To tackle this question, we need to consider the properties of gases and how they behave under standard conditions. At normal temperature and pressure (NTP), both hydrogen gas (H₂) and oxygen gas (O₂) occupy the same volume, which is 1 cm³ in this case. Let's analyze each statement one by one.

Analyzing the Statements

Statement a: The number of molecules is the same in both gases

This statement is true. At NTP, one mole of any ideal gas occupies approximately 22.4 liters. Since both gases are at the same volume of 1 cm³, we can calculate the number of molecules using Avogadro's number (approximately 6.022 x 10²³ molecules/mole). The number of molecules in 1 cm³ of hydrogen and oxygen can be calculated as follows:

• For hydrogen (H₂): The molar mass is about 2 g/mol, and the density at NTP is roughly 0.09 g/L. Thus, 1 cm³ of H₂ contains about 0.00009 g, which corresponds to a very small fraction of a mole.
• For oxygen (O₂): The molar mass is about 32 g/mol, and the density at NTP is approximately 1.43 g/L. Therefore, 1 cm³ of O₂ contains about 0.00143 g, which also corresponds to a small fraction of a mole.

However, since both gases are at the same volume, the number of molecules will be proportional to the number of moles, which is the same for both gases at this volume. Thus, the number of molecules is indeed the same.

Statement b: The RMS velocity of molecules of both gases is the same

This statement is false. The root mean square (RMS) velocity of gas molecules is given by the formula:

v_rms = √(3RT/M)

Where R is the universal gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas. Since hydrogen has a much lower molar mass (2 g/mol) compared to oxygen (32 g/mol), the RMS velocity of hydrogen will be significantly higher than that of oxygen. Therefore, this statement does not hold true.

Statement c: The internal energy of each gas is the same

This statement is false. The internal energy (U) of an ideal gas is related to its temperature and the number of moles. For a monatomic ideal gas, it can be expressed as:

U = (3/2)nRT

Since the number of moles and the molar mass differ between hydrogen and oxygen, their internal energies will also differ, even if they occupy the same volume at the same temperature. Thus, this statement is incorrect.

Statement d: The average velocity of molecules of each gas is the same

This statement is false. The average velocity of gas molecules is influenced by their mass and temperature. Since hydrogen molecules are lighter than oxygen molecules, they will have a higher average velocity. Therefore, this statement is also not correct.

Summary of Findings

In summary, the only correct statement is:

• a) The number of molecules is the same in both gases.

Statements b, c, and d are incorrect due to the differences in molar mass and the resulting physical properties of the gases. Understanding these concepts helps clarify the behavior of gases under various conditions and their molecular characteristics.