Carnot’s equations

Because the temperature of the finite cold reservoir (contents of the refrigerator) is a variable, use differential forms of Carnot s equations, Eqs. (5.7) and (5.8) ... 

In these equations Qc and Qu refer to the reservoirs. With dQu = Cl dTc, the first of Carnot s equations becomes ... 

Combine this equation with the second of Carnot s equations ... 

Equations (5.7) and (5.8) are known as Carnot s equations. In Eq. (5.7) the smallest possible value of QC is zero the corresponding value of Tc is the absolute zero of temperature on the Kelvin scale. As mentioned in Sec. 1.4, this occurs at -273.15°C. Equation (5.8) shows that the thermal efficiency of a Carnot engine can approach unity only when TH approaches infinity or Tc approaches zero. On earth nature provides heat reservoirs at neither of these conditions all heat engines therefore operate at thermal efficiencies less than unity. The cold reservoirs naturally available are the atmosphere, lakes and rivers, and the oceans, for which Tc = 300 K. Practical hot reservoirs are objects such as furnaces maintained at high temperature by combustion of fossil fuels and nuclear reactors held at high temperature by fission of radioactive elements, for which T = 600 K. With these values,... 

Carnot s equations, 146-147 Carnot s theorem, 142-143 Chemical potential, 298, 302, 303 as equilibrium criterion, 298-299, 503 for ideal gas, 302 for ideal solution, 303 Chemical reaction equilibrium constant for, 504-516 equilibrium conversion of, 518-528, 533-542 heat effects of, 116-133 reaction coordinate for, 497-501 reversible, 41-42, 505-507 standard property changes for, 125, 505 stoichiometry, 497-501... 

In approaclringtlris problem, use the differential fomi of Carnot s equation,... 

The Carnot s equations for a completely reversible heat engine are... 

In the operations constituting a Carnot s cycle, changes of Q and T occur separately. In the majority of cases both these changes occur together, so that the temperature of the working substance may be regarded as a function of the time. Equation (4) therefore requires extension, and this was effected by Lord Kelvin in May, 1854, in the following way ... 

In 1879 Lord Kelvin introduced the term nwtivity for the possession, the waste of which is called dissipation at constant temperature this is identical with Maxwell s available energy. He showed in a paper On Thermodynamics founded on Motivity and Energy Phil. Mag., 1898), that all the thermodynamic equations could be derived from the properties of motivity which follow directly from Carnot s theorem, without any explicit introduction of the entropy. 

Equation (6.16), which includes Equation (6.6), is a mathematical statement of Carnot s theorem ... 

Carcass method Carnot cycle Carothers Casson s equation Catalyst (general)... 

Relationship (9.1) is applied to the machine in the Carnot process. During the isothermal expansion step we must deliver that quantity entropy, in the course of expansion. If we would not do so, then we eould not hold the temperature in the machine. The machine requires thus the subsequent delivery of the entropy. The entropy becomes in any case larger after Maxwell s equation, if we increase the volume. 

FIGURE 10.2.3. Carnot s cycle for ideal gas. Substituting the above equations and Eq. (18) into Eq. (23) results in W = RTilnl W 72ln ... 

Carnot s reversible engine-consists of an ideal gas that operates between a hot reservoir and a cold reservoir, at temperatures 0i and 02 respectively. Until their identity has been is established, we shall use 0 for the temperature that appears in the ideal gas equation and T for the absolute temperature (which, as we shall see in the next section, is defined by the efficiency of a reversible cycle). Thus, the ideal gas equation is written as pV = NRQ, in which 0 is the temperature measured by noting the change of some quantity such as volume or pressure. (Note that measuring temperature by volume expansion is purely empirical each... 

Clausius Rudolf Julius Emmanuel (1822-1888) Ger. math., reconciled Carnot s theory of heat to equivalence of heat and work (2nd Law of thermodynamics), changes of state (Clausius-Clapeyron equation), contributed theory of electrolysis... 

The Clapeyron equation is named after Benoit-Pierre-Emile Clapeyron, 1799-1864, a French engineer who translated Carnot s cycle into the language of calculus. 

When Gibbs first turned his attention to thermodynamics in the early 1870 s, the subject had already achieved a certain level of maturity. The essential step had been taken in 1850 by Rudolf Clausius, when he argued that two laws are needed to reconcile Carnot s principle about the motive power of heat with the law of energy transformation and conservation. Efforts to understand the second of the two laws finally led Clausius in 1865 to his most concise and ultimately most fruitful analytical formulation. In effect, two basic quantities, internal energy and entropy, are defined by the two laws of thermodynamics. The internal energy U is that function of the state of the system whose differential is given by the equation expressing the first law,... 

The second law of thermodynamics was actually postulated by Carnot prior to the development of the first law. The original statements made concerning the second law were negative—they said what would not happen. The second law states that heat will not flow, in itself, from cold to hot. While no mathematical relationships come directly from the second law, a set of equations can be developed by adding a few assumptions for use in compressor analysis. For a reversible process, entropy, s, can be defined in differential form as... 

Because the gas in the Carnot cycle starts and ends at the same state, the system s entropy does not change during a cycle. Now apply the second law to the universe for the case of the Carnot cycle. Because the processes are reversible, the entropy of the universe does not change by Equation 2b. This can be written ... 

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