Question

The normal boiling points of acetylene \(\left(\mathrm{C}_{2} \mathrm{H}_{2}\right)\) and \(\mathrm{NO}_{2}\) are \(-84^{\circ} \mathrm{C}\) and \(21^{\circ} \mathrm{C}\), respectively. (a) At \(25^{\circ} \mathrm{C}\) and 1 atm, which gas would you expect to have a molar volume closer to the ideal value? (b) If you want to reduce the deviations from ideal behavior, in what direction would you change the temperature? The pressure?

Short answer

b) How can we change the temperature and pressure to reduce the deviations from ideal behavior for these gases?

Step-by-step solution

General Concept of Ideal Gas Behavior

Gases that obey the ideal gas law, \(PV=nRT\), are considered to behave ideally. Under certain conditions, real gases may deviate from the ideal gas behavior due to intermolecular forces and molecular size effects.

(a) Comparing Molar Volumes at 25°C and 1 atm

Since both gases are compared at the same temperature and pressure, we can use the ideal gas law to calculate their respective molar volumes and compare them. The ideal gas law is given by \(PV=nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. Since we're working at 1 atm pressure, to calculate the molar volume at 25°C, we can rearrange the equation to: \(V_m = \dfrac{RT}{P}\), where \(V_m\) is the molar volume. For both gases, we have: \(P = 1\: \text{atm}\), \(R = 0.0821\frac{\text{L atm}}{\text{mol K}}\), \(T = 25°\text{C} + 273.15=\: 298.15\text{K}\) Now, we can calculate the molar volumes: \(V_{m_{C_{2}H_{2}}} = \dfrac{(0.0821\frac{\text{L atm}}{\text{mol K}})(298.15\text{K})}{1 \text{atm}} = 24.45 \text{ L/mol}\) \(V_{m_{NO_{2}}} = \dfrac{(0.0821\frac{\text{L atm}}{\text{mol K}})(298.15\text{K})}{1 \text{atm}} = 24.45 \text{ L/mol}\) Both gases have the same molar volume under these conditions, which indicates that they are behaving ideally.

(b) Reducing Deviations from Ideal Behavior

To reduce deviations from ideal behavior in gases, we should change the conditions in such a way that intermolecular forces and molecular size effects can be minimized. Generally, this can be achieved by increasing the temperature and decreasing the pressure. 1. Increasing the temperature: Higher temperatures tend to give gases more kinetic energy, reducing the significance of intermolecular forces and making the gas behavior more ideal. 2. Decreasing the pressure: Lower pressures reduce the probability of close interactions between molecules, again minimizing the effects of intermolecular forces and leading to more ideal behavior.

Key concepts

Molar Volume

Molar volume, a crucial concept in understanding gases, refers to the volume occupied by one mole of a substance at a given temperature and pressure. It's particularly relevant in the context of the ideal gas law, where it is denoted as \(V_m\).

For an ideal gas, the molar volume can be calculated using the equation \(V_m = \frac{RT}{P}\), as shown in the textbook solution. Here, \(R\) is the universal gas constant with a value of 0.0821 L atm/mol K, \(\text{P}\) is the pressure in atmospheres, and \(T\) is the absolute temperature in Kelvin.

At standard temperature and pressure (0°C and 1 atm), the molar volume for an ideal gas is 22.414 L/mol. However, the molar volume can vary depending on the conditions, as well as deviations from ideal behavior due to intermolecular forces and the finite size of gas particles.

Deviations from Ideal Behavior

Real gases often deviate from ideal behavior due to interactions and particle volume, which are not accounted for by the ideal gas law. These deviations become significant especially at high pressures and low temperatures, where the particles are closer together, and their intermolecular forces become more pronounced.

The ideal gas law assumes that gas molecules do not attract or repel each other and occupy no space; however, this is not true for real gases. The Van der Waals equation is an example of how we can adjust the ideal gas law to account for these deviations by including terms for the volume of the molecules and the strength of their attractions.

Impact of Temperature and Pressure on Deviations

To minimize deviations from ideal behavior, increasing the temperature can impart more kinetic energy to the molecules, overcoming intermolecular attractions. Conversely, decreasing the pressure allows particles to be more widely spaced, reducing the impact of these forces and the particle volume.

Intermolecular Forces

Intermolecular forces are the attractive or repulsive forces between molecules that influence their physical properties and behavior. They range from weak van der Waals forces, like London dispersion forces, to stronger forces such as dipole-dipole interactions and hydrogen bonds.

In the context of gases, these forces are particularly important as they can cause real gases to deviate from the expected behavior predicted by the ideal gas law. For example, molecules with stronger intermolecular forces tend to have lower actual molar volumes than predicted because the attractive forces pull the molecules closer together.

Role in Deviations from Ideal Gas Law

Strong intermolecular forces can reduce the volume and increase the pressure of a gas sample, leading to significant deviations from the ideal gas law, particularly under conditions of low temperature and high pressure.

Real Gases vs. Ideal Gases

The distinction between real gases and ideal gases is fundamental in understanding gas behavior. While ideal gases are hypothetical and follow the ideal gas law in all conditions, real gases exhibit properties that can cause them to deviate under certain conditions.

The ideal gas law \(PV=nRT\) assumes that gas particles have no volume and do not interact with one another. However, real gases have a finite volume, and their particles experience intermolecular forces, which the ideal gas law does not consider.

Understanding Real Gas Behavior

The behavior of real gases approaches that of an ideal gas under conditions of high temperature and low pressure, where the effects of intermolecular forces and particle volume are minimal. To predict the behavior of real gases more accurately under various conditions, equations of state such as the Van der Waals equation are used.