Cathinone

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Cathinone 

 

Cathinone

Clinical data
ATC code	none
Legal status
Legal status	AU: S9 (Prohibited substance) CA: Schedule III DE: Anlage I (Authorized scientific use only) UK: Class C US: Schedule I
Pharmacokinetic data
Elimination half-life	0.7–2.3 h
Identifiers
CAS Number	71031-15-7
PubChem CID	62258
DrugBank	DB01560
ChemSpider	56062
UNII	540EI4406J
KEGG	C08301
ChEBI	CHEBI:4110
ChEMBL	ChEMBL2104047
CompTox Dashboard (EPA)	DTXSID0050427
ECHA InfoCard	100.163.927
Chemical and physical data
Formula	C9H11NO
Molar mass	149.19 g/mol g·mol−1
3D model (JSmol)	Interactive image

Cathinone /ˈkæθɪnoʊn/ (also known as benzoylethanamine, or β-keto-amphetamine) is a monoamine alkaloid found in the shrub Catha edulis (khat) and is chemically similar to ephedrine, cathine, methcathinone and other amphetamines. It is probably the main contributor to the stimulant effect of Catha edulis. Cathinone differs from many other amphetamines in that it has a ketone functional group. Other phenethylamines that share this structure include the stimulants methcathinone, MDPV, mephedrone and the antidepressant bupropion.

History

Discovery

Khat has been cultivated in the Horn of Africa and Arabian Peninsula region of the world for thousands of years. It is most commonly chewed for the euphoric effect it produces. The active ingredient was first proposed in 1930, when cathine was identified as a predominant alkaloid in the plant. Cathine was thought to be the main active ingredient in khat until the 1960s, when it was found that the amount of cathine in the khat leaves is insufficient to produce the effects observed. In 1975, the United Nations Narcotic Laboratory analyzed khat leaves from Yemen, Kenya and Madagascar and found the presence of a different alkaloid, cathinone. Cathinone is a similar molecule to cathine, but is much more abundant in younger plants. This finding caused scientists to speculate about whether cathinone was the true active ingredient in khat.

A study was conducted in 1994 to test the effects of cathinone. Six volunteers who had never chewed khat were given an active khat sample and a cathinone-free placebo sample. The researchers analyzed the participants’ moods, activity levels and blood pressure before and after consuming the khat or placebo. This analysis showed that cathinone produced amphetamine-like symptoms, leading the researchers to confirm that cathinone, not cathine, is the active ingredient in khat leaves.

Cultural significance

Man chewing khat

Over 20 million people in the Arabian Peninsula and East Africa chew khat leaves daily. It is an important piece of the culture and economy in this region, especially in Ethiopia (where khat is said to have originated), Kenya, Djibouti, and Yemen. Men usually chew it during parties or other social gatherings while smoking cigarettes and drinking tea. Farmers and other workers also use khat in the afternoon to reduce fatigue and hunger as the day goes on. It functions like the caffeine in a strong cup of coffee as an anti-fatigue drug. Students and drivers have been known to use it to stay alert for longer periods of time.

In order to produce its desired effects, khat leaves should be chewed fresh. The fresh leaves have a higher concentration of cathinone. Waiting too long after cultivation to chew the leaf will allow the cathinone to break down into its less potent form, cathine. Because of the need for quick chewing, it is a habit that has historically been prevalent only where the plant grows. However, in the recent years with improvements in road and air transport, khat chewing has spread to all corners of the world.

The cultivation of khat in Yemen is a highly profitable industry for farmers. Khat plants will grow differently depending on the climate they are grown in and each one will produce different amounts of cathinone. It generally doesn’t grow as well as in coastal, hot climates. In Yemen, the khat plant is named after the region in which it is grown. The Nehmi khat plant has the highest known concentration of cathinone, 342.5 mg/100g.

Legality

Internationally, cathinone is a Schedule I drug under the Convention on Psychotropic Substances. Circa 1993, the DEA added cathinone to the Controlled Substances Act's Schedule I.

The sale of khat is legal in some jurisdictions, but illegal in others). Substituted cathinones were also often used as the key ingredient of recreational drug mixes commonly known as "bath salts" in the United States.

The table below shows the legality of khat and cathinone in various countries:

Region	Regulation
Eritrea	Illegal
Ethiopia	Legal
Somalia	Legal
Djibouti	Legal
Kenya	Khat is legal but cathinone and cathine are classified as Class C substances
South Africa	Khat is a protected plant
China	Illegal
Israel	Legal – The khat plant leaves are allowed to be chewed and beverages containing khat are legal, but it is illegal to sell pills based on cathinone extracts
Malaysia	Illegal
Saudi Arabia	Illegal
Yemen	Khat is legal but the cultivation and selling of the plant is regulated by the government
Denmark	Illegal
Finland	Illegal
France	Khat is prohibited as a stimulant
Germany	Khat is illegal but a derivative of cathinone is available upon prescription
Ireland	Illegal unless authorized
Netherlands	Cathinone and cathine have been illegal but khat was announced as illegal in 2012
Norway	Illegal
Poland	Illegal
Sweden	Illegal
Switzerland	Illegal
United Kingdom	Illegal
Canada	Illegal to obtain unless approved by a medical practitioner
United States	Illegal
Australia	Khat is regulated under the Australian Customs Service and a special permit is needed to import it for personal use
New Zealand	Illegal
Georgia	The khat plant itself is allowed to be sold and chewed, but it is illegal to sell or make beverages containing khat
Bulgaria	Illegal under List I - "Plants and substances with a high risk to the public health due to their harmful effect of misuse, prohibited for use in human and veterinary medicine"

Biological effects

Mechanism of action

Cathinone has been found to stimulate the release of dopamine and inhibit the reuptake of epinephrine, norepinephrine and serotonin in the central nervous system (CNS). These neurotransmitters are all considered monoamines and share the general structure of an aromatic ring and an amine group attached by a two-carbon separator. Because cathinone is a hydrophobic molecule, it can easily cross cell membranes and other barriers, including the blood-brain barrier. This property allows it to interact with the monoamine transporters in the synaptic cleft between neurons. Cathinone induces the release of dopamine from brain striatal preparations that are prelabelled either with dopamine or its precursors.

The metabolites of cathinone, cathine and norephedrine also possess CNS stimulation, but create much weaker effects. The effects of cathinone on the body can be countered by a preceding administration of a dopamine-receptor antagonist. The antagonist will keep the neuron at its resting state, so the cathinone cannot cause extraneous release of dopamine or other neurotransmitters.

Cathinone can also affect the parasympathetic nervous system (PSNS) by blocking adrenergic receptors and inhibiting smooth muscle contraction. It can also induce dry mouth, blurred vision and increased blood pressure and heart rate.

Pharmacology

Khat leaves are removed from the plant stalk and are kept in a ball in the cheek and chewed. Chewing releases juices from the leaves, which include the alkaloid cathinone. The absorption of cathinone has two phases: one in the buccal mucosa and one in the stomach and small intestine. The stomach and small intestine are very important in the absorption of ingested alkaloids. At approximately 2.3 hours after chewing khat leaves, the maximum concentration of cathinone in blood plasma is reached. The mean residence time is 5.2 ± 3.4 hours. The elimination half-life of cathinone is 1.5 ± 0.8 hours. A two-compartment model for absorption and elimination best describes this data. However, at most, only 7% of the ingested cathinone is recovered in the urine. This indicates that the cathinone is being broken down in the body. Cathinone has been shown to selectively metabolize into R,S-(-)-norephedrine and cathine. The reduction of the ketone group in cathinone will produce cathine. This reduction is catalyzed by enzymes in the liver. The spontaneous breakdown of cathinone is the reason it must be chewed fresh after cultivation.

Effects on health

The first documentation of the khat plant being used in medicine was in a book published by an Arabian physician in the 10th century. It was used as an antidepressant because it led to feelings of happiness and excitement. Chronic khat chewing can also create drug dependence, as shown by animal studies. In such studies, monkeys were trained to push a lever to receive the drug reward. As the monkeys' dependence increased, they pressed the lever at an increasing frequency.

Khat chewing and the effects of cathinone on the body differ from person to person, but there is a general pattern of behavior that emerges after ingesting fresh cathinone:

1. Feelings of euphoria that last for one to two hours
2. Discussion of serious issues and increased irritability
3. The chewer's imagination is very active
4. Depressive stage
5. Irritability, loss of appetite and insomnia

There are other effects not related to the CNS. The chewer can develop constipation and heartburn after a khat session. Long-term effects of cathinone can include gum disease or oral cancer, cardiovascular disease and depression. The withdrawal symptoms of cathinone include, hot flashes, lethargy and a great urge to use the drug for at least the first two days.

Chemistry

Biosynthesis

Mechanism of the Non-Beta Oxidation pathway for the biosynthesis of S-Cathinone in the Khat plant

The synthesis of cathinone in khat begins with L-phenylalanine and the first step is carried out by L-phenylalanine ammonia lyase (PAL), which cleaves off an ammonia group and creates a carbon-carbon double bond, forming cinnamic acid. After this, the molecule can either go through a beta-oxidative pathway or a non-beta-oxidative pathway. The beta-oxidative pathway produces benzoyl-CoA while the non-beta-oxidative pathway produces benzoic acid. Both of these molecules can be converted to 1-phenylpropane-1,2-dione by a condensation reaction catalyzed by a ThDP-dependent enzyme (Thiamine diphosphate-dependent enzyme) with pyruvate and producing CO2. 1-phenylpropane-1,2-dione goes through a transaminase reaction to replace a ketone with an ammonia group to form (S)-cathinone. (S)-Cathinone can then undergo a reduction reaction to produce the less potent but structurally similar cathine or norephedrine, which are also found in the plant.

Aside from the beta- and non-beta-oxidative pathways, the biosynthesis of cathinone can proceed through a CoA-dependent pathway. The CoA-dependent pathway is actually a mix between the two main pathways as it starts like the beta-oxidative pathway and then when it loses CoA, it finishes the synthesis in the non-beta-oxidative pathway. In this pathway, the trans-cinnamic acid produced from L-phenylalanine is ligated to a Coenzyme A (CoA), just like the beginning of the beta-oxidative pathway. It then undergoes hydration at the double bond. This product then loses the CoA to produce benzaldehyde, an intermediate of the non-beta-oxidative pathway. Benzaldehyde is converted into benzoic acid and proceeds through the rest of the synthesis.

Synthetic production

Synthesize enantiomerically pure S-Cathinone

 

Racemic cathinone from propiophenone via the α-brominated intermediate

Two mechanism of synthesizing Cathinone

Cathinone can be synthetically produced from propiophenone through a Friedel-Crafts Acylation of propionic acid and benzene. The resulting propiophenone can be brominated, and the bromine can be substituted with ammonia to produce a racemic mixture of cathinone. A different synthetic strategy must be employed to produce enantiomerically pure (S)-cathinone. This synthetic route starts out with the N-acetylation of the optically active amino acid, S-alanine. Then, phosphorus pentachloride (PCl₅) is used to chlorinate the carboxylic acid forming an acyl chloride. At the same time, a Friedel-Crafts acylation is preformed on benzene with aluminum chloride catalyst. Finally, the acetyl protecting group is removed by heating with hydrochloric acid to form enantiomerically pure S-(-)-cathinone.

Structure

Bupropion: a cathinone derivative

Cathinone can be extracted from Catha edulis, or synthesized from α-bromopropiophenone (which is easily made from propiophenone). Because cathinone is both a primary amine and a ketone, it is very likely to dimerize, especially as a free base isolated from plant matter.

The structure of cathinone is very similar to that of other molecules. By reducing the ketone, it becomes cathine if it retains its stereochemistry, or norephedrine if its stereochemistry is inverted. Cathine is a less potent version of cathinone and cathinone's spontaneous reduction is the reason that older khat plants are not as stimulating as younger ones. Cathinone and amphetamine are closely related in that amphetamine is only lacking the ketone C=O group. Cathinone is structurally related to methcathinone, in much the same way as amphetamine is related to methamphetamine. Cathinone differs from amphetamine by possessing a ketone oxygen atom (C=O) on the β (beta) position of the side chain. The corresponding alcohol, cathine, is a less powerful stimulant. The biophysiological conversion from cathinone to cathine is to blame for the depotentiation of khat leaves over time. Fresh leaves have a greater ratio of cathinone to cathine than dried ones, therefore having more psychoactive effects.

There are many cathinone derivatives that include the addition of an R group to the amino end of the molecule. Some of these derivatives have medical uses as well. Bupropion is one of the most commonly prescribed antidepressants and its structure is Cathinone with a tertiary butyl group attached to the nitrogen and chlorine attached to the benzene ring meta- (chemistry) to the main carbon chain.

Other cathinone derivatives are strong psychoactive drugs. One such drug is methylone, a drug structurally similar to MDMA.

Energy quality

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Energy_quality 

 

Heat, a form of energy, is partly potential energy and partly kinetic energy

 

Energy quality is the contrast between different forms of energy, the different trophic levels in ecological systems and the propensity of energy to convert from one form to another. The concept refers to the empirical experience of the characteristics, or qualia, of different energy forms as they flow and transform. It appeals to our common perception of the heat value, versatility, and environmental performance of different energy forms and the way a small increment in energy flow can sometimes produce a large transformation effect on both energy physical state and energy. For example the transition from a solid state to liquid may only involve a very small addition of energy. Methods of evaluating energy quality are sometimes concerned with developing a system of ranking energy qualities in hierarchical order.

Introduction

Since before antiquity there has been deep philosophical, aesthetic and scientific interest in the contrast of quality with quantity. In some respects the history of modern and postmodern thought can be characterized by the phenomenological approach to these two concepts. A central question has been whether the many different qualitative aspects of the world can be understood in terms of rational quantities, or whether the qualitative and quantitative are irreconcilable: that is, there is no "rational quality", or quale ratio. Many scientists and analytic philosophers say they are not, and therefore consider some qualitative phenomena like, for instance, spirituality, and astrology to be unquantifiable, unanalysable by scientific methods, and therefore ungrounded in physical reality. The notion of energy quality therefore has a tendency to be linked with phenomena many scientists consider unquantifiable, or at least incommunicable, and are consequently dismissed out of hand.

At the same time many people have also recognised qualitative differences in the way things can be done by different entities (both physical and biological). Humans, for example have qualitatively different capacities than many other mammals, due, in part, to their opposable thumb. In the attempt to formalise some of the qualitative differences, entities were grouped according to distinguishing features or capacities. Different schools of thought used different methods to make distinctions. Some people chose taxonomic and genome structure, while others chose energetic function as the basis of classifications. The former are often associated with biology, while the latter with the trophic food chain analysis of ecology. These can be considered attempts to formalise quantitative, scientific studies of the qualitative differences between entities. The efforts were not isolated to biology and ecology, since engineers were also interested in quantifying the amount of work that qualitatively different sources of energy could provide.

Ohta

According to Ohta (1994, pp. 90–91) the ranking and scientific analysis of energy quality was first proposed in 1851 by William Thomson under the concept of "availability". This concept was continued in Germany by Z. Rant, who developed it under the title, "die Exergie" (the exergy). It was later continued and standardised in Japan. Exergy analysis now forms a common part of many industrial and ecological energy analyses. For example, I.Dincer and Y.A. Cengel (2001, p. 132) state that energy forms of different qualities are now commonly dealt with in steam power engineering industry. Here the "quality index" is the relation of exergy to the energy content (Ibid.). However energy engineers were aware that the notion of heat quality involved the notion of value – for example A. Thumann wrote, "The essential quality of heat is not the amount but rather its 'value'" (1984, p. 113) – which brings into play the question of teleology and wider, or ecological-scale goal functions. In an ecological context S.E. Jorgensen and G.Bendoricchio say that exergy is used as a goal function in ecological models, and expresses energy "with a built-in measure of quality like energy" (2001, p. 392).

Energy quality evaluation methods

There appear to be two main kinds of methodology used for the calculation of energy quality. These can be classed as either receiver or donor methods. One of the main differences that distinguishes these classes is the assumption of whether energy quality can be upgraded in an energy transformation process.

Receiver methods: view energy quality as a measure and indicator of the relative ease with which energy converts from one form to another. That is, how much energy is received from a transformation or transfer process. For example, A. Grubler [1] used two types of indicators of energetic quality pars pro toto: the hydrogen/carbon (H/C) ratio, and its inverse, the carbon intensity of energy. Grubler used the latter as an indicator of relative environmental quality. However Ohta says that in multistage industrial conversion systems, such as a hydrogen production system using solar energy, the energy quality is not upgraded (1994, p. 125). 

Donor methods: view energy quality as a measure of the amount of energy used in an energy transformation, and that goes into sustaining a product or service (H.T.Odum 1975, p. 3). That is how much energy is donated to an energy transformation process. These methods are used in ecological physical chemistry, and ecosystem evaluation. From this view, in contrast with that outlined by Ohta, energy quality is upgraded in the multistage trophic conversions of ecological systems. Here, upgraded energy quality has a greater capacity to feedback and control lower grades of energy quality. Donor methods attempt to understand the usefulness of an energetic process by quantifying the extent to which higher quality energy controls lower quality energy.

Energy quality in physical-chemical science (direct energy transformations)

Constant energy form but variable energy flow

T.Ohta suggested that the concept of energy quality may be more intuitive if one considers examples where the form of energy remains constant but the amount of energy flowing, or transferred is varied. For instance if we consider only the inertial form of energy, then the energy quality of a moving body is higher when it moves with a greater velocity. If we consider only the heat form of energy, then a higher temperature has higher quality. And if we consider only the light form of energy then light with higher frequency has greater quality (Ohta 1994, p. 90). All these differences in energy quality are therefore easily measured with the appropriate scientific instrument.

Variable energy form, but constant energy flow

The situation becomes more complex when the form of energy does not remain constant. In this context Ohta formulated the question of energy quality in terms of the conversion of energy of one form into another, that is the transformation of energy. Here, energy quality is defined by the relative ease with which the energy transforms, from form to form.

If energy A is relatively easier to convert to energy B but energy B is relatively harder to convert to energy A, then the quality of energy A is defined as being higher than that of B. The ranking of energy quality is also defined in a similar way. (T.Ohta 1994, p. 90).

Nomenclature: Prior to Ohta's definition above, A.W.Culp produced an energy conversion table describing the different conversions from one energy to another. Culp's treatment made use of a subscript to indicate which energy form is being talked about. Therefore, instead of writing "energy A", like Ohta above, Culp referred to "J_e", to specify electrical form of energy, where" J" refers to "energy", and the "e"subscript refers to electrical form of energy. Culps notation anticipated Scienceman's (1997) later maxim that all energy should be specified as form energy with the appropriate subscript.

Energy quality in biophysical economics (indirect energy transformations)

The notion of energy quality was also recognised in the economic sciences. In the context of biophysical economics energy quality was measured by the amount of economic output generated per unit of energy input (C.J. Cleveland et al. 2000). The estimation of energy quality in an economic context is also associated with embodied energy methodologies. Another example of the economic relevance of the energy quality concept is given by Brian Fleay. Fleay says that the "Energy Profit Ratio (EPR) is one measure of energy quality and a pivotal index for assessing the economic performance of fuels. Both the direct and indirect energy inputs embodied in goods and services must be included in the denominator." (2006; p. 10) Fley calculates the EPR as the energy output/energy input.

DIFFERENT HIERARCHICAL RANKS OF ENERGY FORM QUALITY HIGHEST QUALITY Ohta Ranking Odum Ranking Electromagnetic Information Mechanical Human Services Photon Protein Food Chemical Electric Power Heat Food, Greens, Grains River-water potential Consolidated Fuels River Chemical energy Mechanical Tide Gross Photosynthesis Average wind Sunlight LOWEST QUALITY

Ranking energy quality

Energy abundance and relative transformation ease as measure of hierarchical rank and/or hierarchical position

Ohta sought to order energy form conversions according to their quality and introduced a hierarchical scale for ranking energy quality based on the relative ease of energy conversion (see table to right after Ohta, p. 90). It is evident that Ohta did not analyse all forms of energy. For example, water is left out of his evaluation. It is important to note that the ranking of energy quality is not determined solely with reference to the efficiency of the energy conversion. This is to say that the evaluation of "relative ease" of an energy conversion is only partly dependent on transformation efficiency. As Ohta wrote, "the turbine generator and the electric motor have nearly the same efficiency, therefore we cannot say which has the higher quality" (1994, p. 90). Ohta therefore also included, 'abundance in nature' as another criterion for the determination energy quality rank. For example, Ohta said that, "the only electrical energy which exists in natural circumstances is lightning, while many mechanical energies exist." (Ibid.).

Transformity as an energy measure of hierarchical rank

Like Ohta, H.T.Odum also sought to order energy form conversions according to their quality, however his hierarchical scale for ranking was based on extending ecological system food chain concepts to thermodynamics rather than simply relative ease of transformation . For H.T.Odum energy quality rank is based on the amount of energy of one form required to generate a unit of another energy form. The ratio of one energy form input to a different energy form output was what H.T.Odum and colleagues called transformity: "the EMERGY per unit energy in units of emjoules per joule" (H.T.Odum 1988, p. 1135).

Carnot's theorem (thermodynamics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Carnot%27s_theorem_(thermodynamics) 

 

Carnot's theorem, developed in 1824 by Nicolas Léonard Sadi Carnot, also called Carnot's rule, is a principle that specifies limits on the maximum efficiency any heat engine can obtain. The efficiency of a Carnot engine depends solely on the temperatures of the hot and cold reservoirs.

Carnot's theorem states that all heat engines between two heat reservoirs are less efficient than a Carnot heat engine operating between the same reservoirs. Every Carnot heat engine between a pair of heat reservoirs is equally efficient, regardless of the working substance employed or the operation details.

The maximum efficiency is the ratio of the temperature difference between the reservoirs and the temperature of the hot reservoir, expressed in the equation ${\displaystyle \eta _{\text{max}}=1-{\frac {T_{\mathrm {C} }}{T_{\mathrm {H} }}}}$, where T_C and T_H are the absolute temperatures of the cold and hot reservoirs, respectively, and the efficiency ${\displaystyle \eta }$ is the ratio of the work done by the engine to the heat drawn out of the hot reservoir.

Carnot's theorem is a consequence of the second law of thermodynamics. Historically, it was based on contemporary caloric theory, and preceded the establishment of the second law.

Proof

An impossible situation: A heat engine cannot drive a less efficient (reversible) heat engine without violating the second law of thermodynamics.

The proof of the Carnot theorem is a proof by contradiction, or reductio ad absurdum, as illustrated by the figure showing two heat engines operating between two reservoirs of different temperature. The heat engine with more efficiency (${\displaystyle \eta _{M}}$) is driving a heat engine with less efficiency (${\displaystyle \eta _{L}}$), causing the latter to act as a heat pump. This pair of engines receives no outside energy, and operates solely on the energy released when heat is transferred from the hot and into the cold reservoir. However, if ${\displaystyle \eta _{M}>\eta _{L}}$, then the net heat flow would be backwards, i.e., into the hot reservoir:

${\displaystyle Q_{\text{hot}}^{\text{out}}=Q<{\frac {\eta _{M}}{\eta _{L}}}Q=Q_{\text{hot}}^{\text{in}}.}$

It is generally agreed that this is impossible because it violates the second law of thermodynamics.

We begin by verifying the values of work and heat flow depicted in the figure. First, we must point out an important caveat: the engine with less efficiency (${\displaystyle \eta _{L}}$) is being driven as a heat pump, and therefore must be a reversible engine. If the less efficient engine (${\displaystyle \eta _{L}}$) is not reversible, then the device could be built, but the expressions for work and heat flow shown in the figure would not be valid.

By restricting our discussion to cases where engine (${\displaystyle \eta _{L}}$) has less efficiency than engine (${\displaystyle \eta _{M}}$), we are able to simplify notation by adopting the convention that all symbols, ${\displaystyle Q}$ and ${\displaystyle W}$ represent non-negative quantities (since the direction of energy flow never changes sign in all cases where ${\displaystyle \eta _{L}\leqslant \eta _{M}}$). Conservation of energy demands that for each engine, the energy which enters, ${\displaystyle E_{in}}$, must equal the energy which exits, ${\displaystyle E_{out}}$:

${\displaystyle E_{\text{in}}^{M}=Q=(1-\eta _{M})Q+\eta _{M}Q=E_{\text{out}}^{M},}$

${\displaystyle E_{\text{in}}^{L}=\eta _{M}Q+\eta _{M}Q\left({\frac {1}{\eta _{L}}}-1\right)={\frac {\eta _{M}}{\eta _{L}}}Q=E_{\text{out}}^{L},}$

The figure is also consistent with the definition of efficiency as ${\displaystyle \eta =W/Q_{h}}$ for both engines:

${\displaystyle \eta _{M}={\frac {W_{M}}{Q_{h}^{M}}}={\frac {\eta _{M}Q}{Q}}=\eta _{M},}$

${\displaystyle \eta _{L}={\frac {W_{L}}{Q_{h}^{L}}}={\frac {\eta _{M}Q}{{\frac {\eta _{M}}{\eta _{L}}}Q}}=\eta _{L}.}$

It may seem odd that a hypothetical heat pump with low efficiency is being used to violate the second law of thermodynamics, but the figure of merit for refrigerator units is not efficiency, ${\displaystyle W/Q_{h}}$, but the coefficient of performance (COP), which is ${\displaystyle Q_{c}/W}$. A reversible heat engine with low thermodynamic efficiency, ${\displaystyle W/Q_{h}}$ delivers more heat to the hot reservoir for a given amount of work when it is being driven as a heat pump.

Having established that the heat flow values shown in the figure are correct, Carnot's theorem may be proven for irreversible and the reversible heat engines.

Reversible engines

To see that every reversible engine operating between reservoirs ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ must have the same efficiency, assume that two reversible heat engines have different values of ${\displaystyle \eta }$, and let the more efficient engine (M) drive the less efficient engine (L) as a heat pump. As the figure shows, this will cause heat to flow from the cold to the hot reservoir without any external work or energy, which violates the second law of thermodynamics. Therefore both (reversible) heat engines have the same efficiency, and we conclude that:

All reversible engines that operate between the same two heat reservoirs have the same efficiency.

This is an important result because it helps establish the Clausius theorem, which implies that the change in entropy is unique for all reversible processes.,

${\displaystyle \Delta S=\int _{a}^{b}{\frac {dQ_{\text{rev}}}{T}},}$

over all paths (from a to b in V-T space). If this integral were not path independent, then entropy, S, would lose its status as a state variable.

Irreversible engines

If one of the engines is irreversible, it must be the (M) engine, placed so that it reverse drives the less efficient but reversible (L) engine. But if this irreversible engine is more efficient than the reversible engine, (i.e., if ${\displaystyle \eta _{M}>\eta _{L}}$), then the second law of thermodynamics is violated. And, since the Carnot cycle represents a reversible engine, we have the first part of Carnot's theorem:

No irreversible engine is more efficient than the Carnot engine operating between the same two reservoirs.

Definition of thermodynamic temperature

The efficiency of the engine is the work divided by the heat introduced to the system or

${\displaystyle \eta ={\frac {w_{cy}}{q_{H}}}={\frac {q_{H}-q_{C}}{q_{H}}}=1-{\frac {q_{C}}{q_{H}}}}$

(1)

where w_cy is the work done per cycle. Thus, the efficiency depends only on q_C/q_H. 

Because all reversible engines operating between the same heat reservoirs are equally efficient, all reversible heat engines operating between temperatures T₁ and T₂ must have the same efficiency, meaning the efficiency is a function only of the two temperatures:

${\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})}$

(2)

In addition, a reversible heat engine operating between temperatures T₁ and T₃ must have the same efficiency as one consisting of two cycles, one between T₁ and another (intermediate) temperature T₂, and the second between T₂ and T₃. This can only be the case if

${\displaystyle f(T_{1},T_{3})={\frac {q_{3}}{q_{1}}}={\frac {q_{2}q_{3}}{q_{1}q_{2}}}=f(T_{1},T_{2})f(T_{2},T_{3}).}$

Specializing to the case that ${\displaystyle T_{1}}$ is a fixed reference temperature: the temperature of the triple point of water. Then for any T₂ and T₃,

${\displaystyle f(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16\cdot f(T_{1},T_{3})}{273.16\cdot f(T_{1},T_{2})}}.}$

Therefore, if thermodynamic temperature is defined by

${\displaystyle T=273.16\cdot f(T_{1},T)\,}$

then the function viewed as a function of thermodynamic temperature, is

${\displaystyle f(T_{2},T_{3})={\frac {T_{3}}{T_{2}}},}$

and the reference temperature T₁ has the value 273.16. (Of course any reference temperature and any positive numerical value could be used—the choice here corresponds to the Kelvin scale.) 

It follows immediately that

${\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})={\frac {T_{C}}{T_{H}}}}$

(3)

Substituting Equation 3 back into Equation 1 gives a relationship for the efficiency in terms of temperature:

${\displaystyle \eta =1-{\frac {q_{C}}{q_{H}}}=1-{\frac {T_{C}}{T_{H}}}}$

(4)

Applicability to fuel cells and batteries

Since fuel cells and batteries can generate useful power when all components of the system are at the same temperature (${\displaystyle T=T_{H}=T_{C}}$), they are clearly not limited by Carnot's theorem, which states that no power can be generated when ${\displaystyle T_{H}=T_{C}}$. This is because Carnot's theorem applies to engines converting thermal energy to work, whereas fuel cells and batteries instead convert chemical energy to work. Nevertheless, the second law of thermodynamics still provides restrictions on fuel cell and battery energy conversion.