Question

Two statements are given below: Statement-I: The ratio of the molar volume of a gas to that of an ideal gas at constant temperature and pressure is called the compressibility factor. 
Statement-II: The RMS velocity of a gas is directly proportional to the square root of \( T(K) \). 

• A. Both statement-I and statement-II are correct
• B. Both statement-I and statement-II are not correct
• C. Statement-I is correct but statement-II is not correct
• D. Statement-I is not correct but statement-II is correct

Hint:

- The compressibility factor (\( Z \)) indicates how much a real gas deviates from an ideal gas. - The RMS velocity (\( v_{\text{rms}} \)) is proportional to the square root of temperature, meaning higher temperatures result in higher molecular speeds.

Answer 1

The Correct Option is A

Explanation:

Statement-I: The compressibility factor (Z) is defined as the ratio of the molar volume of a gas to the molar volume of an ideal gas at the same temperature and pressure. Mathematically, it is expressed as:

Z = \(\frac{V_{\text{real}}}{V_{\text{ideal}}}\)

where \(V_{\text{real}}\) is the molar volume of the real gas and \(V_{\text{ideal}}\) is the molar volume of the ideal gas. This statement is correct as it directly defines the compressibility factor.

Statement-II: The root mean square (RMS) velocity of a gas is given by:

\(v_{\text{RMS}} = \sqrt{\frac{3RT}{M}}\)

where \(R\) is the gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass. This formula shows that the RMS velocity is directly proportional to the square root of the absolute temperature \(T\). Therefore, this statement is also correct.

In conclusion, both statement-I and statement-II are correct.

Answer 2

Statement-I: The ratio of the molar volume of a gas to that of an ideal gas at constant temperature and pressure is called the compressibility factor.

This statement is correct. The compressibility factor (Z) is defined as the ratio of the molar volume of a real gas to the molar volume of an ideal gas under the same conditions. It helps in determining how much a gas deviates from ideal behavior.

Statement-II: The RMS velocity of a gas is directly proportional to the square root of T (K).

This statement is also correct. The root mean square (RMS) velocity of a gas is given by the equation: \( v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of the gas molecule. As shown, RMS velocity is directly proportional to the square root of the temperature.

Since both statements are correct, the correct answer is:

Both statement-I and statement-II are correct.