Question

Calculate \(\Delta A\) for the isothermal compression of \(2.95 \mathrm{mol}\) of an ideal gas at \(325 \mathrm{K}\) from an initial volume of \(60.0 \mathrm{L}\) to a final volume of \(20.5 \mathrm{L}\). Does it matter whether the path is reversible or irreversible?

Short answer

The change in Helmholtz Free Energy \(\Delta A\) for the isothermal compression of \(2.95 \ \mathrm{mol}\) of an ideal gas at \(325 \ \mathrm{K}\) from an initial volume of \(60.0 \ \mathrm{L}\) to a final volume of \(20.5 \ \mathrm{L}\) is approximately \(-4133 \ \mathrm{J}\). It does not matter if the process is reversible or irreversible, as Helmholtz Free Energy is a state function and depends only on the initial and final states of the system.

Step-by-step solution

List the known variables and determine the unknown variable

We know the following variables: - \(n = 2.95 \ \mathrm{mol}\) - \(R = 8.314 \ \mathrm{J \ mol^{-1} \ K^{-1}}\) (Ideal gas constant) - \(T = 325 \ \mathrm{K}\) - \(V_i = 60.0 \ \mathrm{L}\) - \(V_f = 20.5 \ \mathrm{L}\) We need to determine the change in Helmholtz free energy, \(\Delta A\).

Convert volumes to SI units

To use the ideal gas constant in SI units (Joules per mole per Kelvin), we need to convert the given volumes from liters to cubic meters (m³). We'll use the conversion factor 1 L = 0.001 m³: \(V_i = 60.0 \ \mathrm{L} \times 0.001 \frac{\mathrm{m}^3}{\mathrm{L}} = 0.060 \ \mathrm{m}^3\) \(V_f = 20.5 \ \mathrm{L} \times 0.001 \frac{\mathrm{m}^3}{\mathrm{L}} = 0.0205 \ \mathrm{m}^3\)

Calculate the change in Helmholtz free energy

Now we substitute the known variables into the formula for the change in Helmholtz free energy: \(\Delta A = nRT\ln\frac{V_f}{V_i} = 2.95 \ \mathrm{mol} \times 8.314 \frac{\mathrm{J}}{\mathrm{mol \ K}} \times 325 \ \mathrm{K} \times \ln\frac{0.0205 \ \mathrm{m}^3}{0.060 \ \mathrm{m}^3}\) Calculating the value: \(\Delta A \approx -4133 \ \mathrm{J}\)

Determine if the path matters

Since Helmholtz Free Energy is a state function, it only depends on the initial and final states of the system and not on the path taken. Therefore, it does not matter if the isothermal compression process is reversible or irreversible; the change in Helmholtz free energy will remain the same. So, in this case, the path does not matter, and the change in Helmholtz Free Energy \(\Delta A\) for the isothermal compression of \(2.95 \ \mathrm{mol}\) of an ideal gas at \(325 \ \mathrm{K}\) from an initial volume of \(60.0 \ \mathrm{L}\) to a final volume of \(20.5 \ \mathrm{L}\) is approximately \(-4133 \ \mathrm{J}\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure (P), volume (V), temperature (T), and amount (n) of an ideal gas through the gas constant (R). The law is usually stated as:
\(PV = nRT\).
This equation allows us to understand how gases will behave under different conditions, assuming the gas can be approximated as 'ideal'. An 'ideal' gas is one that follows the kinetic molecular theory closely, with gas particles that have negligible volume and do not exert intermolecular forces on one another.

In the context of the textbook exercise, the Ideal Gas Law isn't directly used to calculate the change in Helmholtz free energy, but understanding this law is still essential as Helmholtz free energy calculations for an ideal gas involve state variables (P, V, n, T) that are components of this fundamental equation.

Isothermal Compression

Isothermal compression is a thermodynamic process that takes place at a constant temperature (iso- meaning 'same', thermal- meaning 'temperature'). During this process, as the volume of a gas is decreased (compression), work is done on the gas, which can lead to an increase in pressure if the temperature is kept constant.

For an ideal gas undergoing isothermal compression, the pressure and volume are inversely related, which can be seen from the rearrangement of the Ideal Gas Law \(PV = nRT\), where the product of pressure and volume remains constant if the temperature (T) and the number of moles (n) are fixed. The concept of isothermal compression is significant in the exercise as it defines the conditions under which the change in Helmholtz free energy takes place. Calculating the change in free energy during an isothermal process requires us to consider how the volume changes while maintaining the same temperature.

State Function

A state function is a property of a system that depends only on its current state, not on the path or process taken to reach that state. This means that the value of a state function is the same regardless of how the system got there, whether through a reversible or an irreversible path. Common examples of state functions include internal energy, enthalpy, entropy, and the Helmholtz free energy.

In the given textbook solution, Helmholtz free energy \(\Delta A\) is a state function. The problem demonstrates that the change in Helmholtz free energy for the isothermal compression of a gas depends only on the initial and final volumes and not on the particulars of the process itself. Hence, knowing that it is a state function simplifies calculations and analysis in thermodynamics because it assures us that it does not matter whether the path is reversible or irreversible; the change in the value of \(\Delta A\) will be the same.