Problem 28 - Chemistry: The Central Science

Question

The pressure on \(0.850\) mol of neon gas is increased from \(1.25\) atm to \(2.75\) atm at \(100^{\circ} \mathrm{C}\). Assuming the gas to be ideal, calculate \(\Delta S\) for this process.

Step-by-Step Solution

Answer

The change of entropy (\(\Delta S\)) for this isothermal process can be calculated using the formula \(\Delta S = nR * \ln(P2/P1)\), where n = 0.850 mol, R = 0.0821 L * atm / (mol * K), P1 = 1.25 atm, and P2 = 2.75 atm. Plugging in the values, we get \(\Delta S \approx 0.0550\) L * atm / K.

Step 1: Identify the given parameters

We are given the following parameters: - Number of moles (n) = 0.850 mol - Initial pressure (P1) = 1.25 atm - Final pressure (P2) = 2.75 atm - Temperature (T) = 100°C Since all calculations must be in Kelvin, we need to convert the temperature from Celsius to Kelvin: T = 100°C + 273.15 = 373.15 K

Step 2: Use the formula for the entropy change of an ideal gas during an isothermal process

Now we'll use the formula for the entropy change during an isothermal process: ΔS = nR * ln(P2/P1) Where n = 0.850 mol, R = 0.0821 L * atm / (mol * K), P1 = 1.25 atm, and P2 = 2.75 atm.

Step 3: Calculate the entropy change

Plug the values into the formula: ΔS = (0.850 mol) * (0.0821 L * atm / (mol * K)) * ln(2.75 atm / 1.25 atm) ΔS ≈ 0.0698 * ln(2.2) ΔS ≈ 0.0698 * 0.7885 ΔS ≈ 0.0550 L * atm / K The entropy change (ΔS) for this process is approximately 0.0550 L * atm / K.

Key Concepts

Ideal Gas

An ideal gas is a theoretical concept that represents a gas composed of many randomly moving particles that are not subject to any force except during elastic collisions. The ideal gas law, which is a cornerstone in understanding ideal gases, is expressed as:\[ PV = nRT \]Here, \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

• The concept of an ideal gas helps us model real gases under many conditions though no real gas perfectly fits these criteria.
• Neon, like other noble gases, behaves very closely to an ideal gas at standard temperature and pressure.

Even though ideal gases are a simplification, using this model allows us to predict the behavior of gases under various situations. It is particularly useful in calculations involving entropy changes during different gas processes.

Isothermal Process

An isothermal process occurs at a constant temperature. In such a process, the system exchanges heat with its surroundings to maintain the temperature constant despite changes in volume and pressure. In the realm of gases, particularly ideal gases, the work done and the heat exchange during an isothermal process are intimately linked.

• For an isothermal process involving an ideal gas, the change in internal energy is zero. This is due to the direct relation of internal energy with temperature, which remains unchanged.
• Entropy change in an isothermal process can be calculated using the formula:\[ \Delta S = nR \ln\left( \frac{P_2}{P_1} \right) \]This equation reflects how entropy is influenced by the pressure ratio when the temperature holds steady.

Understanding isothermal processes is crucial for analyzing systems where temperature remains constant, such as in the given example that calculates the entropy change (\( \Delta S \)) of neon gas during such a process at 100°C.

Neon Gas

Neon gas, a member of the noble gas family, is famous for its lack of color, taste, and distinct chemical inertness due to its full electron shells. This makes neon excellent for uses such as in neon signs and high-voltage indicators.

• Due to its monoatomic nature and noble gas characteristics, neon behaves almost ideally under standard conditions, which aids in applying the ideal gas law accurately.
• In the condition described in the exercise, neon's properties allow for an effective demonstration of entropy calculations in isothermal processes for an ideal gas.

While neon is often used in visual technologies, its gaseous properties make it a great candidate for educational purposes in physics and chemistry, especially in demonstrating theoretical concepts like those involving entropy changes in ideal gases.