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# 物理代写|热力学代写Thermodynamics代考|ENES232 How Do I Calculate Expansivity and Compressibility?

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## 物理代写|热力学代写Thermodynamics代考|How Do I Calculate Expansivity and Compressibility?

The isobaric (constant pressure) expansivity or coefficient of thermal expansion $\alpha$ is defined by
$$\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p=\left(\frac{\partial \ln V}{\partial T}\right)_p .$$
$\alpha$ is usually positive but for water at temperatures between $273.15 \mathrm{~K}$ and $277.13 \mathrm{~K}$, it is negative. The isothermal compressibility $\kappa_T$ is defined by
$$\varkappa_T=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_T=-\left(\frac{\partial \ln V}{\partial p}\right)_T,$$
and by the $-1$ rule (provided by Equation 1.153) is related to $\alpha$
$$\frac{\alpha}{\kappa_T}=\left(\frac{\partial p}{\partial T}\right)_V .$$
The isentropic (constant entropy) compressibility $\kappa_S$ is defined by
$$\kappa_S=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_S .$$

## 物理代写|热力学代写Thermodynamics代考|What Can I Gain from Measuring the Speed of Sound in Fluids?

While the speed of sound in a phase is important in its own right in a number of applications, most of the interest in this quantity arises from its relationship with the thermodynamic properties of isotropic, Newtonian fluids and isotropic elastic solids. For fluid phases, as these usually support only a single longitudinal sound mode, the sound propagation speed $u$ is given by (Herzfeld and Litovitz 1959)
$$u^2=\left(\frac{\partial p}{\partial \rho}\right){\mathrm{s}}=\frac{1}{\rho \kappa_S}=\frac{\gamma}{\rho \kappa_T} .$$ In Equation 3.90, all the symbols have been previously defined, including $\gamma=C_p / C_V$, which is now given in the form of $C_p$ and $C_V$, the molar isobaric and isochoric heat capacities, respectively (see Question 1.8.3). Equation $3.90$ is strictly valid only in the limits of vanishing amplitude and vanishing frequency (Herzfeld and Litovitz 1959; Morse and Ingard 1968; Goodwin and Trusler 2003). Although it is usually not possible to achieve both of these conditions directly in experiment, extrapolation of measurements made at finite amplitude and frequency are generally adequate. Equation $3.90$ shows that the isentropic compressibility may be obtained from measurements of the speed of sound and the density, and that the isothermal compressibility may also be obtained if $\gamma$ is known. Equation $3.90$ forms the basis of almost all experimental determinations of the isentropic compressibility and is a convenient route to $\gamma$. For independent variables of either $(T, p)$ or $\left(T, \rho_n\right)$, where $\rho_n$ is the amount-of-substance density (which we distinguish from the mass density, $\rho$ ), Equation $3.90$ can be recast for $(T, p)$ as $$u^2=\frac{1}{M}\left[\left(\frac{\partial \rho_n}{\partial p}\right)_T-\frac{T}{\rho_n^2 C_p}\left(\frac{\partial \rho_n}{\partial T}\right)_p^2\right]^{-1},$$ and for $\left(T, \rho_n\right)$ $$u^2=\frac{1}{M}\left[\left(\frac{\partial p}{\partial \rho_n}\right)_T+\frac{T}{\rho_n^2 C_V}\left(\frac{\partial p}{\partial T}\right){\rho_n}^2\right]$$

## 物理代写|热力学代写热力学代考|我如何计算可膨胀性和可压缩性?

$$\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p=\left(\frac{\partial \ln V}{\partial T}\right)_p .$$
$\alpha$定义，它通常为正，但对于温度介于$273.15 \mathrm{~K}$和$277.13 \mathrm{~K}$之间的水，它是负的。等温压缩性$\kappa_T$由
$$\varkappa_T=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_T=-\left(\frac{\partial \ln V}{\partial p}\right)_T,$$

$$\frac{\alpha}{\kappa_T}=\left(\frac{\partial p}{\partial T}\right)_V .$$

$$\kappa_S=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_S .$$ 定义

## 物理代写|热力学代写热力学代考|从测量流体中的声速中我能得到什么?

$$u^2=\left(\frac{\partial p}{\partial \rho}\right){\mathrm{s}}=\frac{1}{\rho \kappa_S}=\frac{\gamma}{\rho \kappa_T} .$$给出，在公式3.90中，所有的符号都已预先定义，包括$\gamma=C_p / C_V$，现在分别以$C_p$和$C_V$的形式给出，摩尔等压热容和等声热容(见问题1.8.3)。方程$3.90$仅在消失振幅和消失频率的极限下严格有效(Herzfeld and Litovitz 1959;莫尔斯和英格德1968年;Goodwin和Trusler 2003)。虽然在实验中通常不可能直接达到这两个条件，但对有限振幅和有限频率下的测量进行外推通常是足够的。方程$3.90$表明，可以通过声速和密度的测量得到等熵压缩率，如果知道$\gamma$，也可以得到等温压缩率。方程$3.90$构成了几乎所有等熵压缩性实验测定的基础，是通往$\gamma$的方便途径。对于$(T, p)$或$\left(T, \rho_n\right)$的自变量，其中$\rho_n$是物质的量密度(我们将其与质量密度$\rho$区分开来)，对于$(T, p)$，可以将公式$3.90$重设为$$u^2=\frac{1}{M}\left[\left(\frac{\partial \rho_n}{\partial p}\right)_T-\frac{T}{\rho_n^2 C_p}\left(\frac{\partial \rho_n}{\partial T}\right)_p^2\right]^{-1},$$，对于\$\left(T, \rho_n\right)$$u^2=\frac{1}{M}\left[\left(\frac{\partial p}{\partial \rho_n}\right)_T+\frac{T}{\rho_n^2 C_V}\left(\frac{\partial p}{\partial T}\right){\rho_n}^2\right]$$

## MATLAB代写

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