Question

When the temperature drops from \(20^{\circ} \mathrm{C}\) to \(10^{\circ} \mathrm{C},\) the pressure of a cylinder of compressed \(\mathrm{N}_{2}\) drops by \(3.4 \%\). The same temperature change decreases the pressure of a propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) cylinder by \(42 \% .\) Explain the difference in behavior.

Short answer

The significant difference in pressure drop between nitrogen and propane cylinders as the temperature drops from 20°C to 10°C can be attributed to the stronger intermolecular attractions in propane due to its larger size and higher polarizability, causing it to deviate significantly from ideal gas behavior. Nitrogen, on the other hand, behaves more like an ideal gas, resulting in only a small change in pressure (3.4% for nitrogen vs. 42% for propane).

Step-by-step solution

Ideal Gas Law and Van der Waals equation

The Ideal Gas Law is given by the equation: PV = nRT where P is pressure, V is volume, n is the number of moles of the gas, R is the gas constant, and T is the temperature in Kelvin. This equation describes how the pressure, volume, and temperature of an ideal gas are related, but real gases deviate from ideal behavior. For real gases, a more accurate equation is the Van der Waals equation: \(\left(P + \frac{a}{V^{2}}\right)(V - b) = nRT\) The Van der Waals equation takes into account intermolecular attraction and molecular volume, represented by the constants 'a' and 'b', respectively. Next, let's understand how these equations would predict the behavior of nitrogen and propane under the given conditions.

Nitrogen and Propane comparison

First, let's find out the molar mass for Nitrogen (N2) and Propane (C3H8). Nitrogen (N2): 2 * 14 = 28 g/mol Propane (C3H8): (3 * 12) + (8 * 1) = 44 g/mol Now, let's remember that the pressure drop in Nitrogen (N2) was only 3.4% and in Propane (C3H8) was 42%. This suggests that Nitrogen behaves more like an ideal gas, while Propane's behavior deviates substantially from ideal gas behavior.

Understanding the difference in behavior

The difference in behavior between the two gases in response to a temperature change could be due to several factors: 1. Polarizability: Propane has a larger electron cloud because it has more electrons than nitrogen. As a result, it can be polarized more easily, leading to stronger dispersion forces between its molecules. 2. Molecular size: Propane has a larger molecular size than nitrogen, leading to more significant intermolecular attractions and requiring an additional volume to accommodate its molecules. Whereas the Ideal Gas Law doesn't take intermolecular forces and molecular volume into account, the Van der Waals equation does. Therefore, the Van der Waals equation would predict the observed behavior more accurately than the Ideal Gas Law for both gases in this specific case. However, these intermolecular attractions and molecular volumes are still relatively small for nitrogen, which is why the change is pressure is much smaller (3.4%) than for propane (42%). In summary, the difference in behavior between nitrogen and propane cylinders when the temperature drops from 20°C to 10°C can be attributed to the stronger intermolecular attractions in propane due to its larger size and higher polarizability, causing it to deviate significantly from ideal gas behavior. Nitrogen, on the other hand, behaves more like an ideal gas, resulting in only a small change in pressure.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a cornerstone of chemistry and physics. It helps to predict the behavior of gases under various conditions by correlating temperature, pressure, volume, and the amount of gas. Simply put, the law can be expressed by the formula:
\[ PV = nRT \]
where P stands for pressure, V for volume, n for moles of gas, R is the ideal gas constant, and T is temperature in Kelvin. This elegant equation assumes gases are made of non-interacting particles in constant, random motion, with negligible volume compared to the container they’re in.

It works particularly well at high temperatures and low pressures where the behavior of real gases tends to mimic that of ideal gases. While it simplifies the real world, it is not entirely accurate because it ignores intermolecular forces and the volume of gas particles themselves.

Van der Waals equation

The Van der Waals equation modifies the Ideal Gas Law to take a closer look at real gases by considering molecular size and intermolecular forces. It can be written as:
\[ \left(P + \frac{a}{V^{2}}\right)(V - b) = nRT \]
In this equation, P is the pressure of the gas, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. The constants a and b are specific to each gas reflecting the gas's unique intermolecular forces and volume occluded by gas particles, respectively.

a considers how particles attract one another, while b accounts for the actual volume occupied by gas molecules, both of which are disregarded in the Ideal Gas Law. The Van der Waals equation is more equipped to explain the behavior of gases, like propane, which do not act ideally under all conditions.

Intermolecular Forces

Intermolecular forces are the forces of attraction and repulsion between molecules that affect their physical properties. There are several kinds of intermolecular forces, with van der Waals forces being one of the most significant when discussing gas behavior. These include London dispersion forces, dipole-dipole interactions, and hydrogen bonds.

London dispersion forces are weak and arise from the temporary dipoles in molecules, becoming more pronounced in larger molecules with more electrons, like propane. Dipole-dipole interactions occur between polar molecules, and hydrogen bonding, the strongest of the three, requires the presence of an electronegative atom like oxygen, nitrogen, or fluorine. These forces are critical in explaining why real gases deviate from ideal gas behavior, particularly at lower temperatures and higher pressures.

Molar Mass

In chemistry, the molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It can be calculated by adding the atomic masses of all atoms in a molecule. For example, in the above exercise, nitrogen (N2) has a molar mass of 28 g/mol and propane (C3H8) has a molar mass of 44 g/mol.

Knowing the molar mass of a gas is crucial for converting between the number of moles and mass, which is essential when using the Ideal Gas Law or the Van der Waals equation for calculations. The molar mass also indirectly influences a gas's behavior under different conditions, as heavier molecules, like propane, tend to have stronger intermolecular forces than lighter gases, like nitrogen, contributing to their deviation from the ideal gas behavior.