Question

Consider the statements (a) to (e) below. Assume that \(\mathrm{H}_{2}(\mathrm{g})\) and \(\mathrm{O}_{2}(\mathrm{g})\) behave ideally. State whether each of the following statements is true or false. For each false statement, explain how you would change it to make it a true statement. (a) Under the same conditions of temperature and pressure, the average kinetic energy of \(\mathrm{O}_{2}\) molecules is less than that of \(\mathrm{H}_{2}\) molecules. (b) Under the same conditions of temperature and pressure, \(\mathrm{H}_{2}\) molecules move faster, on average, than \(\mathrm{O}_{2}\) molecules. (c) The volume of \(1.00 \mathrm{mol}\) of \(\mathrm{H}_{2}(\mathrm{g})\) at \(25.0^{\circ} \mathrm{C}\) 1.00 atm is \(22.4 \mathrm{L}\) (d) The volume of \(2.0 \mathrm{g} \mathrm{H}_{2}(\mathrm{g})\) is equal to the volume of \(32.0 \mathrm{g} \mathrm{O}_{2}(\mathrm{g}),\) at the same temperature and pressure. (e) In a mixture of \(\mathrm{H}_{2}\) and \(\mathrm{O}_{2}\) gases, with partial pressures \(P_{\mathrm{H}_{2}}\) and \(P_{\mathrm{O}_{2}^{\prime}}\) respectively, the total pressure is the larger of \(P_{\mathrm{H}_{2}}\) and \(P_{\mathrm{O}_{2}}\).

Short answer

(a) False, corrected: 'Under the same conditions of temperature and pressure, the average kinetic energy of \(O_2\) molecules is the same as that of \(H_2\) molecules.' (b) True. (c) False, corrected: 'The volume of 1.00 mol of \(H_2(g)\) at \(0.0ºC\) and 1.00 atm is \(22.4 L\)'. (d) True. (e) False, corrected: 'In a mixture of \(H_2\) and \(O_2\) gases, with partial pressures \(P_{H_{2}}\) and \(P_{O_{2}}\) respectively, the total pressure is \(P_{H_{2}} + P_{O_{2}}\)'.

Step-by-step solution

Statement (a) Analysis

Under the same conditions of temperature and pressure, the average kinetic energy of gases is same, irrespective of their molecular mass. Therefore, the statement (a) is false. A correct statement would be, 'Under the same conditions of temperature and pressure, the average kinetic energy of \(O_2\) molecules is the same as that of \(H_2\) molecules.'

Statement (b) Analysis

As per the kinetic theory of gases, lighter gas molecules move faster than heavier gas molecules at the same temperature. Therefore, the statement \(H_2\) molecules move faster, on average, than \(O_2\) molecules under the same conditions of temperature and pressure, is true.

Statement (c) Analysis

As per Avogadro's law, 1 mol of an ideal gas indeed occupies a volume of \(22.4 L\) at standard temperature and pressure (0ºC and 1 atm). However, the statement mentions a temperature of \(25.0ºC\), which is not standard. Thus, the statement is false. A correct version would state, 'The volume of 1.00 mol of \(H_2(g)\) at \(0.0ºC\) and 1.00 atm is \(22.4 L\)'.

Statement (d) Analysis

According to Avogadro's principle, equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules. Therefore, the volume of \(2.0 g H_2(g)\) (equivalent to 1 mol) is equal to the volume of \(32.0 g O_2(g)\) (also equivalent to 1 mol) at the same temperature and pressure, making this statement true.

Statement (e) Analysis

According to Dalton's law of partial pressures, the total pressure of a gas mixture is the sum of the partial pressures of the individual gases in the mixture. Thus, the statement is false. A correct statement would be: 'In a mixture of \(H_2\) and \(O_2\) gases, with partial pressures \(P_{H_{2}}\) and \(P_{O_{2}}\) respectively, the total pressure is \(P_{H_{2}} + P_{O_{2}} \)'.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that describes the behavior of gases under ideal conditions. It expresses the relationship between pressure, volume, temperature, and number of moles of a gas using the formula: \[ PV = nRT \]where:

• \( P \) is the pressure of the gas, typically in atmospheres (atm),
• \( V \) is the volume in liters (L),
• \( n \) is the number of moles of gas,
• \( R \) is the ideal gas constant, approximately 0.0821 L⋅atm/mol⋅K, and
• \( T \) is the temperature in Kelvin.

This law assumes that gases behave ideally, meaning they follow the principles laid out by the Kinetic Molecular Theory: gas molecules have no volume and experience no intermolecular forces. The Ideal Gas Law is used to calculate changes in one state variable when the others are fixed.
Keep in mind that real gases deviate from ideal behavior at high pressures and low temperatures, where intermolecular forces and the volume of gas molecules become significant.

Avogadro's Principle

Avogadro's Principle is an important concept in understanding the behavior of gases. It states that equal volumes of gases, at the same temperature and pressure, contain the same number of molecules not considering the type or size of the molecules. This means that 1 mole of any gas at standard temperature and pressure (STP) - 0°C and 1 atm - will occupy 22.4 liters.
A practical implication of Avogadro's Principle can be seen in comparing gases like hydrogen and oxygen. At a given temperature and pressure, equal volumes of these gases will contain an equal number of molecules. This concept also helps us understand molar volume, which is the volume occupied by one mole of a gas at STP, and supports the calculation of volumes for gases in chemical reactions, assuming ideal conditions.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures explains how the pressure in a gas mixture is determined. According to this law, the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases.
Mathematically, it is expressed as: \[ P_{\text{total}} = P_1 + P_2 + P_3 + \, ... \, + P_n \]where each \( P_i \) is the partial pressure of a specific gas in the mixture.
This principle is very helpful when calculating pressures in systems where gases are combined, such as air, which is a mixture of nitrogen, oxygen, and other gases. Dalton's Law reinforces that the pressure contributed by each gas is independent of the others, making it possible to predict the behavior of gas mixtures under various conditions.

Kinetic Energy of Gases

The kinetic energy of gases is a key concept that explains their movement and temperature. It is based on the Kinetic Molecular Theory, which states that gas temperature is directly proportional to the average kinetic energy of its molecules.
The average kinetic energy (KE) of a gas molecule can be calculated using the formula:\[ \text{KE}_{\text{avg}} = \frac{3}{2} kT \]where \( k \) is the Boltzmann constant, and \( T \) is the temperature in Kelvin.
According to the theory, this implies that two different gases at the same temperature will have the same average kinetic energy, regardless of their molecular mass. However, lighter molecules, such as hydrogen, will move faster than heavier molecules, like oxygen. This understanding helps in explaining diffusion, effusion, and the general behavior of gases.