Question

I need good report about (Applications of differential equations in thermodynamic)

At least 4-5 pages please and would be better if its not handwriting

Answer 1

Applications of differential equations in thermodynamics

As a basic subject, higher mathematics has been widely applied in various disciplines and all aspects of life. This report, based on higher mathematics thoughts, introduces basic concepts of higher mathematics such as differential calculus to analyze the application of higher mathematics in thermodynamics. Here the continuity and total differential properties of thermodynamic state functions are summarized, and the derivation of thermodynamic formula is accomplished through these properties and the principle of calculus. This report discusses the relation and difference between the process function and the state function to avoid the confusion of concepts in the process of thermodynamics calculation. It also points out the significance of higher mathematics in studying thermodynamics and the necessary of using differential calculus to solve and deal with chemical problems. The application of differential calculus can provide a better understanding on physical thermodynamics and cultivate the ability to solve practical problems with higher mathematics knowledge and thoughts.

Introduction

Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties measured in a laboratory or production process. Thermodynamics is based on a fundamental set of postulates, that became the laws of thermodynamics. One of the fundamental thermodynamic equations is the description of thermodynamic work in analogy to mechanical work, or weight lifted through an elevation against gravity, as defined in 1824 by French physicist Sadi Carnot. Carnot used the phrase motive power for work. In the footnotes to his famous On the Motive Power of Fire, he states: “We use here the expression motive power to express the useful effect that a motor is capable of producing. This effect can always be likened to the elevation of a weight to a certain height. It has, as we know, as a measure, the product of the weight multiplied by the height to which it is raised.” With the inclusion of a unit of time in Carnot's definition, one arrives at the modern definition for power.

Four laws of thermodynamics ( zeroth, first, second, third law) has made the the pillar of physical and chemical thermodynamics with the aid of Differential calculus. Starting from the 4 laws of thermodynamics we can see the use of Differential equation indifferent part of thermodynamics. For example entire process of heat conduction Convection and radiation, is completely expressible in terms of Differential equations. Here in this report starting with the application of differential equations in the law of thermodynamics we will extend the application towards other part also.

​​​

Applications

First order equations

Just as with the internal energy version of the fundamental equation, the chain rule can be used on the above equations to find k+2 equations of state with respect to the particular potential. If Φ is a thermodynamic potential, then the fundamental equation may be expressed as:

where the are the natural variables of the potential. If is conjugate then we have the equations of state for that potential, one for each set of conjugate variables.

Only one equation of state will not be sufficient to reconstitute the fundamental equation. All equations of state will be needed to fully characterize the thermodynamic system. What is commonly called "the equation of state" is just the "mechanical" equation of state involving the Helmholtz potential and the volume:

For an ideal gas, this becomes the familiar PV=NkBT.

Euler integrals

Because all of natural variables of the internal energy U are extensive quantities, it follows from Euler's homogeneous function theorem that

Substituting into the expressions for the other main potentials we have the following expressions for the thermodynamic potentials:

Gibbs–Duhem relationship

Differentiating the Euler equation for the internal energy and combining with the fundamental equation for internal energy, it follows that:

which is known as the Gibbs-Duhem relationship. The Gibbs-Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with r components, there will be r+1 independent parameters, or degrees of freedom. For example, a simple system with a single component will have two degrees of freedom, and may be specified by only two parameters, such as pressure and volume for example. The law is named after Willard Gibbs and Pierre Duhem.

Second order equations

Maxwell relations

Maxwell relations are equalities involving the second derivatives of thermodynamic potentials with respect to their natural variables. They follow directly from the fact that the order of differentiation does not matter when taking the second derivative. The four most common Maxwell relations are:

Material properties

Second derivatives of thermodynamic potentials generally describe the response of the system to small changes. The number of second derivatives which are independent of each other is relatively small, which means that most material properties can be described in terms of just a few "standard" properties. For the case of a single component system, there are three properties generally considered "standard" from which all others may be derived

Compressibility at constant temperature or constant entropy

Specific heat (per-particle) at constant pressure or constant volume

Coefficient of thermal expansion

These properties are seen to be the three possible second derivative of the Gibbs free energy with respect to temperature and pressure.

Thermodynamic Property Relations

Properties such as pressure, volume, temperature, unit cell volume, bulk modulus and mass are easily measured. Other properties are measured through simple relations, such as density, specific volume, specific weight. Properties such as internal energy, entropy, enthalpy, and heat transfer are not so easily measured or determined through simple relations. Thus, we use more complex relations such as Maxwell relations, the Clapeyron equation, and the Mayer relation.

Maxwell relations in thermodynamics are critical because they provide a means of simply measuring the change in properties of pressure, temperature, and specific volume, to determine a change in entropy. Entropy cannot be measured directly. The change in entropy with respect to pressure at a constant temperature is the same as the negative change in specific volume with respect to temperature at a constant pressure, for a simple compressible system. Maxwell relations in thermodynamics are often used to derive thermodynamic relations.

The Clapeyron equation allows us to use pressure, temperature, and specific volume to determine an enthalpy change that is connected to a phase change. It is significant to any phase change process that happens at a constant pressure and temperature. One of the relations it resolved to is the enthalpy of vaporization at a provided temperature by measuring the slope of a saturation curve on a pressure vs. temperature graph. It also allows us to determine the specific volume of a saturated vapor and liquid at that provided temperature. In the equation below, L represents the specific latent heat, T represents temperature, and represents the change in specific volume.

APPLICATION OF DIFFERENTIAL EQUATION IN NEWTON’S LAW OF COOLING

It is a model that describes, mathematically, the change in temperature of an object in a given environment. The law states that the rate of change (in time) of the temperature is proportional to the difference between the temperature T of the object and the temperature Te of the environment surrounding the object.

dT/dt = – k (T – Te)

Heat equation

In physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity (such as heat) evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation.

For a function u(x,y,z,t) of three spatial variables (x,y,z) and the time variable t, the heat equation is

where is a real coefficient called the diffusivity of the medium. Using Newton's notation for derivatives, and the notation of vector calculus, the heat equation can be written in compact form as

Where denotes the Laplace operator, and is the time derivative of u. One