Problem 97 - A Complete Resource Book in Chemistry for JEE Main

Question

The root mean square velocity of one mole of a monoatomic gas having molar mass \(\mathrm{M}\) is \(u_{\mathrm{rms}}\). The relation between the average kinetic energy (E) of the gas and \(u_{r m}\) is (a) \(\mathrm{u}_{\mathrm{rm}}=\sqrt{(3 \mathrm{E} / 2 \mathrm{M})}\) (b) \(\mathrm{u}_{\mathrm{ms}}=\sqrt{(2 \mathrm{E} / 3 \mathrm{M})}\) (c) \(\mathrm{u}_{\mathrm{mas}}=\sqrt{(2 \mathrm{E} / \mathrm{M})}\) (d) \(u_{\operatorname{me}}=\sqrt{(E / 3 M)}\)

Step-by-Step Solution

Answer

The correct relation is (c) \(u_{rms} = \sqrt{\frac{2E}{M}}\).

Step 1: Formula for Root Mean Square Velocity

The root mean square (RMS) velocity of a gas is given by the formula: \[u_{rms} = \sqrt{\frac{3RT}{M}}\] where \(R\) is the gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass.

Step 2: Formula for Average Kinetic Energy

The average kinetic energy \(E\) for one mole of a monoatomic gas is given by: \[E = \frac{3}{2}RT\].

Step 3: Relate RMS Velocity with Kinetic Energy

We know that \(u_{rms} = \sqrt{\frac{3RT}{M}}\) and \(E = \frac{3}{2}RT\). Solve for \(RT\) from the kinetic energy formula: \[RT = \frac{2E}{3}\].

Step 4: Substitute to Find u_{rms} in Terms of E

Substitute \(RT = \frac{2E}{3}\) into the RMS velocity formula: \[u_{rms} = \sqrt{\frac{3(\frac{2E}{3})}{M}} = \sqrt{\frac{2E}{M}}\]. This matches option (c).

Key Concepts

Kinetic Energy of Gases

Understanding the kinetic energy of gases is crucial in exploring the behavior of gases under various conditions. In physics and chemistry, kinetic energy refers to the energy that an object possesses due to its motion. For gases, this concept helps explain how their temperature relates to their internal energy.In the context of monoatomic gases, the average kinetic energy per mole is given by the formula: \[ E = \frac{3}{2} RT \]where:

• \(E\) is the average kinetic energy,
• \(R\) is the universal gas constant (approximately 8.314 J/mol·K), and
• \(T\) is the temperature in Kelvin.

This formula essentially means that the energy is dependent on the temperature of the gas. Higher temperatures imply higher kinetic energy, meaning the gas particles move more swiftly.When considering the kinetic energy of gases, we assume ideal behavior, where intermolecular forces are negligible, and collisions are perfectly elastic. This way, the temperature is a measure of average kinetic energy of the particles.

Molar Mass

Molar mass, a fundamental property of substances, refers to the mass of one mole of a substance and is expressed in grams per mole (g/mol). It plays a significant role when studying gases, particularly in the context of the root mean square velocity and other properties that depend on the mass of gas molecules.To find the molar mass, one usually adds the atomic masses of all atoms in a molecule. For example, the molar mass of oxygen (O₂) is calculated by adding the atomic masses of two oxygen atoms, each approximately 16 g/mol, which results in a molar mass of 32 g/mol.The molar mass affects the root mean square velocity, given by the formula:\[ u_{rms} = \sqrt{\frac{3RT}{M}} \]Here:

• \(M\) represents the molar mass,
• \(R\) is the gas constant, and
• \(T\) is the temperature in Kelvin.

This reveals that heavier gases (with higher molar masses) tend to have lower root mean square velocities at a given temperature. Thus, molar mass plays a critical role in understanding how quickly or slowly different gases move under the same conditions.

Gas Laws

Gas laws are a set of relationships that describe how gases behave under various conditions. These laws—derived from experimental observations—help predict the properties of gases and establish connections between variables such as temperature, pressure, volume, and number of particles.Some of the crucial gas laws include:

• **Boyle's Law**: This law states that the pressure of a fixed amount of gas at constant temperature is inversely proportional to its volume. Mathematically, it's expressed as \(PV = \text{constant}\).
• **Charles's Law**: According to Charles's Law, the volume of a fixed quantity of gas is directly proportional to its temperature (in Kelvin) when the pressure is kept constant. This is given by \(\frac{V}{T} = \text{constant}\).
• **Avogadro's Law**: This law posits that equal volumes of gases, at the same temperature and pressure, contain equal numbers of molecules, \(\frac{V}{n} = \text{constant}\), where \(n\) is the number of moles.
• **Ideal Gas Law**: Combining the previous laws, the ideal gas law provides a more general relationship: \(PV = nRT\). Here, \(n\) is the number of moles of the gas.

These principles are foundational for understanding gas behaviors in conditions ranging from everyday atmospheric pressure to high-pressure research settings. They provide insight into practical applications such as understanding the behavior of atmospheric gases, industrial applications, and even the function of airbags!