Thermodynamics

Authors and Affiliations

1. Kaiserslautern, Germany

   Wolfgang Demtröder

Corresponding author

Correspondence to Wolfgang Demtröder .

Summary

• The temperature of a body is given either as absolute temperature T in Kelvin or as Celsius temperature \(T_{\mathrm{C}}/\,{}^{\circ}\text{C}\) or in the US as Fahrenheit temperature. The relations are

  $$\displaystyle\begin{aligned}\displaystyle T/\mathrm{K}&\displaystyle=T_{\mathrm{C}}/\,{}^{\circ}\text{C}+273.15,\\ \displaystyle T/\mathrm{F}&\displaystyle=(9/5)T_{\mathrm{C}}\,{}^{\circ}\text{C}+32\\ \displaystyle&\displaystyle=(9/5)[T/\mathrm{K}-273.15)+32\\ \displaystyle&\displaystyle=(9/5)T/\mathrm{K}-459.67{\;}.\end{aligned}$$

  For temperature measurements all quantities can be used, that depend on the temperature (expansion of a liquid volume, electric resistance, thermo-voltage, conductivity of semiconductors).

• The thermal expansion of bodies is caused by the non-harmonic interaction potential between neighbouring atoms.

• The absolute temperature is determined with the gas thermometer, where the increase of the gas pressure with temperature in a constant volume is proportional to the temperature increase.

• The thermal energy of a body is determined by the kinetic and potential energy of the atoms or molecules. The temperature increase \(\Delta T\) of the system is proportional to the supplied heat energy \(\Delta Q=C\cdot\Delta T\).

• The molar heat capacity for a constant volume of a gas \(C_{V}=R\cdot f/2\), is equal to the product of gas constant \(R= k\cdot N_{\mathrm{A}}\) times one half of the number f of degrees of freedom of the atoms or molecules in the gas.

• The molar heat capacity at constant pressure is \(C_{p}=C_{V}+R\)

• The transition from the solid to the liquid phase requires the molar melting energy \(W=\Lambda_{\mathrm{m}}\) per mole. During the melting the potential energy of the atoms or molecules increases while the kinetic energy stays constant. Similar the transition from the liquid to the gaseous phase needs the energy per mole \(W=\Lambda_{\mathrm{e}}\) (heat of evaporation).

• Thermal energy can be transported from one area to another

  • by heat conduction

  • by convection

  • by thermal radiation

• The amount of heat transported per second by heat conduction in the direction r through the area A is \(\,\mathrm{d}Q/\,\mathrm{d}t=-\lambda\cdot A\cdot(\mathrm{grad}\,T)_{\boldsymbol{r}}\), i. e. the product of heat conductivity λ, area A and temperature gradient in the direction of r.

• For metals the heat conductivity is proportional to the electrical conductivity, which indicates that the electrons are mainly responsible for both conductivities.

• The thermodynamic state of a system is unambiguously determined by the state variables pressure p, volume V and temperature T. For ν moles of an ideal gas in the volume V the general gas equation is

  $$\displaystyle p\cdot V=\nu\cdot R\cdot T{\;}.$$

  The number of internal state variables in real gases is given by Gibbs’ phase rule (10.134).

• The entropy S of a system is a measure for the number of possible ways the state of the system can be realized. The change of the entropy is \(\,\mathrm{d}S=\,\mathrm{d}Q/T\) where \(\,\mathrm{d}Q\) is the heat energy supplied to or by the system.

• The first law of thermodynamics \(\Delta U=\Delta Q+\Delta W\) describes the energy conservation. The change \(\Delta U\) of internal energy \(U=N\cdot(f/2)kT\) of a system with N atoms or molecules equals the sum of supplied heat \(\Delta Q\) and mechanical work \(\Delta W\) performed on or by the system. For real gases is \(U=E_{\mathrm{kin}}+E_{\mathrm{pot}}\), because the interaction energy between the atoms has to be taken into account.

• Special processes in a system of an ideal gas are:isochoric processes (\(V=\mathrm{const}\)) \(\Rightarrow\,\mathrm{d}U=C_{V}\cdot\,\mathrm{d}T\),isobaric processes (\(p=\mathrm{const}\)) \(\Rightarrow\,\mathrm{d}U=\,\mathrm{d}Q-p\,\mathrm{d}V\),isothermal processes (\(T=\mathrm{const}\)) \(\Rightarrow p\cdot V=\mathrm{constant}\),adiabatic processes (\(\,\mathrm{d}Q=0\)) \(\Rightarrow\,\mathrm{d}U=\,\mathrm{d}W\) and \(p\cdot V^{\kappa}=\mathrm{constant}\) with \(\kappa=C_{p}/C_{V}=\) adiabatic index.

• The second law of thermodynamics states that at the conversion of heat into mechanical energy at most the fraction \(\eta=(T_{1}-T_{2})/T_{1}\) can be converted when the heat reservoir is cooled from the temperature T ₁ to T ₂.

• The entropy \(S=k\cdot\ln P\) is a measure for the number P of realization possibilities for a system with a given temperature T and total energy E.

• Reversible processes are ideal processes where a system passes a cycle of processes and reaches its initial state without any losses. An example is the Carnot Cycle where the system passes through two isothermal and two adiabatic processes.

• For reversible processes the entropy remains constant. For all irreversible processes the entropy increases and the free energy \(F=U-T\cdot S\) decreases.

• The entropy S approaches zero for \(T\to 0\) (third law of thermodynamics).

• For real gases the Eigen-volume of the atoms and the interaction between the atoms cannot be neglected as for ideal gases. The equation of state \(p\cdot V=\nu\cdot R\cdot T\) of ideal gases is modified to the van der Waals equation \((p+a/V^{2})\cdot(V-b)=R\cdot T\), where \(a/V^{2}\) denotes the internal pressure and \(b/4\) the Eigen-volume of the \(N_{\mathrm{A}}\) molecules per mole.

• The heat of evaporation of a liquid \(\Lambda=T\cdot\,\mathrm{d}p_{\mathrm{s}}/\,\mathrm{d}T\cdot(V_{\mathrm{v}}-V_{\mathrm{l}})\) is due to the mechanical work necessary to enlarge the volume \(V_{\mathrm{l}}\) of the liquid to the much larger volume \(V_{\mathrm{v}}\) of the vapour against the external pressure and against the internal attractive forces between the molecules. The second contribution is much larger than the first one.

• In a \(p(T)\) phase diagram the liquid and gaseous phases are separated by the vapour phase curve and the liquid and solid phase by the melting curve. The two curves intersect in the triple point (\(T_{\mathrm{tr}},p_{\mathrm{tr}}\)) where all three phases can coexist.

• The vapour pressure of a liquid is lowered by addition of solvable substances, which increases the evaporation temperature. Also the melting temperature can be lowered.

Problems

10.1

Give a physically intuitive explanation, why the thermal expansion coefficient for liquids is larger than that of solids.

10.2

Prove example 2 in Sect. 10.1.2.

10.3

A container with \(1\,\mathrm{mol}\) helium and a container of equal size with \(1\,\mathrm{mol}\) nitrogen are heated with the same heat power of \(10\,\mathrm{W}\). Calculate after which time the temperature of the gas in the containers has risen from \({20}\) to \(100\,{}^{\circ}\text{C}\). The heat capacity of the containers is \(10\,\mathrm{W}\mathrm{s}/\mathrm{K}\). How long does it take, until \(T=1000\,{}^{\circ}\text{C}\) is reached, when we assume that the vibrational degrees of freedom of \(\mathrm{N}_{2}\) can be excited already at \(T=500\,{}^{\circ}\text{C}\)? All heat losses should be neglected.

10.4

Give a vivid and a mathematical justification for the time dependent temperature function \(T(t)\) during the mixing experiment of Fig. 10.12.

10.5

A container (\(m=0.1\,\mathrm{k}\mathrm{g}\)) with \(10\,\mathrm{mol}\) air at room temperature rests on the ground. What is the probability that it lifts by itself \(10\,\mathrm{c}\mathrm{m}\) above ground? Such an event would cause a cooling (conversion of thermal into potential energy). How large is the decrease of the temperature? (Specific heat of the gas is \((5/2)R\), that of the container is \(1\,\mathrm{k}\mathrm{J}/(\mathrm{k}\mathrm{g}\cdot\mathrm{K})\).)

10.6

A volume of \(1\,\mathrm{d}\mathrm{m}^{3}\) of helium under standard conditions (\(p_{0}=1\,\mathrm{bar}\), \(T_{0}=0\,{}^{\circ}\text{C}\)) is heated up to the temperature \(T=500\,\mathrm{K}\). What is the entropy increase for isochoric and for isobaric heating?

10.7

The critical temperature for \(\mathrm{C}\mathrm{O}_{2}\) (\(M=44\,\mathrm{g}/\mathrm{mol}\)) is \(T_{\mathrm{c}}=304.2\,\mathrm{K}\) and the critical pressure \(p_{\mathrm{c}}=7.6\cdot 10^{6}\,\mathrm{Pa}\), its density at the critical point is \(\varrho=46\,\mathrm{k}\mathrm{g}/\mathrm{m}^{3}\). What are the van der Waals constants a and b?

10.8

What is the entropy increase \(\Delta S_{1}\) when \(1\,\mathrm{k}\mathrm{g}\) water is heated from \({0}\) to \(50\,{}^{\circ}\text{C}\)? Compare \(\Delta S_{1}\) with the entropy increase \(\Delta S_{2}\) when \(0.5\,\mathrm{k}\mathrm{g}\) water of \(0\,{}^{\circ}\text{C}\) is mixed with \(0.5\,\mathrm{k}\mathrm{g}\) of \(100\,{}^{\circ}\text{C}\).

10.9

A power station delivers the mechanical work W ₁ when water vapour of \(600\,{}^{\circ}\text{C}\) drives a turbine and cools down to \(100\,{}^{\circ}\text{C}\).

1. a)

   What is the Carnot efficiency?

2. b)

   How many % of the output energy can one win, when the water of \(100\,{}^{\circ}\text{C}\) is used for heating and cools down to \(30\,{}^{\circ}\text{C}\)?

10.10

A hot solid body (\(m=1\,\mathrm{k}\mathrm{g}\), \(c=470\,\mathrm{J}/(\mathrm{k}\mathrm{g}\cdot\mathrm{K})\), \(T=300\,{}^{\circ}\text{C}\)) is immersed into \(10\,\mathrm{k}\mathrm{g}\) of water at \(20\,{}^{\circ}\text{C}\).

1. a)

   What is the final temperature?

2. b)

   What is the entropy increase?

10.11

Calculate the pressure that a wire with \(1\,\mathrm{m}\mathrm{m}\) diameter exerts onto an ice block with a width of \(10\,\mathrm{c}\mathrm{m}\) (according to Fig. 10.80) when both ends are connected with a mass \(m=5\,\mathrm{k}\mathrm{g}\). What is the increase of the melting temperature? What is the heat supplied to the ice block by the wire, if the outside temperature and the wire temperature are \(300\,\mathrm{K}?\) How much ice can be melted per second by the wire?

10.12

Calculate from the diagram of Fig. 10.64b the theoretical efficiency of the Otto-motor.

10.13

Show that for a periodically supplied heat at x = 0 Eq. 10.42 is a solution of the Eq. 10.38b for one-dimensional heat conduction.

10.14

What is the maximum power an upwind power plant can deliver (area \(5\,\mathrm{k}\mathrm{m}^{2}\), temperature below the glass roof \(T=50\,{}^{\circ}\text{C}\), height of the tower \(100\,\mathrm{m}\), outside temperature \(20\,{}^{\circ}\text{C}\) at the top of the chimney).

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