Problem 26 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The ratio of universal gas constant and molar mass of gas is called molar gas constant. The value of molar gas constant is greater for (a) He (b) \(\mathrm{N}_{2}\) (c) \(\mathrm{H}_{2}\) (d) same for all

Step-by-Step Solution

Answer

The value of molar gas constant is greatest for (c) H_{2}, because it has the smallest molar mass.

Step 1: Understanding Molar Gas Constant

The molar gas constant (R) is the universal gas constant divided by the molar mass (M) of the gas, given by the expression: R = R_u / M, where R_u is the universal gas constant.

Step 2: Identifying the Molar Masses of Gases

To compare the molar gas constant of different gases, identify the molar masses of the given gases. Helium (He) has a molar mass of approximately 4 g/mol. Nitrogen (_{2}) as a diatomic molecule has a molar mass of approximately 28 g/mol. Hydrogen (_{2}) as a diatomic molecule has a molar mass of approximately 2 g/mol.

Step 3: Comparing Molar Gas Constants

Since R is inversely proportional to the molar mass (M), the gas with the smallest molar mass will have the largest molar gas constant. Among He, _{2}, and _{2O}, hydrogen (_{2}) has the smallest molar mass, thus the largest molar gas constant.

Key Concepts

Universal Gas Constant

The universal gas constant, often denoted as R_u, is a fundamental constant in the equations of chemical thermodynamics and physical chemistry. It is the foundation of the ideal gas law, where PV = nR_uT, linking pressure (P), volume (V), number of moles (n), and temperature (T).

This constant provides a bridge between macroscopic and molecular properties of gases. The value of R_u is 8.314 J/(mol·K), and it remains constant regardless of the gas being considered. What changes with different gases, however, is how R_u relates to the specific molar gas constant for a gas due to variations in molar mass.

Molar Mass of Gas

Understanding the molar mass of a gas is crucial when it comes to determining its specific molar gas constant. The molar mass (M) is the mass of one mole of a substance and is expressed in grams per mole (g/mol). It varies for different gases based on their molecular composition.

For example, the molar mass of hydrogen (H₂), the lightest gas, is about 2 g/mol, whereas for helium (He) it is 4 g/mol. Comparatively, heavier gases like nitrogen (N₂) have larger molar masses, such as 28 g/mol. The molar mass directly impacts a gas' behavior under different conditions and is inversely proportional to the molar gas constant R, which is why gases with lower molar masses will have higher specific molar gas constants.

Chemical Thermodynamics

Chemical thermodynamics deals with the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. The principles behind chemical thermodynamics allow scientists to predict the spontaneity of reactions, calculate energy exchanges, and understand the behavior of gases under varying conditions.

In the context of gases, thermodynamics involves understanding how the temperature, pressure, volume, and quantity of a gas relate to each other, as represented by the ideal gas law and interpreted through constants like the universal gas constant. This field is essential for the development of energy-efficient processes, engines, refrigeration systems, and much more, representing the profound relationship between energy, matter, and temperature change.