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# 物理代写|热力学作业代写THERMODYNAMICS代考|EGR248 What Are Partial Molar Quantities?

5出现了热力学的其他提法。统计热力学，或称统计力学，关注于从粒子的微观行为对其集体运动的统计预测。1909年，Constantin Carathéodory以公理的形式提出了一种纯数学的方法，这种描述通常被称为几何热力学

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## 物理代写|热力学作业代写THERMODYNAMICS代考|What Are Partial Molar Quantities?

When a homogeneous mixture of several chemicals $A, B, C \ldots$ is formed there are some mixtures for which the thermodynamic quantities, such as volume and internal energy behave additively. Such mixtures are termed ideal. That means that the resulting total thermodynamic quantity $X$ can be evaluated as the sum, $\Sigma_{\mathrm{B}} n_{\mathrm{B}} X_{\mathrm{B}}^{}$, where $X_{\mathrm{B}}^{}$ is the molar quantity $X$ of pure $B$.

In general, however, mixtures are non-ideal, because of the specific interactions of the different substances at the molecular level. The molar quantity $X$ of pure $B$ then must be replaced by what is called the partial molar quantity $X_{\mathrm{B}}$. In a sense, this represents the molar property of B within the mixture of a certain composition as opposed to B in the pure state; it is therefore defined as the effect of adding a mole of $B$ to a mixture of otherwise unchanged composition.

Thus, the partial molar quantity $X_{B}$ (which is an intensive quantity) of substance $B$ in a mixture is defined by
$$X_{\mathrm{B}}=\left(\partial X / \partial n_{\mathrm{B}}\right){T, p, n{\mathrm{A} \neq \mathrm{B}} \prime}$$
where $n_{\mathrm{A}} \neq n_{\mathrm{B}}$ means all the $n^{\prime}$ s except $n_{\mathrm{B}}$ are held constant; for a pure substance $\mathrm{B}, X_{\mathrm{B}}=$ $X / n_{\mathrm{B}}=X_{\mathrm{m}}$. The total differential of an extensive quantity $X$ of a mixture can thus be written as
$$\mathrm{d} X=(\partial X / \partial T){p, n{\mathrm{B}}} \mathrm{d} T+(\partial X / \partial p){T, n{\mathrm{B}}} \mathrm{d} p+\sum_{\mathrm{B}} X_{\mathrm{B}} \mathrm{d} n_{\mathrm{B}}$$
and, by the use of Euler’s theorem (see Question 1.10.2), as
$$X=\sum_{\mathrm{B}} n_{\mathrm{B}} X_{\mathrm{B}}$$
or recast as
$$X_{\mathrm{m}}=\sum_{\mathrm{B}} x_{\mathrm{B}} X_{\mathrm{B}}$$

## 物理代写|热力学作业代写THERMODYNAMICS代考|What Are Mixtures, Solutions and Molality?

Mixture is the word reserved for systems (whether they be gases, liquids and solids) containing more than one substance; all components in the mixture are treated equally. On the other hand, the term solution is reserved for liquids or solids containing more than one substance where one substance is deemed to be a solvent and the others are solutes; these entities are not treated in the same way. If the sum of the mole fractions of the solutes is small compared with unity the solution is termed dilute.

The composition of a solution is usually expressed in terms of the molalities of the solutes. The molality of a solute $\mathrm{B} m_{\mathrm{B}}$ in a solvent A of molar mass $M_{\mathrm{A}}$ is defined by
$$m_{\mathrm{B}}=n_{\mathrm{B}} /\left(n_{\mathrm{A}} M_{\mathrm{A}}\right),$$
and is related to the mole fraction $x_{\mathrm{B}}$ by
$$x_{\mathrm{B}}=m_{\mathrm{B}} \frac{\mathrm{M}{\mathrm{A}}}{1+\Sigma{\mathrm{B}} m_{\mathrm{B}}},$$
or
$$m_{\mathrm{B}}=\frac{x_{\mathrm{B}}}{M_{A}\left(1-\Sigma_{\mathrm{B}} x_{\mathrm{B}}\right)} .$$

## 物理代写|热力学作业代写THERMODYNAMICS代考|What Are Partial Molar Quantities?

$$X_{\mathrm{B}}=\left(\partial X / \partial n_{\mathrm{B}}\right) T, p, n \mathrm{~A} \neq \mathrm{B}$$

$$\mathrm{d} X=(\partial X / \partial T) p, n \mathrm{Bd} T+(\partial X / \partial p) T, n \mathrm{Bd} p+\sum_{\mathrm{B}} X_{\mathrm{B}} \mathrm{d} n_{\mathrm{B}}$$

$$X=\sum_{\mathrm{B}} n_{\mathrm{B}} X_{\mathrm{B}}$$

$$X_{\mathrm{m}}=\sum_{\mathrm{B}} x_{\mathrm{B}} X_{\mathrm{B}}$$

## 物理代写|热力学作业代写 THERMODYNAMICS代考|What Are Mixtures, Solutions and Molality?

$$m_{\mathrm{B}}=n_{\mathrm{B}} /\left(n_{\mathrm{A}} M_{\mathrm{A}}\right)$$

$$x_{\mathrm{B}}=m_{\mathrm{B}} \frac{\mathrm{MA}}{1+\Sigma \mathrm{B} m_{\mathrm{B}}},$$

$$m_{\mathrm{B}}=\frac{x_{\mathrm{B}}}{M_{A}\left(1-\Sigma_{\mathrm{B}} x_{\mathrm{B}}\right)}$$

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