Question

From the third law, the entropy of a substance at temperature \(T\) is given as, (a) \(S_{T}=\frac{1}{3} C_{P}\left(\right.\) at \(\left.T_{1}\right)+\int_{T_{1}}^{T} \frac{C_{p}}{T} d T\) (b) \(S=C_{p}\left(\right.\) at \(\left.T_{1}\right)+\int_{\tau_{1}}^{T} C_{p} \ln d T\) (c) \(S=C_{p} \ln T+\int_{\tau_{1}}^{T} \frac{C_{p}}{T} d T\) (d) none of the above

Short answer

The correct answer is (d) none of the above.

Step-by-step solution

Understanding Concepts

Entropy is a thermodynamic quantity representing the unavailability of a system's thermal energy for conversion into mechanical work. For an ideal gas, the entropy change from a temperature \(T1\) to \(T\) with constant pressure is given by: \[ ΔS = S_T - S_{T1} = ∫_T^{T1} C_P/T dT \]If we assume the initial entropy \(S_{T1}\) as zero, the entropy at temperature \(T\) becomes: \[ S_T = ∫_T^{T1} C_P/T dT \] The above formula is the correct representation for entropy. Now let us look at the options and find the correct one.

Comparing with Given Options

Comparing above formula with the given options, the option (a) includes \(\frac{1}{3}\) \(C_P\) which is not present in the correct formula and the integral limit is incorrect, so option (a) is incorrect. In option (b), there is logarithm in the integral which is not part of the correct formula, so option (b) is incorrect.Option (c) includes \(C_p \ln T\) which is not present in the correct formula and the integral limit is incorrect, so option (c) is incorrect.Since none of the options (a), (b), and (c) fit with the correct formula, option (d) seems to be the correct choice.

Key concepts

Thermodynamic Entropy

Thermodynamic entropy is a fundamental concept that quantifies the disorder or randomness of a system. It plays a crucial role in understanding how energy transformations occur, particularly in heat engines and refrigerators. Entropy increases as a system's molecules have more freedom of movement and more ways in which their positions and velocities can be arranged. In essence, it is a measure of the number of microscopic configurations that correspond to a system’s macroscopic state.

When a system undergoes a thermodynamic process, such as heating or expansion, its entropy can change. For instance, when a solid melts into a liquid, there is an increase in entropy since the liquid state has more disorder than the solid state. Understanding entropy is crucial not only to physicists but also to engineers, who use this concept to optimize energy systems and processes.

Third Law of Thermodynamics

The third law of thermodynamics states that as the temperature of a system approaches absolute zero, the entropy of a perfect crystal of any pure substance approaches zero. This law has profound implications. It implies that it is impossible to reach absolute zero in a finite number of steps because entropy would have decreased to absolute zero, an unattainable state.

This law provides a reference point for entropy measurement known as the entropy reference point. By calculating the entropy difference between this zero point and any other state, we can understand how much disorder or thermal energy has been added to or removed from the system. However, this doesn't mean that entropy at temperatures above absolute zero is zero; rather, that the entropy change is zero as a system approaches this unattainable temperature.

Entropy Calculation

To calculate the change in entropy for a thermodynamic process, the formula \[ \Delta S = \int_{T_{1}}^{T} \frac{C_P}{T} dT \]is commonly used, where \( \Delta S \) is the change in entropy, \( C_P \) is the heat capacity at constant pressure, \( T_1 \) is the initial temperature, and \( T \) is the final temperature. It's noteworthy that for an ideal gas, the change in entropy can also be calculated when volume is held constant, using the heat capacity at constant volume \( C_V \) instead.

To carry out this calculation, one often performs an integration of the heat capacity over the temperature range of interest. The integral accounts for the variation in system disorder as the particles within it absorb heat. However, the actual computation can get complicated if the heat capacity changes with temperature; in such cases, the integral might need to be evaluated numerically or using specific heat capacity functions.

Ideal Gas Entropy Change

For an ideal gas, the change in entropy can be determined using a specific relationship derived from the first and second laws of thermodynamics. An ideal gas is a theoretical gas composed of many randomly moving point particles that do not interact except when they collide elastically.

The change in entropy of an ideal gas during a process from an initial state to a final state is given by the expression \[ \Delta S = C_P \ln\left(\frac{T}{T_1}\right) - R \ln\left(\frac{V}{V_1}\right) \]for a process at constant pressure or \[ \Delta S = C_V \ln\left(\frac{T}{T_1}\right) + R \ln\left(\frac{P_1}{P}\right) \]for a process at constant volume, where \( R \) is the ideal gas constant, and \( P \) and \( V \) represent the pressure and volume, respectively. These formulas highlight the direct relationship between entropy change and the logarithmic relationships of temperature, volume, and pressure changes in an ideal gas.