Problem 27 - Chemistry The Central Science In Si Units

Question

(a) What sign for \(\Delta S\) do you expect when the volume of \(0.200 \mathrm{~mol}\) of an ideal gas at \(27^{\circ} \mathrm{C}\) is 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?

Step-by-Step Solution

Answer

94S is positive. For 94S, the calculation gives approximately 1.022 J/K. Temperature specification is not needed for isothermal processes.

Step 1: Understanding Entropy Change Sign

Entropy (94S) is a measure of disorder or randomness. When an ideal gas expands isothermally, its entropy generally increases because the molecules have more space to move, leading to higher disorder. Thus, we expect the 94S to be positive when the volume of an ideal gas increases isothermally.

Step 2: Applying the Entropy Change Formula

For an isothermal expansion of an ideal gas, the change in entropy (94S) is given by the formula:\[\Delta S = nR\ln\left(\frac{V_f}{V_i}\right)\]where:- \(n\) is the number of moles,- \(R\) is the ideal gas constant \(8.314 \frac{J}{mol\cdot K}\),- \(V_f\) is the final volume,- \(V_i\) is the initial volume.

Step 3: Substitute the Values

Substitute the given values into the entropy change formula. Using \(n = 0.200 \, mol\), \(V_i = 10.0 \, L\), and \(V_f = 18.5 \, L\), the calculation becomes:\[\Delta S = 0.200 \times 8.314 \times \ln\left(\frac{18.5}{10.0}\right)\]

Step 4: Calculate the Natural Logarithm

Calculate the natural logarithm:\[\ln\left(\frac{18.5}{10.0}\right) = \ln(1.85) \approx 0.615\]

Step 5: Calculate 94S

Now compute the final value of 94S:\[\Delta S = 0.200 \times 8.314 \times 0.615 \approx 1.022 \, \frac{J}{K}\]

Step 6: Temperature Dependence of Entropy Change

For an isothermal process involving an ideal gas, the temperature value is not directly needed in the 94S formula as long as the process is isothermal, because the temperature remains constant. Hence, specifying a different temperature wouldn
9t alter the technique of calculation for such a process.

Key Concepts

Isothermal Expansion

Isothermal expansion is when a gas expands while maintaining a constant temperature. In this type of expansion, the internal energy doesn't change because the temperature remains the same. During isothermal expansion of an ideal gas, the gas molecules spread out over a larger volume. This ensures that the energy added to the system is used entirely to do work. Since the temperature is constant, any heat input results in work done by the gas. For example, imagine a balloon that starts to expand when you blow into it, but stays at room temperature. That's similar to what happens in an isothermal expansion process. This concept also applies to real-life systems such as engines where maintaining constant temperature can ensure efficient work conversion. To summarize, isothermal expansion involves:

• Constant temperature throughout the expansion process
• Molecules having more space to move
• Work being done by the gas, due to the energy added

Ideal Gas Law

The ideal gas law is a fundamental equation that describes the behavior of an ideal gas. It is expressed as:\[ PV = nRT \]Here, \(P\) stands for pressure, \(V\) for volume, \(n\) for the number of moles, \(R\) for the ideal gas constant (which is 8.314 J/mol·K), and \(T\) for temperature in Kelvin.In the context of an isothermal process, because the temperature \(T\) remains constant, the pressure and volume become inversely proportional: if volume increases, pressure decreases, and vice versa. This relationship helps predict how a gas will behave when subjected to different volumes or pressures.The ideal gas law assumes no interactions between gas particles and idealizes gas properties. It serves as a great approximation in many real-world situations, especially under non-extreme conditions. Though not perfect, it provides valuable insights into gas behavior.Consideration of:

• Pressure (P) and volume (V) changes, when temperature (T) is constant
• The role of the number of moles (n) in determining volume
• Practical use in predicting gas behaviors in controlled environments

Entropy Change Calculation

Entropy is a thermodynamic quantity indicating the amount of disorder in a system. During an isothermal expansion of an ideal gas, the change in entropy can be calculated using the formula:\[ \Delta S = nR\ln\left(\frac{V_f}{V_i}\right) \]Where \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(V_f\) and \(V_i\) are the final and initial volumes respectively.In an isothermal expansion, the entropy change \(\Delta S\) is usually positive because the gas molecules move into a larger volume, increasing randomness. For example, with 0.200 moles expanding from an initial 10 L to a final 18.5 L, the entropy change can be calculated with these steps:- Use the formula with given values: \(\Delta S = 0.200 \times 8.314 \times \ln(1.85)\)- Compute \(\ln(1.85)\) which is approximately 0.615.- Multiply the results to find \(\Delta S\approx 1.022 \, J/K\).This shows how entropy increases in expansion, highlighting the concept of energy distribution in a system. Understanding entropy changes is significant for predicting system behavior during energy exchange processes.