The universal gas constant (R) appears in the ideal gas law, \(PV = nRT\). Its value depends on the units used for pressure (P), volume (V), amount of substance (n), and temperature (T). We need to identify the value of R expressed in SI units.
C) 8.314 J K⁻¹ mol⁻¹ is the correct value of the universal gas constant in SI units.
The question asks about Avogadro's law, which describes the relationship between the volume of a gas and the number of molecules it contains, under specific conditions of temperature and pressure. We need to recall the statement of Avogadro's law to identify the correct option.
C) Equal number of molecules
Ideal gas behavior is based on certain assumptions, primarily that gas particles have negligible volume and no intermolecular forces. Real gases deviate from ideal behavior when these assumptions break down. Deviation is most significant under conditions where intermolecular forces become prominent and the volume of gas molecules becomes a significant fraction of the total volume. These conditions are typically low temperature and high pressure.
A) NH₃ at low temperature and high pressure — Ammonia (NH₃) molecules exhibit strong hydrogen bonding, which is a significant intermolecular force. Low temperature reduces the kinetic energy of molecules, making these forces more effective. High pressure forces molecules closer together, making both intermolecular forces and the finite volume of molecules more significant. These combined factors lead to the greatest deviation from ideal gas behavior.
The question asks for the compressibility factor (Z) for an ideal gas. We need to recall the definition of the compressibility factor and the ideal gas law to determine its value for an ideal gas.
Thus, the compressibility factor for an ideal gas is exactly 1.
C) Equal to 1. As derived, for an ideal gas, \(PV = nRT\), which leads to \[Z = \frac{PV}{nRT} = \frac{nRT}{nRT} = 1\].
The root mean square (RMS) velocity of gas molecules is a measure of the average speed of the particles in a gas. It is derived from the kinetic theory of gases and depends on the temperature and molar mass of the gas. We need to recall the standard formula for RMS velocity.
Correct Option: D) The root mean square velocity of gas molecules is given by the formula \(v_{\text{rms}} = \sqrt{\frac{3 RT}{M}}\), where \(R\) is the ideal gas constant, \(T\) is the absolute temperature, and \(M\) is the molar mass of the gas. This formula is a direct consequence of the kinetic theory of gases.
To determine the correct order of most probable velocity (\(v_{mp}\)), average velocity (\(v_{avg}\)), and root mean square velocity (\(v_{rms}\)), we need to recall their respective formulas based on the kinetic theory of gases. These velocities are derived from the Maxwell-Boltzmann distribution of molecular speeds.
Recall the formulas for each velocity:
Most Probable Velocity (\(v_{mp}\)): This is the velocity possessed by the maximum number of molecules in a gas at a given temperature. Its formula is:
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]Average Velocity (\(v_{avg}\)): This is the arithmetic mean of the velocities of all the molecules present in the gas. Its formula is:
\[ v_{avg} = \sqrt{\frac{8RT}{\pi M}} \]Root Mean Square Velocity (\(v_{rms}\)): This is the square root of the mean of the squares of the velocities of the individual molecules. It is a measure of the average speed of the particles in a gas, and it is directly related to the kinetic energy of the gas. Its formula is:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]In these formulas, \(R\) is the ideal gas constant, \(T\) is the absolute temperature, and \(M\) is the molar mass of the gas.
Compare the coefficients under the square root: To compare these velocities, we can compare the numerical coefficients multiplying \(\frac{RT}{M}\) inside the square root for each expression:
Calculate the numerical values of the coefficients:
Order the coefficients:
Comparing the numerical values: \(2 < 2.546 < 3\)
Therefore, the order of the velocities is:
\(v_{mp} < v_{avg} < v_{rms}\)
D) \(v_{mp} < v_{avg} < v_{rms}\)