Problem 60 - A Complete Resource Book in Chemistry for JEE Main

Question

What is the ratio of kinetic energies of \(3 \mathrm{~g}\) of hydrogen and 4 grams of oxygen at \(\mathrm{T}(\mathrm{K}) ?\) (a) \(12: 1\) (b) \(6: 1\) (c) \(1 ; 6\) (d) \(24: 1\)

Step-by-Step Solution

Answer

The ratio of kinetic energies of hydrogen and oxygen is 12:1.

Step 1: Understand the formula for kinetic energy per mole

The formula for the average kinetic energy per mole of a gas is given by \( KE = \frac{3}{2} RT \) where \( R \) is the gas constant and \( T \) is the temperature in Kelvin. The kinetic energy is proportional to the temperature for a given amount of gas.

Step 2: Calculate moles of hydrogen

The molar mass of hydrogen (H\(_2\)) is approximately \( 2 \text{ g/mol} \). Given \( 3 \text{ g} \) of hydrogen, the number of moles is \( \frac{3}{2} = 1.5 \text{ moles} \).

Step 3: Calculate moles of oxygen

The molar mass of oxygen (O\(_2\)) is approximately \( 32 \text{ g/mol} \). Given \( 4 \text{ g} \) of oxygen, the number of moles is \( \frac{4}{32} = 0.125 \text{ moles} \).

Step 4: Calculate kinetic energy for hydrogen

Using the formula \( KE = \frac{3}{2} RT \), the kinetic energy per mole is the same for all gases at the same temperature, however, we have \( 1.5 \text{ moles} \) of hydrogen, so the total kinetic energy for hydrogen is \( 1.5 \times \frac{3}{2} RT = \frac{9}{4} RT \).

Step 5: Calculate kinetic energy for oxygen

Using the same formula, for \( 0.125 \) moles of oxygen, the total kinetic energy is \( 0.125 \times \frac{3}{2} RT = \frac{3}{16} RT \).

Step 6: Determine the ratio of kinetic energies

The ratio of the kinetic energies is \( \frac{\frac{9}{4} RT}{\frac{3}{16} RT} = \frac{9}{4} \times \frac{16}{3} = 12:1 \). The RT cancels out in the fraction, simplifying to 12:1.

Key Concepts

Moles Calculation

Calculating moles is an important step in understanding chemical reactions and properties of substances. A mole is a unit that helps express very large quantities of atoms, molecules, or particles. To find the number of moles in a substance, you use the formula:
\[\text{Moles} = \frac{\text{Mass of the substance (g)}}{\text{Molar mass (g/mol)}}\]Calculating moles involves determining the mass of the sample and dividing by its molar mass.

• Molar mass is the mass of one mole of a substance and typically measured in g/mol.
• The molar mass of hydrogen (H\(_2\)) is about 2 g/mol.
• The molar mass of oxygen (O\(_2\)) is around 32 g/mol.

For the exercise, if we have 3 grams of hydrogen, we calculate its moles as follows:
\[\frac{3}{2} = 1.5 \text{ moles of hydrogen}\]For 4 grams of oxygen:
\[\frac{4}{32} = 0.125 \text{ moles of oxygen}\]Understanding moles helps in predicting the behavior of gases and chemical reactions at various conditions.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation used to describe the behavior of gases. It relates the pressure, volume, temperature, and moles of a gas through the formula:
\[PV = nRT\]

• P stands for pressure in atmospheres or pascals.
• V is the volume in liters or cubic meters.
• n is the number of moles of the gas.
• R is the gas constant.
• T is the temperature in Kelvin.

The equation provides a way to compute any of these variables if the others are known, assuming the gas behaves ideally:

• "Ideal" means that the gas particles do not exert forces on one another, apart from elastic collision.
• The volume of individual gas particles is negligible compared to the volume the gas occupies.

The Ideal Gas Law is foundational, especially in situations involving changes in pressure, temperature, or volume of a gas sample.

Gas Constant

The gas constant \( R \) is a key component of the Ideal Gas Law, and it serves as a bridge connecting physical properties in the formula.
Its value is often given as \( R = 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1} \) or equivalently \( R = 8.314 \, \text{J K}^{-1} \text{mol}^{-1} \) depending on the units used.
Gas constants enable calculations such as determining energies or pressures and can be adjusted according to the preferred units. The choice of which version of \( R \) to use depends on the measurement units in the problem at hand.

• When volume is in liters and pressure in atmospheres, use \( R = 0.0821 \text{ L atm K}^{-1} \text{mol}^{-1} \).
• For energy calculations where pressure is expressed in pascals, use \( R = 8.314 \text{ J K}^{-1} \text{mol}^{-1} \).

The universality of the gas constant underscores the interconnectedness of thermodynamic properties across different gases.

Temperature Dependence of Kinetic Energy

Kinetic energy of gas molecules is closely related to temperature. According to kinetic molecular theory, the average kinetic energy of molecules in a gas is directly proportional to the absolute temperature:\[ KE = \frac{3}{2} RT \]This equation shows that as temperature increases, the kinetic energy also rises.

• Kinetic energy, measured in joules, is the energy of motion of particles.
• Temperature, specifically in Kelvin, is tied to the average speed of gas molecules.
• RT signifies the proportionality of energy with temperature for any mole of an ideal gas.

The temperature dependence is why gases expand when heated; molecules move faster and require more space. Grounded in this principle, calculations such as those from the Ideal Gas Law underscore why considerations of temperature are crucial in thermodynamic studies. Understanding this can also help interpret phenomena like thermal expansion and real reactions of gases in different conditions.