \(T\) is the absolute temperature in Kelvin.
• And, \(m\) is the mass of one molecule of the gas.

This formula indicates that RMS velocity depends on the temperature of the gas and the mass of its particles. As temperature increases, the speed of the particles increases too, and vice versa. This relationship is crucial in understanding gas behavior, especially under various thermal conditions.

temperature relationship

The temperature of a gas plays a crucial role in determining its RMS velocity. There is a direct relationship between temperature and RMS velocity.
This relationship is mathematically expressed as: \[ v_{rms} \propto \sqrt{T}\]
This means that the RMS velocity is proportional to the square root of the absolute temperature. If the temperature of the gas doubles, the RMS velocity increases by a factor of the square root of 2.
To apply this concept to our problem in the exercise, when the RMS velocity is doubled, the square root of the new temperature must equal twice the square root of the original temperature. This helped us find that the new temperature is four times the original temperature (option 3). Understanding this relationship allows you to predict how changes in temperature affect the speed of gas molecules.

gas laws

Gas laws are essential in understanding the behavior of gases under different conditions. These laws combine principles from both physics and chemistry to describe how gases interact with changes in temperature, volume, and pressure.

• The most well-known gas laws are Boyle's Law, Charles's Law, and Avogadro's Law, which are encapsulated in the Ideal Gas Law postulate: \[ PV = nRT \] Here:
  • \(P\) is pressure.
  • \(V\) is volume.
  • \(n\) is the number of moles of gas.
  • \(R\) is the gas constant.
  • \(T\) is the absolute temperature.

These laws dictate how gas variables are interrelated. For instance, increasing the temperature of gas while keeping the volume constant increases the pressure. Similarly, changes in temperature can affect molecular speeds and so the RMS velocity. Understanding gas laws is fundamental to applying concepts like RMS velocity in real-world scenarios.

proportionality in physics

Proportionality is a fundamental concept in physics, which tells us how one quantity changes in relation to another. In the context of RMS velocity, we use proportionality to understand how velocity changes with temperature.

• Using the simplified RMS velocity formula, \( v_{rms} \propto \sqrt{T} \), we see a proportional relationship.
• This means that if you know how one quantity changes, you can predict how the other will change.

For example, if the temperature increases by a factor of 4, the RMS velocity increases by a factor of 2 (since \( \sqrt{4} = 2 \)). Proportionality helps in making reliable predictions and solving complex problems by breaking them down into simpler, proportional relationships. Mastering this concept simplifies understanding other physics and chemistry principles.