Homogeneous Systems

Excerpts

• Homogeneous Systems

  For the present, besides $M$, let $\theta$ and $v$ be the independent variables.

• Homogeneous Systems

  Then the pressure $p$, the specific energy $u = \dfrac{U}{M}$, and the specific entropy $\phi = \dfrac{\Phi}{M}$ are functions of $\theta$ and $v$,

• Homogeneous Systems

  Therefore, since $d\theta$ and $dv$ are independent of each other,

• Homogeneous Systems

  These two equations lead to an experimental test of the second law;

• Homogeneous Systems

  This expression vanishes in the case of perfect gases, since then the temperature remains constant.

• Homogeneous Systems

  The numerator of this expression may be found directly from the characteristic equation of the substance. The denominator, however, depends on the amount of heat which the substance absorbs during isothermal reversible expansion.

• Homogeneous Systems

  For gases, $\gamma$ is large; and, in fact, the fewer the number of atoms in a molecule of the gas, the larger does it become.

• Homogeneous Systems

  As $\left(\dfrac{\dd p}{\dd v}\right)_{\theta}$ is necessarily negative, $c_{p}$ is always greater than $c_{v}$, except in the limiting case, when the coefficient of expansion is $= 0$, as in the case of water at $4°$ C.,; then $c_{p} - c_{v} = 0$.

• Homogeneous Systems

  It follows that, for solids and liquids, the difference $c_{p} - c_{v}$ depends rather on the relation between the energy and the volume than on the external work of expansion.

• Homogeneous Systems

  This means that in solids and liquids the energy depends far more on the temperature than on the volume. For gases, $\gamma$ is large; and, in fact, the fewer the number of atoms in a molecule of the gas, the larger does it become.

• Homogeneous Systems

  This equation contains only quantities that can be directly measured, and establishes a relation between the rate of change of the coefficient of thermal expansion of the substance with temperature (*i.e.* the deviation from Gay-Lussac’s law), and the rate of change of the specific heat with pressure.

• Homogeneous Systems

  If, under constant pressure, $v$ were proportional to $\theta$, as in Gay-Lussac’s law, then, by equation % [eqn:(86)](86)%, $\Delta \theta = 0$, as is really the case for perfect gases.

• Homogeneous Systems

  In 4 we defined temperature by means of the gas thermometer, but had to confine that definition to the cases in which the readings of the different gas thermometers (hydrogen, air, etc.) agree as nearly as the desired accuracy of the result requires.

• Homogeneous Systems

  Equations % [eqn:(88)](88)% and % [eqn:(89)](89)%, like Thomson and Joule’s formula, are valid only within certain limits. It is, however, of theoretical interest to see how the different relations necessarily follow from one another.

• Homogeneous Systems

  As soon as accurate measurement of even a single substance has determined $\theta$ as a function of $t$, the question regarding the value of the absolute temperature may be considered as solved for all cases.

Equations

• Homogeneous Systems

  d\phi = \frac{du + p\, dv}{\theta} = \frac{1}{\theta} \left(\frac{\dd u}{\dd \theta}\right)_{v} d\theta + \frac{\left(\dfrac{\dd u}{\dd v}\right)_{\theta} + p}{\theta}\, dv

  The definition of specific entropy gives its differential as the heat term (du + p dv) divided by the absolute temperature, expanded in the independent variables θ and v.

• Homogeneous Systems

  \left(\frac{\dd u}{\dd v}\right)_{\theta} = \theta \left(\frac{\dd p}{\dd \theta}\right)_{v} - p

  The rate of change of specific energy with volume at constant temperature equals θ times the rate of change of pressure with temperature at constant volume, minus the pressure.

• Homogeneous Systems

  \left(\frac{\dd \phi}{\dd \theta}\right)_{v} = \frac{c_{v}}{\theta}

  The temperature rate of change of specific entropy at constant volume equals the specific heat at constant volume divided by absolute temperature.

• Homogeneous Systems

  \left(\frac{\dd \phi}{\dd v}\right)_{\theta} = \left(\frac{\dd p}{\dd \theta}\right)_{v}

  The volume rate of change of specific entropy at constant temperature equals the temperature rate of change of pressure at constant volume.

• Homogeneous Systems

  c_{p} - c_{v} = \theta \left(\frac{\dd p}{\dd \theta}\right)_{v} · \left(\frac{\dd v}{\dd \theta}\right)_{p}

  The difference of the specific heats at constant pressure and constant volume equals θ times the product of two rates of change of pressure and volume with temperature.

• Homogeneous Systems

  c_{p} - c_{v} = -\theta \left(\frac{\dd p}{\dd v}\right)_{\theta} · \left(\frac{\dd v}{\dd \theta}\right)_{p}^{2}

  The difference of the specific heats is written, using the relation between the derivatives of pressure and volume, as -θ times the volume-pressure rate of change times the square of the thermal expansion rate.

• Homogeneous Systems

  \left(\frac{\dd c_{p}}{\dd p}\right)_{\theta} = -\theta \left(\frac{\dd^{2} v}{\dd \theta^{2}}\right)_{p}

  The rate of change of specific heat at constant pressure with pressure equals minus θ times the second temperature derivative of specific volume at constant pressure; it relates measurable quantities.

• Homogeneous Systems

  c_{p} - c_{v} = \left\{\left(\frac{\dd u}{\dd v}\right)_{\theta} + p\right\} \left(\frac{\dd v}{\dd \theta}\right)_{p}

  The difference of specific heats, from the first law, equals the sum of the internal-energy term and the external-work term, multiplied by the thermal expansion rate.

• Homogeneous Systems

  \frac{\theta}{p} · \left(\frac{\dd p}{\dd \theta}\right)_{v} - 1

  The ratio of the internal-energy term to the external-work term, written in terms of the pressure-temperature rate of change, is θ/p times that rate minus one.

• Homogeneous Systems

  \dfrac{c_{p}}{c_{v}} = \gamma

  The ratio of the specific heat at constant pressure to that at constant volume is denoted γ.

• Homogeneous Systems

  \left(\frac{\dd u}{\dd p}\right)_{\theta} = -\theta \left(\frac{\dd v}{\dd \theta}\right)_{p} - p\left(\frac{\dd v}{\dd p}\right)_{\theta}

  The rate of change of specific energy with pressure at constant temperature equals minus θ times the thermal expansion rate minus p times the compressibility rate.

• Homogeneous Systems

  \left(\frac{\dd \phi}{\dd p}\right)_{\theta} = -\left(\frac{\dd v}{\dd \theta}\right)_{p}

  The pressure rate of change of specific entropy at constant temperature equals minus the thermal expansion rate at constant pressure.

• Homogeneous Systems

  \left(\frac{\dd p}{\dd v}\right)_{\theta} = -\frac{1014000}{0.00000295 · v}

  As used for mercury at 0° C, the pressure-volume rate of change at constant temperature is given numerically in atmospheres, with 0.00000295 the compressibility coefficient in atmospheres and 1014000 the absolute pressure of one atmosphere.

• Homogeneous Systems

  \Delta \theta = \frac{\theta \left(\dfrac{\dd v}{\dd \theta}\right)_{p} - v}{c_{p}}\, \Delta p

  For a small pressure drop in a throttling expansion, the temperature change equals the change in pressure times θ times the thermal expansion rate minus v, divided by c_p.

• Homogeneous Systems

  \Delta \theta = \frac{\alpha}{\theta^{2}}\, \Delta p

  Thomson and Joule's empirical formula: the temperature change in the throttling experiment equals a constant α over θ squared times the pressure change.

• Homogeneous Systems

  \theta \left(\frac{\dd v}{\dd \theta}\right)_{p} - v = c_{p} \frac{\alpha}{\theta^{2}}

  Combining the throttling relation with the Thomson and Joule formula gives θ times the thermal expansion rate minus v equal to c_p times α over θ squared.

• Homogeneous Systems

  c_{p} = \frac{c_{p}^{(0)}}{\left(1 - \dfrac{3\alpha p}{\theta^{3}}\right)^{\efrac{2}{3}}}

  The specific heat at constant pressure, deduced from the Thomson and Joule formula with the ideal-state limit, equals its zero-pressure value divided by a power of (1 - 3αp/θ³).

• Homogeneous Systems

  c_{p} = \theta^{2} · f(\theta^{3} - 3\alpha p)

  The general solution of the differential equation for c_p is θ² times an arbitrary function of θ³ - 3αp.

• Homogeneous Systems

  v = \frac{c_{p}^{(0)} \theta}{3p} \left(\sqrt[3]{1 - \frac{3\alpha p}{\theta^{3}}} + \beta\right)

  The characteristic equation of the gas deduced from the Thomson and Joule experiments gives specific volume in terms of θ and p, with β a constant of integration fixed from the density at 0° C and atmospheric pressure.

• Homogeneous Systems

  p_{1}v_{1} - p_{2}v_{2} = W

  The external work per unit mass done in the throttled flow equals the difference of the pressure-volume products before and after the throttle.

• Homogeneous Systems

  W = -\Delta(pv)

  For small changes, the external work per unit mass in the throttled flow equals minus the change of the product pv.

• Homogeneous Systems

  \Delta u = W + Q = -\Delta(pv)

  By the first law, with no heat added (Q = 0), the change of specific energy equals the work term, which is minus the change of pv.

• Homogeneous Systems

  \left(\frac{q}{dv}\right)_{t} = \left(\frac{\dd u}{dv}\right)_{t} + p

  By the first law, the ratio of heat absorbed in isothermal reversible expansion to the change of volume equals the internal-energy rate of change plus the pressure.

• Homogeneous Systems

  \theta = \frac{100 e^{J}}{e^{J_{1}} - 1}

  The absolute temperature is found from J and J₁, which are integrals fixed by the readings of an arbitrary thermometer between the freezing and boiling points of water.

• Homogeneous Systems

  \alpha = \frac{1}{\theta_{0}} = \frac{e^{J_{1}} - 1}{100}

  The coefficient of thermal expansion of a perfect gas, independent of any gas thermometer, equals the reciprocal of the absolute freezing-point temperature, expressed through J₁.

• Homogeneous Systems

  \theta = t + \frac{1}{\alpha'}

  For a perfect gas used as the arbitrary thermometer, the absolute temperature equals the thermometer reading t plus the reciprocal of the expansion coefficient α'.

• Homogeneous Systems

  \left(\frac{\dd v}{\dd t}\right)_{p} = \alpha' v_{0}

  At constant pressure, the rate of change of specific volume with the thermometer reading t equals the expansion coefficient α' times v₀, the specific volume at the melting point of ice.

Problems

No exercises in this chapter.