Question

(a) What sign for \(\Delta S\) do you expect when the volume of 0.200 mol of an ideal gas at \(27^{\circ} \mathrm{Cis}\) increased isothermally from an initial volume of 10.0 \(\mathrm{L} ?(\mathbf{b})\) If the final volume is 18.5 \(\mathrm{L}\) , calculate the entropy change for the process. (c) Do you need to specify the temperature to calculate the entropy change?

Short answer

(a) For an isothermal expansion, the sign of the entropy change \(\Delta S\) is positive since the volume increases and the gas spreads out, increasing its randomness. (b) Using the formula \(\Delta S = nR\ln{\frac{V_f}{V_i}}\) and the given values, the entropy change for the process is approximately \(25.6 \, \text{J/K}\). (c) No, specifying the temperature is not necessary to calculate the entropy change for an isothermal process of an ideal gas, as the formula depends only on the ratio of volumes, the number of moles, and the gas constant.

Step-by-step solution

Part (a) Sign of Entropy Change

In an isothermal expansion, if the volume increases, then the entropy increases because the gas spreads out, and its randomness increases. For this problem, we have an expansion since the final volume is greater than the initial volume. Therefore, the sign of \(\Delta S\) is positive.

Part (b) Calculate the Entropy Change

Now, we will use the formula for entropy change in an isothermal process to find its value: \[\Delta S = nR\ln{\frac{V_f}{V_i}}\] Given: \(n = 0.200 \, \text{mol}\) \(R = 8.314 \, \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}\) \(V_i = 10.0 \, \text{L}\) \(V_f = 18.5 \, \text{L}\) Converting the volumes to SI units: \(V_i = 0.010 \, \text{m}^3\) \(V_f = 0.0185 \, \text{m}^3\) Now substitute the values into the entropy change formula and calculate \(\Delta S\): \[\Delta S = (0.200 \, \text{mol}) (8.314 \, \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}) \ln{\frac{0.0185 \, \text{m}^3}{0.010 \, \text{m}^3}}\] \[\Delta S \approx 25.6 \, \text{J/K}\]

Part (c) Temperature Specification

The temperature of the process does not need to be specified to calculate the entropy change in an isothermal process for an ideal gas. The formula we used for entropy change, \(\Delta S = nR\ln{\frac{V_f}{V_i}}\), does not have a term for the temperature. For an ideal gas, the entropy change in an isothermal process depends only on the ratio of the final volume to the initial volume, the number of moles of gas, and the gas constant.

Key concepts

Isothermal Expansion of an Ideal Gas

When an ideal gas undergoes an isothermal expansion, the temperature of the system remains constant. Isothermal means 'same temperature', so even as the gas expands, the energy in the form of internal temperature does not change. However, the volume of the gas does increase, allowing the gas particles more freedom to move and spread out. This expansion under constant temperature is a classic physics problem illustrating how a gas can do work or absorb heat without altering its internal energy.

Since temperature remains steady, the pressure of the gas must decrease according to the ideal gas law, which states that pressure is inversely proportional to volume when temperature and the amount of gas (in moles) remain unchanged. An easy way to visualize this is to imagine a piston enclosing a gas: if the plunger is slowly pulled out under constant thermal conditions, the gas expands, the pressure drops, and the overall disorder, or entropy, of the system increases.

Calculation of Entropy Change

Entropy is a measure of disorder or randomness in a system, and calculating changes in entropy is vital in understanding thermodynamic processes. To calculate the entropy change for an isothermal expansion of an ideal gas, a specific formula is used: \[\Delta S = nR\ln{\frac{V_f}{V_i}}\] where \(\Delta S\) is the change in entropy, \(n\) is the number of moles of the gas, \(R\) is the universal gas constant, and \(V_f\) and \(V_i\) are the final and initial volumes, respectively.

The logarithmic relationship indicates that the change in entropy is directly proportional to the ratio of the change in volume. This mathematical relationship shows us that as the volume increases, the change in entropy will be positive, highlighting that entropy tends to increase in a system where gases expand isothermally.

Entropy and Randomness Relationship

The concept of entropy is deeply intertwined with the idea of randomness or disorder within a system. Essentially, entropy measures how spread out or dispersed a system's energy is among its possible states. In the context of an ideal gas, the more volume the gas occupies, the more ways the gas particles can be arranged, leading to greater disorder and higher entropy.

The isothermal expansion of an ideal gas serves as a perfect example: as the gas expands into a larger volume without a change in temperature, its particles can occupy more positions, and the randomness of their positions and momenta increases. Therefore, the entropy change, which we calculate as being positive, reflects this increase in disorder. The relationship between entropy and randomness is fundamental to the second law of thermodynamics, which states that the entropy of isolated systems cannot decrease over time; it either increases or remains constant.

Ideal Gas Law Applications

The ideal gas law is a cornerstone of thermodynamics and physical chemistry, providing a close approximation of the behavior of gases under various conditions. It is expressed as the equation \(PV = nRT\), where \(P\) stands for pressure, \(V\) for volume, \(n\) for moles of gas, \(R\) for the ideal gas constant, and \(T\) for temperature.

Applications of the ideal gas law are extensive in sciences and engineering. It is used to predict how a gas will respond to changes in pressure, temperature, or volume and is a foundational principle in calculating work in thermodynamic cycles, such as in engines or refrigerators. Notably, the law also simplifies the calculation of entropy changes in isothermal processes, as seen in the example of isothermal expansion, since it does not require the knowledge of temperature for this specific process.