Question

Root mean square velocity of a molecule is \(v\) at pressure \(P\). If pressure is increased two times, then the rms velocity becomes (A) \(3 \mathrm{~V}\) (B) \(2 \mathrm{v}\) (C) \(0.5 \mathrm{~V}\) (D) \(\mathrm{v}\)

Short answer

In conclusion, the root mean square velocity when the pressure is increased two times is \(v'_{rms} = \sqrt{2}\cdot v_{rms}\). None of the given options (A, B, C, or D) are correct.

Step-by-step solution

1. Recall the root mean square speed formula

The root mean square speed (rms speed) is given by the following formula: \(v_{rms} = \sqrt{\frac{3kT}{m}}\) where: - \(v_{rms}\) is the root mean square speed - \(k\) is the Boltzmann constant - \(T\) is the temperature in Kelvin - \(m\) is the mass of one molecule of the gas

2. Express the pressure in terms of the rms speed

We will use the ideal gas law to relate pressure to the rms speed. 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 gas constant - \(T\) is the temperature in Kelvin Since the number of molecules \(N\) is related to the number of moles \(n\) by the formula \(N = nN_A\), where \(N_A\) is Avogadro's number, we can write the ideal gas law as: \(PV = \frac{NkT}{N_A}\) Now, we can write the pressure formula in terms of the rms speed: \(P = \frac{Nkm}{N_A} \cdot \frac{v_{rms}^2}{3}\) where we have replaced the temperature \(T\) by the rms speed using the formula \(v_{rms}^2 = 3kT/m\)

3. Determine the new root mean square speed when the pressure is increased two times

Let's call the new pressure and rms speed \(P'\) and \(v'_{rms}\), respectively. The new pressure \(P'\) is two times the initial pressure \(P\): \(P' = 2P\) Now, with the new pressure and rms speed, the pressure formula becomes: \(P' = \frac{Nkm}{N_A}\cdot \frac{(v'_{rms})^2}{3}\) Since we already know that \(P' = 2P\), we can equate the two pressure formulas and solve for the new rms speed: \(\frac{Nkm}{N_A}\cdot \frac{(v'_{rms})^2}{3} = 2 \cdot \frac{Nkm}{N_A} \cdot \frac{v_{rms}^2}{3}\) By simplifying the equation, we get: \((v'_{rms})^2 = 2\cdot(v_{rms})^2\) Finally, we take the square root of both sides: \(v'_{rms} = \sqrt{2}\cdot v_{rms}\) Since \(\sqrt{2}\) is not equal to any of the given options, we can conclude that none of the given options (A, B, C, or D) are correct.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics, used to describe the behavior of ideal gases. It is formulated as \(PV = nRT\).
This equation relates pressure \(P\), volume \(V\), number of moles \(n\), and temperature \(T\) of a gas, with \(R\) being the universal gas constant.

Key Points of the Ideal Gas Law:

• It assumes that gas molecules exhibit no attraction to each other and have negligible volume.
• The law is only strictly accurate for ideal gases, but it can be applied approximately to real gases under many conditions.
• By manipulating this formula, we can explore relationships between these properties, such as how changing pressure might affect volume or temperature.

The Ideal Gas Law provides the fundamental framework for understanding behaviors of gases in various physical situations, making it essential in many scientific and engineering applications.

Boltzmann Constant

The Boltzmann Constant is a crucial link between macroscopic and microscopic phenomena. It appears in the root mean square speed formula for gases as well as in several other physical relationships. The constant \(k\) has a value of \(1.38 \times 10^{-23} \, \text{J/K}\).

Applications of the Boltzmann Constant:

• It translates temperature, a macroscopic property, into energy, which is felt on the molecular scale.
• It is pivotal in statistical mechanics, providing insights into the distribution of particles and their energy.
• In the formula \(v_{rms} = \sqrt{\frac{3kT}{m}}\), it helps describe the average speed of particles in a gas due to their thermal energy.

The Boltzmann Constant serves as a cornerstone for connecting the large-scale, observable behavior of materials with the smaller-scale atomic actions.

Pressure and Temperature Relation

Pressure and temperature are closely related in the behavior of gases. When analyzing gas laws and equations, understanding this relation is crucial for understanding results like changes in speed or volume when conditions are altered.

Important Factors:

• Increased temperature generally leads to increased pressure if the volume remains constant, as outlined by Gay-Lussac's Law.
• The Ideal Gas Law, \(PV = nRT\), helps demonstrate how these factors adjust one another. Any change in temperature (\(T\)) naturally influences pressure (\(P\)) in the system.
• The relationship also affects root mean square speed, since higher temperatures lead to faster molecular speeds and thus potential changes in pressure.

Understanding this relationship is essential for predicting how gases will react to changes in environmental conditions.

Gas Molecule Behavior

Understanding how gas molecules behave allows scientists and engineers to predict how gases will respond to changes in their environment, like shifts in temperature or pressure.

Characteristics of Gas Molecules:

• Gas molecules are in constant motion and collide frequently with each other and with the walls of their container.
• The speed of gas molecules is primarily determined by the temperature of the system; higher temperatures result in faster moving molecules.
• Root mean square velocity provides a way to calculate an average speed of these molecules, accounting for the varying speeds of individual particles.

This behavior helps illustrate why gases expand to fill any container, why they are compressible, and how they can exert pressure. A deep understanding of these characteristics is essential for applications ranging from meteorology to engine design.