Show that \(T \mathrm{~d} s=c_{x}(\partial T / \partial Y)_{x} \mathrm{~d} Y+c_{Y}(\partial T / \partial x)_{Y} \mathrm{~d} x\), where \(x=X / n\) is the amount of extensive variable, \(X\), per mole, \(c_{x}\) is the heat capacity per mole at constant \(x\), and \(c_{Y}\) is the heat capacity per mole at constant \(Y\).

Heat capacity is a measure of the amount of heat energy required to change the temperature of a substance by a certain amount. Specifically, it is the heat energy needed to raise the temperature of one mole of a substance by one degree Celsius or Kelvin. Heat capacity can vary depending on whether the measurement is made at constant volume (denoted as \(c_v\)) or constant pressure (denoted as \(c_p\)). In the context of the given exercise, heat capacity per mole at constants x and Y are denoted as \(c_x\) and \(c_Y\) respectively, where x and Y are specific variables related to the problem.

• \(c_x\): Heat capacity at constant x
• \(c_Y\): Heat capacity at constant Y

Heat capacity is crucial in understanding how much energy is needed to attain a specific temperature change. This concept is integral to the equation provided in the problem, as it helps to express differential relationships involving temperature and entropy changes. In thermodynamic systems, these capacities help in defining how systems store energy and how they respond to energy inputs.

Entropy is a fundamental concept in thermodynamics representing the degree of disorder or randomness in a system. It quantifies the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state. An increase in entropy generally indicates an increase in disorder and energy dispersal within a system.

• Defined as a state function: entropy depends only on the state's initial and final conditions, not on the process taken to get from one to the other.
• Units: Entropy is typically measured in Joules per Kelvin (J/K).

In the given exercise, the differential change in entropy, denoted as \(ds\), plays a key role. The relationship between heat capacities and entropy is expressed through partial derivatives and the differentials of temperature and specific variables x and Y. This relationship is essential to thermodynamic processes, such as determining the efficiency and feasibility of thermodynamic cycles.

Differential equations in thermodynamics describe how variables change with respect to each other. These equations are vital tools for modeling and predicting behavior in physical systems.

• Differential equations provide relationships between state variables (such as temperature, pressure, entropy) and changes within a system.
• In the problem, differential equations are used to express changes in entropy (\(ds\)) and temperature (\(dT\)) in terms of other variables.

The specific differential expression derived in the exercise links heat capacities with the partial derivatives of temperature concerning specific variables x and Y. This mathematical formulation allows us to see how small changes in state variables impact the overall state of the system, providing insights into the system's thermodynamic behavior.

Thermodynamic relationships form the backbone of how we understand physical systems' behavior under different conditions. These relationships are often expressed through equations and laws that connect various thermodynamic properties, such as pressure, volume, temperature, entropy, and energy.

• State functions: Properties like entropy, enthalpy, and internal energy depend only on the system's state, not on how it reached that state.
• Partial derivatives: These derivatives help articulate how a change in one variable influences another while keeping a third variable constant.

In the context of our exercise, the relationship involves connecting heat capacities at constant conditions (x and Y) with entropy and temperature changes. By manipulating and combining differential equations, we can derive more meaningful expressions that describe the interconnected behavior of thermodynamic properties. Understanding these relationships allows us to predict how energy transfers and transformations occur within a system.