# 物理代写|热力学代写thermodynamics代考|Using T-ds Relationships

#### Doug I. Jones

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## 物理代写|热力学代写thermodynamics代考|Using T-ds Relationships

Finding the change in entropy associated with a process requires integration of an appropriate equation for $d S$. (I know you were hoping to avoid calculus, but here it is.) To perform the integration, you must know the kind of path the process takes. Common paths are isothermal, isobaric, isochoric, isenthalpic, isentropic, and adiabatic. I discuss these process paths in Chapter 2.
The path taken by a real process differs somewhat from these ideal-process, constant-property paths. In the section “Working with T-s Diagrams,” I show that the integration for an isothermal path is quite easy. When the temperature varies in a process, you need a mathematical relationship between the heat transfer $(\delta Q)$ and the temperature $(T)$ to perform the integration of $d S$. Two important relationships, known as the Gibbs equations, are used to define relationships between entropy, heat transfer, and temperature.
The energy equation developed in Chapter 5, written in differential form, gives a mathematical relationship between heat transfer $(\delta Q)$, internal energy $(d U)$, and work $(\delta W)$ for an internally reversible process as follows:
$$\delta Q_{\text {rev }}=d U+\delta W_{\text {rev }}$$
You get the first Gibbs equation by replacing $\delta Q_{\text {rev }}$ with $d U+\delta W_{\text {rev }}$ in the equation that defines entropy:
$$d S=\left(\frac{\delta Q}{T}\right)_{\text {tim. Rer. }}$$
The first Gibbs equation relates the change in entropy of a system (dS) to internal energy $(d U)$ and boundary work where $\delta W_{\mathrm{rev}}=P d V$, as follows:
$$d S=\frac{d U}{T}+\frac{P d V}{T}$$
Intemal energy is the energy in a material related to its molecular activity. I discuss internal energy in Chapter 2. Boundary work occurs when a boundary in a system moves, such as a piston moving in a cylinder (see Chapter 5).
The second Gibbs equation relates the entropy change of a system to enthalpy and flow work. Enthalpy $(H)$ is a property that combines internal energy $(U)$ plus the product of pressure and specific volume $(P V)$ as I discuss in Chapter 2. Flow work is associated with the work done by flowing fluids in a process, such as a turbine or a compressor in a gas turbine engine (see Chapter 5). The second Gibbs equation is written as follows:
$$d S=\frac{d H}{T}-\frac{V d P}{T}$$
I discuss how these relationships are used to calculate changes in entropy for several different thermodynamic systems in the following sections.

## 物理代写|热力学代写thermodynamics代考|Calculating Entropy Change

In this section, I discuss how to use and modify the two Gibbs equations from the preceding section to determine the entropy change for processes involving solids, liquids, gases, saturated liquid-vapor mixtures, and ideal gases. You can find the enthalpy change of a process by integrating either of the Gibbs equations between the initial and final states of a process.

To perform the integration of the Gibbs equations, you must know the relationship between internal energy or enthalpy and temperature for a substance. You must also know the relationship between the pressure, volume, and temperature of a substance to complete the integration. For an ideal gas, the ideal-gas law can be used. For other substances, you need to use tabulated data, which can be found in the appendix.
For pure substances
Entropy is a property, and you can find the value of entropy for a substance from thermodynamic tables just like any other property, such as internal energy or enthalpy. You need any two independent intensive properties – for example, temperature and pressure or internal energy and specific volumeto determine the value of entropy for a substance. Figure 83 shows you what I mean. You can find the entropy of several different substances from the thermodynamic property tables in the appendix. I discuss intensive properties in Chapter 2.

Although the third law of thermodynamics states the entropy of a substance is zero at absolute zero temperature, thermodynamic tables usually define entropy as being equal to zero at a more convenient reference temperature. This definition means that at temperatures below the reference temperature, entropy can have negative values. The reference temperature for water is 0.01 degree Celsius at the saturated liquid state. For refrigerant R-134a as a saturated liquid, the reference temperature is -40 degrees Celsius. The reference temperature for ideal gases is absolute zero temperature. Negative values for entropy aren’t significant, because only the change in entropy for a thermodynamic process is important. Check out the thermodynamic property tables in the appendix to see these reference values for yourself.

Figure $8-3$ shows that the entropy of a compressed liquid, $s(T, P)$, at temperature $T$ and pressure $P$, can be approximated by the entropy of the saturated liquid, $s,(T)$, at the given temperature $T: s(T, P)=s_f(T)$.
For example, you can look up the entropy of saturated liquid water at 10 megapascals pressure and 260 degrees Celsius in Table A- 2 of the appendix to find that it’s 2.870 kilojoules per kilogram-Kelvin. This calculation requires a bit of interpolation of the table. I show you how to interpolate tables in Chapter 3. Or, you can look up the entropy of saturated water at 260 degrees Celsius in Table A-3 of the appendix and find that it has a value of 2.884 kilojoules per kilogram-Kelvin. You can see that the entropy for the saturated liquid is not that different from the entropy of the compressed liquid at the same temperature, so it’s a reasonable approximation.

# 热力学代写

## 物理代写|热力学代写thermodynamics代考|Using T-ds Relationships

$$\delta Q_{\text {rev }}=d U+\delta W_{\text {rev }}$$

$$d S=\left(\frac{\delta Q}{T}\right){\text {tim. Rer. }}$$ 第一个吉布斯方程将系统的熵变(dS)与热力学能$(d U)$和边界功($\delta W{\mathrm{rev}}=P d V$)联系起来，如下:
$$d S=\frac{d U}{T}+\frac{P d V}{T}$$

$$d S=\frac{d H}{T}-\frac{V d P}{T}$$

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