Question

Answer the following true or false. If false, explain why. (a) The change of entropy of a closed system is the same for every process between two specified states. (b) The entropy of a fixed amount of an ideal gas increases in every isothermal compression. (c) The specific internal energy and enthalpy of an ideal gas are each functions of temperature alone but its specific entropy depends on two independent intensive properties. (d) One of the \(T d s\) equations has the form \(T d s=d u-p d v\). (e) The entropy of a fixed amount of an incompressible substance increases in every process in which temperature decreases.

Short answer

(a) True. (b) False - entropy decreases in isothermal compression. (c) True. (d) True. (e) False - entropy decreases with temperature decrease.

Step-by-step solution

- Analyze statement (a)

The change of entropy for a closed system between two specified states is a property change, which depends only on the initial and final states and not on the process path. This statement is true.

- Analyze statement (b)

For an isothermal compression of an ideal gas, the volume decreases while temperature remains constant. According to the entropy change formula for an ideal gas, \[\frac{\text{d}S}{\text{d}V} = \frac{C_v}{T} - \frac{R}{V}\], since the volume decreases, the entropy change \(\text{d}S\) is negative. Thus, the entropy of an ideal gas decreases in isothermal compression. This statement is false.

- Analyze statement (c)

Specific internal energy \(u\) and enthalpy \(h\) of an ideal gas are only functions of temperature according to these relations: \(u = u(T)\) and \(h = h(T)\). However, entropy \(s\) is a function of both temperature and volume or temperature and pressure, i.e., \(s = s(T, V)\) or \(s = s(T, P)\). This statement is true.

- Analyze statement (d)

The T d s equation is derived from first and second laws of thermodynamics. One form of the \(T d s\) equation for a reversible process is \(T d s = d u + p d v\). This matches the form given in the statement, so this statement is true.

- Analyze statement (e)

For incompressible substances, changes in entropy are largely dependent on temperature because they do not change volume under pressure. When temperature decreases for an incompressible substance, the randomness or disorder typically decreases, hence entropy decreases. This statement is false.

Key concepts

entropy change

Entropy change in thermodynamics is a crucial aspect of understanding how systems evolve. Entropy can be thought of as a measure of disorder or randomness within a system.
For a closed system, the change in entropy depends only on the initial and final states, meaning it is a state function. This implies it does not depend on the path taken to reach from one state to another.
For example, whether you compress a gas abruptly or slowly to the same final state, the entropy change will remain identical if the initial and final states are the same.
This fundamental concept is crucial for solving many problems in thermodynamics and provides insight into the principles governing energy transfer and efficiency.

ideal gas properties

Ideal gases follow specific laws and properties which simplify their analysis. Key among these is the ideal gas law stated as \(\text{PV = nRT}\), where *P* is pressure, *V* is volume, *n* is the number of moles, *R* is the universal gas constant, and *T* is temperature.

The internal energy \(u\) and enthalpy \(h\) of an ideal gas are solely functions of temperature: \(u = u(T)\) and \h = h(T). \ The state functions depend on temperature independent of pressure or volume.
Entropy \s, however, depends on both temperature and another independent variable such as volume or pressure: \(s = s(T, V)\) or \(s = s(T, P)\).
For an isothermal process involving an ideal gas, when compressed, the volume of the gas decreases and since the temperature remains the same, the entropy decreases.

TdS equations

The T dS equations form the foundation for understanding entropy changes in thermodynamics. These equations combine the first and second laws of thermodynamics to link entropy change with other properties. One commonly used form is \(T dS = du + p dV\), applicable to reversible processes.

Here, *T* represents temperature, *dS* is the change in entropy, *du* is the change in internal energy, *p* is pressure, and *dV* is the change in volume.
These relationships help in computing the entropy change for varying thermodynamic processes. By integrating these equations, we can derive expressions for entropy changes based on measurable properties like temperature, pressure, and volume.

incompressible substances

Incompressible substances, like liquids and solids, retain constant volume regardless of pressure changes. This characteristic makes their analysis somewhat simpler compared to gases.
For incompressible substances, entropy change is predominantly influenced by temperature changes.

As temperature decreases, the molecular motion within the substance slows down, reducing entropy. Conversely, as temperature increases, molecular activity rises, and entropy increases.
Hence, understanding temperature's effect on entropy is vital for analyzing processes involving incompressible substances.
It is important to note that changes in entropy due to pressure for incompressible substances are negligible, making them unique in thermodynamic studies.