Problem 173 - A Complete Resource Book in Chemistry for JEE Main

Question

The entropy change involved in the isothermal reversible expansion of 2 moles of an ideal gas from a volume of \(10 \mathrm{dm}^{3}\) to a volume of \(100 \mathrm{dm}^{3}\) at \(27^{\circ} \mathrm{C}\) is: (a) \(35.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) (b) \(38.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) (c) \(45.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) (d) \(23.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\)

Step-by-Step Solution

Answer

The entropy change is approximately \(38.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\), correct answer is (b).

Step 1: Understand the formula for entropy change

Entropy change \(\Delta S\) during the isothermal reversible expansion of an ideal gas 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 \(8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\), \(V_f\) is the final volume, and \(V_i\) is the initial volume.

Step 2: Plug in the known values

In this exercise, the number of moles \(n = 2\), the initial volume \(V_i = 10 \mathrm{~dm}^3\), and the final volume \(V_f = 100 \mathrm{~dm}^3\). The formula becomes: \[\Delta S = 2 \times 8.314 \ln\left(\frac{100}{10}\right)\]

Step 3: Calculate the natural logarithm term

Calculate the term inside the logarithm: \[\ln\left(\frac{100}{10}\right) = \ln(10)\approx 2.302\]

Step 4: Calculate the entropy change

Substitute the value from Step 3 back into the formula to find \(\Delta S\): \[\Delta S = 2 \times 8.314 \times 2.302 \approx 38.297 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\]

Step 5: Compare with given options

Round \(38.297 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) and compare it to the given options. The closest option is \(38.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) which is option (b).

Key Concepts

Isothermal Expansion

Isothermal expansion is a process in which an ideal gas expands at a constant temperature. During this process, the internal energy of the gas remains unchanged because, for an ideal gas, the internal energy is dependent only on temperature.
To maintain a constant temperature, the system must exchange heat with the surroundings.
This heat exchange ensures that the energy lost by the gas, as it does work during expansion, is compensated by an equal amount of energy gained from the surroundings.

• The most distinguishing feature of isothermal processes is their temperature constancy.
• In isothermal expansion, the pressure and volume of the gas change reciprocally while maintaining a fixed temperature.
• It's a common process studied in thermodynamics because it provides insights into how gases behave under controlled temperatures.

Understanding isothermal expansions is crucial for many applications, such as understanding engines and refrigeration cycles where heat conversion at constant temperatures is essential.

Ideal Gas

The concept of an ideal gas is a fundamental one in physics and chemistry that helps us understand how gases behave under different conditions. An ideal gas is a hypothetical gas whose particles occupy no volume and have no interactions with one another.
This simplification allows us to use a set of laws, known as the ideal gas laws, which describe the behavior of gases under various conditions of pressure, volume, and temperature.

• The ideal gas law is usually stated as: \[PV = nRT\]where:
  • \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature.
• While no real gas behaves as an ideal gas under all conditions, many gases approximate this behavior under high temperature and low pressure where interactions between molecules are minimal.
• Understanding ideal gases helps in making calculations for reactions and processes simpler, allowing scientists to predict the behavior of gases in various scenarios.

Understanding the ideal gas concept is crucial for studying real-life gas behaviors and predicting their responses in controlled environments.

Reversible Process

A reversible process is a theoretical concept used in thermodynamics to help us understand how systems change. In a reversible process, the system changes in such a way that the process can be reversed without leaving any trace on the surrounding environment.
This means that both the system and surroundings can be returned to their original states without any net change.

• In real-life, truly reversible processes do not exist due to natural irreversibilities like friction and air resistance.
• A reversible process is considered the most efficient way to convert work and heat, and is often used as a benchmark to define the performance limits of engines and refrigerators.
• The key characteristic of a reversible process is that it occurs in infinitesimally small increments, allowing the system to remain in thermodynamic equilibrium throughout the change.

Using reversible processes as a model, scientists and engineers can design systems that closely approximate maximum efficiency while understanding the limitations imposed by natural forces.

Natural Logarithm

The natural logarithm is a mathematical function that greatly simplifies computations involving exponential growth and decay, and it's especially useful in thermodynamics. The natural logarithm is denoted by \(\ln\) and has a base of \(e\), which is an irrational number approximately equal to 2.718.
In the context of thermodynamics and the entropy formula, the natural logarithm shows up due to its properties related to ratios and growth, particularly in processes like isothermal expansions.

• The natural logarithm helps in calculating entropy changes when the volume of a gas changes in a process.
• It is particularly useful because it helps convert multiplicative processes into additive ones, making the calculations more manageable.
• A typical scenario where natural logarithms are used includes calculating changes in entropy, pressure, or volume in gases.

Understanding how natural logarithms function is crucial for solving problems related to exponential or continuous growth and is especially relevant in fields that involve changes that compound over time.