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19.1: Thermodynamics and Heat Engines

Difficulty Level: At Grade Created by: CK-12
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Heat is a form of energy transfer. It can change the kinetic energy of a substance. For example, the average molecular kinetic energy of gas molecules is related to temperature. A heat engine turns a portion of the input heat (thermal energy) into mechanical work. A second portion of the input heat must be exhausted in order for the engine to have repetitive motion. Therefore, in a practical engine it is impossible for all the input heat to be converted to work.

Entropy is a measure of disorder, or the variety of ways in which a system can organize itself with the same total energy. The entropy of any isolated system always tends to disorder (i.e. entropy is always increasing). In the universe, the entropy of a subset (like evolution on Earth) can decrease (i.e. more order) but the total entropy of the universe is increasing (i.e. more disorder).

Thermodynamics is the study of heat engines. Any engine or power plant obeys the laws of thermodynamics. The first law of thermodynamics is a statement of conservation of energy. Total energy, including heat, is conserved in any process and in the complete cycle of a heat engine. The second law of thermodynamics as it applies to heat engines gives an absolute limit on the efficiency of any heat engine that goes through repetitious cycles.

Key Concepts

  • The temperature of a gas is a measure of the amount of average kinetic energy that the atoms in the gas possess.
  • The pressure of a gas is the force the gas exerts on a certain area. For a gas in a container, the amount of pressure is directly related to the number and intensity of atomic collisions on a container wall.
  • An ideal gas is a gas for which interactions between molecules are negligible, and for which the gas atoms or molecules themselves store no potential energy. For an “ideal” gas, the pressure, temperature, and volume are simply related by the ideal gas law.
  • Atmospheric pressure (\begin{align*}1 \;\mathrm{atm} = 101,000\end{align*} Pascals) is the pressure we feel at sea level due to the weight of the atmosphere above us. As we rise in elevation, there is less of an atmosphere to push down on us and thus less pressure.
  • When gas pressure-forces are used to move an object then work is done on the object by the expanding gas. Work can be done on the gas in order to compress it.
  • Adiabatic process: a process that occurs with no heat gain or loss to the system in question.
  • Isothermal: a process that occurs at constant temperature (i.e. the temperature does not change during the process).
  • Isobaric: a process that occurs at constant pressure.
  • Isochoric: a process that occurs at constant volume.
  • If you plot pressure on the vertical axis and volume on the horizontal axis, the work done in any complete cycle is the area enclosed by the graph. For a partial process, work is the area underneath the curve, or\begin{align*}P\Delta\!V\end{align*}.
  • In a practical heat engine, the change in internal energy must be zero over a complete cycle. Therefore, over a complete cycle \begin{align*}W = \Delta\!Q\end{align*}.
  • The work done by a gas during a portion of a cycle = \begin{align*}P\Delta\!V\end{align*}, note \begin{align*}\Delta\!V\end{align*}can be positive or negative.
  • The efficiency of any heat engine : \begin{align*}\eta = W/Q_{in}\end{align*}
  • An ideal engine, the most efficient theoretically possible, is called a Carnot Engine. Its efficiency is given by the following formula, where the temperatures are, respectively, the temperature of the exhaust environment and the temperature of the heat input, in Kelvins. In a Carnot engine heat is input and exhausted in isothermal cycles, and the efficiency is \begin{align*} \eta = 1 -\frac{T_{\mathrm{cold}}}{T_\mathrm{hot}} \end{align*}.
  • The Stirling engine is a real life heat engine that has a cycle similar to the theoretical Carnot cycle. The Stirling engine is very efficient (especially when compared to a gasoline engine) and could become an important player in today's world where green energy and efficiency will reign supreme.

Key Equations

\begin{align*}Q = mc \Delta T\end{align*} ; the heat gained or lost is equal to the mass of the object multiplied by its specific heat multiplied by the change of its temperature.

\begin{align*}Q = mL\end{align*} ; the heat lost or gained by a substance due to a change in phase is equal to the mass of the substance multiplied by the latent heat of vaporization/fusion (\begin{align*} L \end{align*} refers to the latent heat)

1 cal = 4.184 Joules ; your food calorie is actually a kilocalorie (Cal) and equal to 4184 J.

\begin{align*} Q_{\text{in}} = Q_{\text{out}} + W +\Delta U \end{align*}

\begin{align*}U\end{align*} is the internal energy of the gas. (This is the first law of Thermodynamics and applies to all heat engines.)

\begin{align*} \left ({\frac{1}{2} m v ^2} \right )_{\text{avg}} = \frac{3}{2}kT \end{align*}

The average kinetic energy of atoms (each of mass \begin{align*}m\end{align*} and average speed \begin{align*}v\end{align*}) in a gas is related to the temperature \begin{align*}T\end{align*} of the gas, measured in Kelvin. The Boltzmann constant \begin{align*}k\end{align*} is a constant of nature, equal to \begin{align*}1.38\times10^{-23} \;\mathrm{J/K}\end{align*}.

\begin{align*} P = \frac{F}{A} \end{align*}

The pressure on an object is equal to the force pushing on the object divided by the area over which the force is exerted. Unit for pressure are \begin{align*}\;\mathrm{N/m}^2\end{align*} (called Pascals)

\begin{align*} PV = NkT \end{align*}

An ideal gas is a gas where the atoms are treated as point-particles and assumed to never collide or interact with each other. If you have \begin{align*}N\end{align*} molecules of such a gas at temperature \begin{align*}T\end{align*} and volume \begin{align*}V\end{align*}, the pressure can be calculated from this formula. Note that \begin{align*}k=1.38\times10^{-23}\;\mathrm{J/K}\end{align*}; this is the ideal gas law

\begin{align*} PV = nRT \end{align*}

\begin{align*}V\end{align*} is the volume, \begin{align*}n\end{align*} is the number of moles; \begin{align*}R\end{align*} is the universal gas constant \begin{align*}= 8.315 \;\mathrm{J/K-n}\end{align*}; this is the most useful form of the gas law for thermodynamics.

\begin{align*}e_c = 1-\frac{T_c}{T_h} && \text{Efficiency of a Carnot (ideal) heat engine}\end{align*}

where \begin{align*} T_c \end{align*} and \begin{align*} T_h \end{align*} are the temperatures of the hot and cold reservoirs, respectively.

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