Question

In Example 5.11 of the text, the molar volume of \(\mathrm{N}_{2}(g)\) at \(\mathrm{STP}\) is given as 22.42 \(\mathrm{Lmol} \mathrm{N}_{2} .\) How is this number calculated? How does the molar volume of He(g) at \(\mathrm{STP}\) compare to the molar volume of \(\mathrm{N}_{2}(g)\) at \(\mathrm{STP}\) (assuming ideal gas behavior)? Is the molar volume of \(\mathrm{N}_{2}(g)\) at 1.000 atm and \(25.0^{\circ} \mathrm{C}\) equal to, less than, or greater than 22.42 \(\mathrm{L} / \mathrm{mol} ?\) Explain. Is the molar volume of \(\mathrm{N}_{2}(g)\) collected over water at a total pressure of 1.000 \(\mathrm{atm}\) and \(0.0^{\circ} \mathrm{C}\) equal to, less than, or greater than 22.42 \(\mathrm{L} / \mathrm{mol}\) ? Explain.

Short answer

The molar volume of N2(g) at STP is 22.42 L/mol, which is the same as the molar volume of He(g) at STP due to their ideal gas behavior. At 1.0 atm and 25.0°C, the molar volume of N2(g) is greater than 22.42 L/mol, specifically 24.47 L/mol. When collected over water at 1.0 atm and 0.0°C, the molar volume of N2(g) is also greater than 22.42 L/mol, calculated as 22.59 L/mol.

Step-by-step solution

Calculate the molar volume of N2(g) at STP

To calculate the molar volume of a gas at STP (Standard Temperature and Pressure), we use the Ideal Gas Law: \(PV=nRT\) where P is pressure in atm, V is volume in liters, n is the number of moles of the gas, R is the ideal gas constant, and T is temperature in Kelvin. At STP, the pressure is 1 atm and the temperature is 273.15 K. In this case, we have 1 mole of N2(g), so we can rearrange the Ideal Gas Law to solve for V: \(V = \frac{nRT}{P}\) Substitute the values for STP and the gas constant (R = \(0.0821 \frac{L \cdot atm}{K \cdot mol}\)): \(V = \frac{(1 \ mol)(0.0821 \frac{L \cdot atm}{K \cdot mol})(273.15 \ K)}{(1 \ atm)} = 22.42 \ L/mol\) The molar volume of N2(g) at STP is 22.42 L/mol.

Compare the molar volume of He(g) at STP to that of N2(g)

Ideal gases at the same temperature and pressure have the same molar volume, regardless of the identity of the gas. Since He(g) and N2(g) are both ideal gases at STP, their molar volumes are the same: Molar volume of He(g) at STP = Molar volume of N2(g) at STP = 22.42 L/mol

Molar volume of N2(g) at 1.0 atm and 25.0°C

To determine if the molar volume of N2(g) at 1.0 atm and 25.0°C is equal to, less than, or greater than 22.42 L/mol, we can calculate the molar volume at these conditions using the Ideal Gas Law. First, convert the temperature to Kelvin: 25.0°C + 273.15 = 298.15 K Now, calculate the molar volume: \(V = \frac{(1 \ mol)(0.0821 \frac{L \cdot atm}{K \cdot mol})(298.15 \ K)}{(1 \ atm)} = 24.47 \ L/mol\) The molar volume of N2(g) at 1.0 atm and 25.0°C is greater than 22.42 L/mol.

Molar volume of N2(g) collected over water at 1.0 atm and 0.0°C

When a gas is collected over water, its total pressure includes the vapor pressure of water. To find the partial pressure of the gas, we can subtract the vapor pressure of water at 0.0°C (which is 0.00604 atm) from the total pressure: Partial pressure of N2(g) = 1.0 atm - 0.00604 atm = 0.99396 atm Now, calculate the molar volume of N2(g) at 0.0°C (which is 273.15 K) and the partial pressure: \(V = \frac{(1 \ mol)(0.0821 \frac{L \cdot atm}{K \cdot mol})(273.15 \ K)}{(0.99396 \ atm)} = 22.59 \ L/mol\) The molar volume of N2(g) collected over water at 1.0 atm and 0.0°C is greater than 22.42 L/mol.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation used to describe the behavior of gases under various conditions. It relates the pressure (P), volume (V), number of moles (n), ideal gas constant (R), and temperature (T) in the following equation: \[ PV = nRT \] This equation tells us how gases behave in a closed system under changes in these variables.

• P is the pressure of the gas, typically measured in atmospheres (atm).
• V is the volume the gas occupies, usually in liters (L).
• n represents the number of moles of the gas.
• R is the ideal gas constant, with a value of 0.0821 \( \frac{L \cdot atm}{K \cdot mol} \).
• T is the temperature in Kelvin (K).

The Ideal Gas Law is especially useful for calculating a gas's molar volume by rearranging the formula to solve for V when conditions are known. This understanding aids in predicting how gases will react to changes in their environment, such as temperature or pressure.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) refers to a set of conditions commonly used in chemistry for the ideal gas calculations. At STP, the temperature is set at 0°C (273.15 K) and the pressure is set at 1 atm. STP provides a standardized reference point to compare gas behaviors, ensuring consistency in experimental results and calculations.

• At STP, ideal gases tend to have a consistent molar volume, roughly 22.42 L/mol, which helps chemists predict gas volume under standard conditions.
• This standardization is crucial for ensuring that experiments and theoretical predictions align.
• Using the Ideal Gas Law under STP conditions simplifies calculations, providing a reliable method to verify the nature of gas behaviors.

Partial Pressure

Partial pressure refers to the pressure that a single component of a gas mixture would exert if it occupied the entire volume alone at the same temperature.

• It is a critical concept used to understand the behavior of gas mixtures.
• The total pressure of a gas mixture is the sum of the partial pressures of all gases present, as stated by Dalton's Law of Partial Pressures.
• In practical scenarios, knowing a gas's partial pressure allows scientists to determine how much of that gas is present in a mixture.

When gases are collected over water, for example, the total pressure includes the vapor pressure of water. To find the gas's partial pressure, one must subtract the water's vapor pressure from the total measured pressure. This procedure is essential for accurate calculations in experiments, as it isolates the effective pressure exerted by the gas of interest.

Gas Behavior

Gas behavior describes how gases interact under various conditions of temperature, pressure, and volume. The Ideal Gas Law gives insights into these interactions by predicting how changes in one variable will affect others.

• At constant temperature, gases will expand when pressure decreases and contract when pressure increases, according to Boyle's Law.
• When volume is held constant, the pressure of a gas will increase with rising temperature (Gay-Lussac's Law).
• Gases tend to move towards occupying any available space to reach uniform distribution.

Accepting the assumptions of ideal gas behavior, most real gases behave ideally under ordinary conditions with moderate temperature and low pressure. However, real deviations occur under extreme conditions, such as high pressure or low temperature, due to interactions between gas particles.