Question

The density of an ideal gas is \(0.03 \mathrm{~g} \mathrm{~cm}^{-3}\), Its pressure is \(106 \mathrm{~g} \mathrm{~cm}^{-1} \mathrm{sec}^{-2}\). What is its \(\mathrm{rms}\) velocity (in \(\left.\mathrm{cm} \mathrm{sec}^{-1}\right) ?\) (a) \(10^{3}\) (b) \(3 \times 10^{4}\) (c) \(10^{8}\) (d) \(10^{4}\)

Short answer

The root mean square velocity is approximately \(10^4\, \mathrm{cm/sec}\).

Step-by-step solution

Understand RMS Velocity Formula

The root mean square (rms) velocity of an ideal gas is given by the formula \(v_{rms} = \sqrt{\frac{3P}{\rho}}\) where \(P\) is the pressure and \(\rho\) is the density of the gas.

Substitute Known Values

In this problem, the density \(\rho = 0.03 \, \mathrm{g/cm}^3\) and the pressure \(P = 106 \, \mathrm{g/cm} \, \mathrm{sec}^{-2}\). Substitute these values into the formula to get: \(v_{rms} = \sqrt{\frac{3 \times 106}{0.03}}\).

Simplify the Expression

First, calculate the fraction: \(\frac{3 \times 106}{0.03} = \frac{318}{0.03}\). Simplify further by performing the division: \(\frac{318}{0.03} = 10600\).

Calculate the RMS Velocity

Now compute the square root of \(10600\): \(v_{rms} = \sqrt{10600} \approx 103\).

Identify the Correct Option

The calculated \(v_{rms}\) is approximately \(103 \times 10\) which is about \(10^{4}\). This matches option (d).

Key concepts

RMS Velocity

The root mean square (rms) velocity is a concept used to describe the speed of particles in an ideal gas. It provides a way to understand how fast the molecules are moving on average. This measurement is essential as it helps us to visualize the kinetic energy of the gas molecules.

• The rms velocity is influenced by two main factors: the temperature of the gas and the mass of the gas molecules.
• It represents an average speed but does not mean that all molecules move at this speed.

By studying rms velocity, we can compare different gases under the same conditions and get insights into their behaviors even without knowing the exact speed of each molecule.

Density of Gas

Density is a measure of how much mass is within a given volume of a substance. For gases, density has a unique role because gases are much less dense than liquids or solids. In the context of the ideal gas law, density is often used to determine how closely molecules are packed together.

• The density of a gas can be affected by pressure and temperature. Higher pressure or lower temperature results in a higher density.
• Density acts as an inverse to volume, as increasing density usually implies a reduction in volume.
• In this problem, the density is given as \(0.03 \, \text{g/cm}^3\), indicating a specific quantity of gas in a fixed volume.

Understanding gas density is crucial for calculations involving buoyancy, gas flow, and thermodynamics.

Pressure of Gas

Pressure is the force exerted by the gas particles colliding against the walls of the container per unit area. In an ideal gas, pressure is directly related to the temperature and the number of molecules.

• Increased temperature or quantity of gas molecules can lead to increased pressure, given the container size is constant.
• It's commonly measured in pascals or atmospheres, but in the given problem, it's expressed in \(\text{g/cm} \, \text{sec}^{-2}\).
• The pressure value here is \(106 \, \text{g/cm} \, \text{sec}^{-2}\), which is used to calculate the rms velocity.

The pressure gives insights on the energy and movement within the gas, crucial for applications like jet propulsion, weather forecasting, and more.

Root Mean Square Velocity Formula

The root mean square velocity formula is a mathematical expression used to calculate the rms velocity, which is the average speed of particles in terms of kinetic theory. For an ideal gas, the formula is:\[v_{rms} = \sqrt{\frac{3P}{\rho}}\]Where:

• \(v_{rms}\) is the root mean square velocity,
• \(P\) is the pressure of the gas,
• \(\rho\) is the density of the gas.

This formula is derived from the ideal gas equation and kinetic theory. It connects the macroscopic properties of gas -- pressure and density -- with the microscopic movement of its molecules. This relation helps in understanding thermal speeds of gas molecules and is pivotal in physics and engineering applications.