Problem 63 - University Physics with Modern Physics

Question

20.63. An object of mass \(m_{1}\) , specific heat capacity \(c_{1}\) , and temperature \(T_{1}\) is placed in contact with a second object of mass \(m_{2},\) specific heat capacity \(c_{2},\) and temperature \(T_{2}>T_{1} .\) As a result, the temperature of the first object increases to \(T\) and the temperature of the second object decreases to \(T^{\prime} .\) (a) Show that the entropy increase of the system is $$ \Delta S=m_{1} c_{1} \ln \frac{T}{T_{1}}+m_{2} c_{2} \ln \frac{T^{\prime}}{T_{2}} $$ and show that energy conservation requires that $$ m_{1} c_{1}\left(T-T_{1}\right)=m_{2} c_{2}\left(T_{2}-T^{\prime}\right) $$ (b) Show that the entropy change \(\Delta S\) , considered as a function of \(T,\) is a maximum if \(T=T,\) which is just the condition of thermodynamic equilibrium. (c) Discuss the result of part (b) in terms of the idea of entropy as a measure of disorder.

Step-by-Step Solution

Answer

The entropy increase is maximized when the system reaches thermal equilibrium, maximizing disorder.

Step 1: Calculate Entropy Change for Each Object

For an object with mass \( m \), specific heat capacity \( c \), and temperature changing from \( T_i \) to \( T_f \), the change in entropy \( \Delta S \) is given by the formula \( \Delta S = m c \ln \left( \frac{T_f}{T_i} \right) \). Using this, we calculate the entropy change for the first object as \( \Delta S_1 = m_1 c_1 \ln \left( \frac{T}{T_1} \right) \) and for the second object as \( \Delta S_2 = m_2 c_2 \ln \left( \frac{T'}{T_2} \right) \). The total entropy change for the system is then \( \Delta S = \Delta S_1 + \Delta S_2 \).

Step 2: Apply Conservation of Energy

According to the conservation of energy, the heat lost by the second object (with mass \( m_2 \)) must equal the heat gained by the first object (with mass \( m_1 \)). This gives us the equation:\[ m_1 c_1 (T - T_1) = m_2 c_2 (T_2 - T') \] This equation ensures that the system is in thermal equilibrium, where energy is neither created nor destroyed but transferred from the hotter object to the cooler one.

Step 3: Analyze Entropy Change as a Function of Final Temperature

The entropy change \( \Delta S(T) \) can be considered as a function of \( T \) and a maximum is found by taking the derivative with respect to \( T \) and setting it to zero. Thus, \( \frac{d}{dT} \left( m_1 c_1 \ln \frac{T}{T_1} + m_2 c_2 \ln \frac{T'}{T_2} \right) = 0 \). Using the relation \( m_1 c_1 (T - T_1) = m_2 c_2 (T_2 - T') \), solve for \( T \) and check that no further increase in entropy occurs beyond this \( T \).

Step 4: Interpret Entropy and Thermodynamic Equilibrium

The result shows that \( \Delta S(T) \) is maximized when the final temperatures are such that the entropy of the system is greatest.This condition is met when the temperatures \( T \) and \( T' \) reach a point of equilibrium where they approach one another. Therefore, the maximum entropy of the system describes a state of maximum disorder or randomness, reaffirming the concept of entropy as a measure of disorder, which increases until equilibrium is achieved.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a property of matter that tells us how much heat is needed to change the temperature of a substance. It is usually given in units of joules per kilogram per degree Celsius (J/kg°C). This value varies depending on the material, indicating how much energy it can absorb before its temperature changes.
When two objects with different specific heat capacities are placed in contact, the amount of energy each can absorb or release will influence how their temperatures change. For instance:

• A substance with a high specific heat capacity can absorb more heat without changing temperature dramatically.
• A substance with a low specific heat capacity will change temperature more easily with the same amount of heat.

Understanding specific heat capacity helps us predict how objects will respond to heat transfer, which is fundamental in solving problems related to entropy change.

Thermodynamic Equilibrium

Thermodynamic equilibrium is a state where no net heat transfer occurs between the objects in contact. Each object in a system has the same temperature. It represents a balance where all parts of the system are in a stable condition.
Achieving thermodynamic equilibrium means:

• The temperatures of interacting objects equalize.
• There's no more heat movement from one object to another.
• The overall energy is conserved within the system as a whole.

When two objects reach thermodynamic equilibrium, their temperature changes no longer affect the entropy of the system, as energy transitions have ceased. In the context of the exercise, reaching equilibrium corresponds to a state where the system entropy is maximized, reflecting maximum disorder.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In thermodynamics, this principle ensures that any energy lost by one object must be gained by another.
For a system of two objects:

• The heat lost by the warmer object must equal the heat gained by the cooler object.
• This ensures that the total energy within the system remains constant.

The equation given in the exercise \[ m_1 c_1 (T - T_1) = m_2 c_2 (T_2 - T') \]demonstrates this principle. It states that the heat exchange between the two objects aligns with the conservation of energy, ultimately leading the system to thermodynamic equilibrium.

Heat Transfer

Heat transfer is the movement of thermal energy from one object or substance to another. It occurs when there is a temperature difference, moving from the hotter object to the cooler one. This process continues until thermal equilibrium is achieved.
The modes of heat transfer include:

• Conduction: Direct heat transfer through a substance without any movement of the substance itself.
• Convection: Heat transfer through a fluid (liquid or gas) caused by the moving fluid itself.
• Radiation: Transfer of heat through electromagnetic waves, such as the heat from the sun.

In the context of the exercise, heat transfer drives the change in temperatures of the two objects, impacting the overall entropy of the system. By understanding how heat transfer works, we can better grasp why entropy changes as energy moves within the system.

Disorder in Thermodynamics

In thermodynamics, disorder refers to the distribution and arrangement of particles within a system. Increasing entropy reflects increasing disorder. This tendency towards disorder is a universal law, known as the Second Law of Thermodynamics.
Entropy measures a system's level of disorder or randomness. When two objects at different temperatures are placed together:

• Heat flows naturally from the hot object to the cooler one.
• Entropy increases until maximum disorder (thermal equilibrium) is achieved.
• This aligns with the natural tendency for systems to evolve towards disorder unless energy input drives order.

As entropy reaches its peak when a system is at equilibrium, the energy distribution becomes more random. This concept of disorder and randomness is crucial for understanding why systems behave the way they do in thermodynamics.