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完美气体中的声速

1816

Pierre-Simon Laplace

（图片仅供参考）

The speed of sound ([latex]c[/latex]) in a 完美气体 is determined by its thermodynamic properties, not its 压力 or density alone. The formula is [latex]c = \sqrt{\gamma R_s T}[/latex], where [latex]\gamma[/latex] is the heat capacity ratio ([latex]c_p/c_v[/latex]), [latex]R_s[/latex] is the specific gas constant, and [latex]T[/latex] is the absolute temperature. Thus, sound travels faster in hotter gas.

The propagation of sound is a mechanical wave that travels through a medium by causing adiabatic (i.e., no heat transfer) compressions and rarefactions. Isaac Newton 最初，皮埃尔-西蒙尝试假设等温过程来计算声速，结果却错了。 Laplace 通过认识到压缩和稀疏发生得如此之快，以至于没有时间与周围环境进行大量的热交换，从而使该过程成为绝热过程，从而纠正了这一问题。

For a perfect gas undergoing an adiabatic process, the relationship between pressure and density is [latex]P \propto \rho^\gamma[/latex]. The speed of sound is generally given by [latex]c = \sqrt{(\partial P / \partial \rho)_S}[/latex], where the derivative is taken at constant entropy (adiabatically). Applying this to the perfect gas model yields [latex]c = \sqrt{\gamma P / \rho}[/latex]. By substituting the perfect gas law in the form [latex]P = \rho R_s T[/latex], we arrive at the more common form [latex]c = \sqrt{\gamma R_s T}[/latex]. This equation reveals the crucial insight that the speed of sound in a gas depends only on its composition (which determines [latex]\gamma[/latex] and [latex]R_s[/latex]) and its absolute temperature.

Acoustics, 空气动力学, 航天, 流体力学, Heat Treatment, 音响工程, 热力学, Ultrasound

UNESCO Nomenclature: 2201

– Acoustics

类型

Physical Law

中断

重大的

用法

广泛使用

前体

Newton’s formula for the speed of sound (isothermal assumption)
绝热过程的概念
ideal gas law
热容比的定义
波动理论

应用程序

aerodynamics and aerospace engineering (calculating mach number)
超音速飞机和火箭的设计
声学与噪声控制工程
利用超声波对材料进行无损检测
用于分析大气现象的气象学

专利：

潜在创新理念

超声波 thermometers for measuring high temperatures in harsh environments
根据声波的传播时间测量气体流量的流量计
基于声学的气体成分传感器
超音速运输的音爆预测和缓解软件
使用超声波分析呼吸气体的医疗诊断工具
使用声波检测泄漏或堵塞的非侵入式管道检测工具
advanced weather prediction models incorporating 声学 wave propagation
使用声速测量的超音速冲压发动机控制系统
用于在特定气体环境中无容器处理材料的声悬浮系统
模拟溶解气泡中声速的水下通信系统

Related to: speed of sound, acoustics, 完美气体, Laplace, adiabatic process, heat capacity ratio, mach number, aerodynamics, gas dynamics, compressibility.

历史背景

盖-吕萨克定律（压力-温度定律）

Also known as Amontons's law, this principle states that for a fixed mass of an ideal gas at a constant volume, its pressure is directly proportional to its absolute temperature. The relationship is expressed mathematically as [latex]P \propto T[/latex] or as a comparison between two states, [latex]\frac{P_1}{T_1} = \frac{P_2}{T_2}[/latex]. This describes gas behavior under isochoric (constant volume) conditions.

道尔顿分压定律

Dalton's Law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. The partial pressure of a gas is the pressure it would exert if it occupied the entire volume alone at the same temperature. The formula is [latex]P_{text{total}} = \sum_{i=1}^{n} p_i[/latex].

电极极化限制

伏打电池的主要缺陷在于电极极化。在运行过程中，铜阴极产生的氢气会在其表面形成一层气泡绝缘层。这层气泡会增加电池的内阻，并减少可用于反应的表面积。因此，电池的电压和电流输出在开始使用后不久就会显著下降。

完美气体中的声速

Regenerator matrix of a Stirling engine demonstrating thermal energy storage in thermodynamics.

斯特林发动机再生式节能器

再生器，或称“节能器”，是罗伯特·斯特林最重要的贡献，也是发动机高效率的关键。它是一个内部热交换器，在循环过程中暂时储存和释放热能。当热气体流向冷端时，它会在再生器内部沉积热量，然后冷气体在返回热端的过程中吸收这些热量。

压力和应变

Engineering stress ([latex]\sigma[/latex]) is the applied load divided by the original cross-sectional area ([latex]A_0[/latex]), while engineering strain ([latex]\epsilon[/latex]) is the change in length ([latex]\Delta L[/latex]) divided by the original length ([latex]L_0[/latex]). These definitions, [latex]\sigma = \frac{F}{A_0}[/latex] and [latex]\epsilon = \frac{\Delta L}{L_0}[/latex], are fundamental for plotting stress-strain curves but assume the specimen's dimensions remain constant during the test.

Electromagnet in a laboratory demonstrating electromagnetism principles.

电磁学和电磁铁

电磁铁是一种由电流产生磁场的磁铁。电流切断后，磁场消失。电磁铁通常由一根绕成线圈的导线组成。磁场强度与电流和线圈匝数成正比。这一原理由汉斯·克里斯蒂安·奥斯特发现。

1802

1810

1816

1816-11-16

1820

1801

1802

1808

1811

1816-11-16

1820

1821

Johann Wilhelm Ritter's experiment with ultraviolet light and silver chloride paper in optics.

紫外线辐射

1801年，约翰·威廉·里特（Johann Wilhelm Ritter）观察到，太阳光谱中紫色端以外的不可见射线比紫光本身更快地使浸有氯化银的纸张变暗。这证明了可见光谱之外存在一种光，他将其命名为“氧化射线”，以与前一年发现的“热射线”（红外线）相对照。

查理定律（体积温度定律）

Also known as the volume-temperature Law or law of volumes, Charles's Law states that for a fixed mass of an ideal gas at constant pressure, the volume it occupies is directly proportional to its absolute temperature. This is expressed as [latex]V \propto T[/latex] or [latex]\frac{V_1}{T_1} = \frac{V_2}{T_2}[/latex]. It explains the tendency of gases to expand when heated under isobaric (constant pressure) conditions.

现代原子理论

原子理论认为，所有物质都由称为原子的离散单元组成。元素由原子核中含有特定数量质子的原子组成。原子曾经被认为是不可分割的，但现在已知它们由亚原子粒子组成：质子、中子和电子。化学反应涉及这些原子的重新排列，这些原子在化学反应过程中保持守恒。

阿伏伽德罗定律

Avogadro's law states that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules. This establishes a direct proportionality between the volume of a gas and the amount of substance (number of moles). The mathematical relationship is expressed as [latex]V \propto n[/latex] or, more commonly, as [latex]V/n = k[/latex], where k is a constant.

Stirling engine demonstrating thermodynamic principles in a laboratory setting.

斯特林循环

The ideal Stirling cycle is a closed regenerative thermodynamic cycle comprising four distinct processes: isothermal expansion, isochoric heat removal, isothermal compression, and isochoric heat addition. The working fluid is permanently contained. Its theoretical thermal efficiency equals the Carnot cycle efficiency, given by [latex]\eta_{th} = 1 - \frac{T_C}{T_H}[/latex], where [latex]T_H[/latex] and [latex]T_C[/latex] are the absolute temperatures of the hot and cold reservoirs.

连续统假设

连续介质假设将流体视为连续物质，而非离散分子。当问题的长度尺度远大于分子间距离时，这种简化是有效的，从而允许在无穷小的点上定义密度和速度等属性。这使得我们能够使用微分方程来描述流体流动的宏观行为。

Laboratory setup for studying magnetic dipole moment in electromagnetism.

磁偶极矩

A magnet's magnetic moment is a vector quantity that characterizes its overall magnetic properties. It can be visualized as a magnetic dipole, a hypothetical pair of north and south poles separated by a distance. The moment points from the south to the north pole. For a current loop, the magnetic moment [latex]\vec{\mu} = I \vec{A}[/latex], where I is the current and A is the area vector.