Find an expression for the change in entropy when two blocks of the same substance and of equal mass, one at the temperature \(T_{\mathrm{h}}\) and the other at \(T_{c}\), are brought into thermal contact and allowed to reach equilibrium. Evaluate the change for two blocks of copper, each of mass \(500 \mathrm{~g}\), with \(C_{p, m}=24.4 \mathrm{~J} \mathrm{~K}^{-1}\) \(\mathrm{mol}^{-1}\), taking \(T_{\mathrm{h}}=500 \mathrm{~K}\) and \(T_{\mathrm{c}}=250 \mathrm{~K}\)

Thermodynamics is a branch of physics that deals with the relations between heat and other forms of energy. It describes how thermal energy is converted to and from other forms of energy and how it affects matter. The key to understanding thermodynamics lies in its four laws which explain concepts such as energy conservation, the direction of heat transfer, and the absolute limits of efficiency in energy exchanges.

In the context of our exercise, we deal with the first law of thermodynamics, which is essentially the law of energy conservation. This law states that energy cannot be created or destroyed, only converted from one form to another. When two blocks of different temperatures are allowed to come into contact, energy will transfer from the hotter block to the colder one until they reach a state of thermal equilibrium.

The second law of thermodynamics is particularly relevant for calculating entropy changes. It states that the total entropy of an isolated system can never decrease over time. In our example, the heat transfer results in an increase in entropy, demonstrating how disorder increases in a spontaneous process - moving from a non-equilibrium to an equilibrium state.

Specific heat capacity, often shortened to specific heat, is a material-dependent property that indicates the amount of heat required to change the temperature of a unit mass of substance by one degree Celsius (or Kelvin). It is denoted by the symbol 'Cp' when measured at constant pressure.

Understanding specific heat is crucial for solving thermodynamics problems involving temperature changes. The specific heat of a substance tells us how 'reluctant' it is to change its temperature. For example, water has a high specific heat, meaning it requires a lot of energy to heat up or cool down. In our exercise, the specific heat capacity of copper is required to determine how much the entropy changes when the block's temperature changes.

To calculate the change in entropy of the copper blocks, we need to know the specific heat capacity of copper, which allows us to link the energy exchange to temperature change and subsequently to entropy change.

The equilibrium temperature is the temperature at which two substances in contact with each other and isolated from external influences will no longer exchange heat, meaning they have reached a state of thermal equilibrium. At this point, there is no net flow of thermal energy between the substances and their temperatures are equal.

Calculating the equilibrium temperature is an essential step when determining the entropy change of a system. As per the exercise, we calculate the final equilibrium temperature by taking the average of the initial temperatures of both copper blocks. This simple approach works because the blocks are of equal mass and made of the same material, hence their heat capacities are equal.

This temperature represents a state of maximum entropy for the system of two blocks, as it is the most disordered state achievable by just the transfer of heat between the two blocks without doing work on the environment or receiving work.

Entropy is a measure of the disorder or randomness in a system and is a central concept in the second law of thermodynamics. It quantifies the number of ways a system can be arranged, often interpreted as the degree of uncertainty or unpredictability.

The formula to calculate the change in entropy, \( \Delta S \), due to a temperature change is given by \( \Delta S = m \cdot Cp \cdot \ln(T_f / T_i) \), where \( m \) is the mass of the substance, \( Cp \) is its specific heat capacity, \( T_f \) is the final equilibrium temperature, and \( T_i \) is the initial temperature of the substance.

In our exercise, we apply this formula separately to each copper block, one initially at a higher temperature and one initially at a lower temperature. Aftermath, we add the entropy change of each block to find the total entropy change for the system. The logarithmic nature of the entropy formula reflects the idea that entropy is related to the exponential increase in the number of microstates accessible to a system as its temperature increases.