MPSC 2026 Prelims Paper-I · Question 55 of 100

Correct — option (4), "Less than one". A heat engine takes in heat Q_H from a hot reservoir, converts part of it into work W, and rejects the remainder Q_C to a cold reservoir. Its thermal efficiency is the fraction of the heat supplied that comes out as work: η = W/Q_H = (Q_H − Q_C)/Q_H = 1 − Q_C/Q_H Read that expression and the answer is already visible. The efficiency would equal one only if Q_C were zero — that is, only if the engine rejected no heat at all and turned every joule it absorbed into work. The second law of thermodynamics says precisely that this cannot happen. In the Kelvin-Planck statement, no process is possible whose sole result is the complete conversion of heat drawn from a single reservoir into work. Some heat must always be discarded to a colder body, so Q_C is strictly greater than zero, so η is strictly less than one. The same conclusion comes out of the Carnot limit. For an engine working between reservoirs at absolute temperatures T_H and T_C, the maximum attainable efficiency is η = 1 − T_C/T_H, and no engine operating between those two temperatures can beat it — that is Carnot's theorem. Efficiency reaches one only if T_C is absolute zero or T_H is infinite, and neither is attainable; the third law of thermodynamics itself rules out reaching absolute zero in any finite number of steps. So the ceiling is always below unity, and a real engine, being irreversible, falls further below it again. The cleanest way to hold the two laws together is by what each forbids. The first law is conservation of energy, and it forbids getting out more work than the heat put in — efficiency greater than one is impossible, which is a perpetual motion machine of the first kind. The second law forbids getting out exactly as much work as the heat put in — efficiency equal to one is impossible, which is a perpetual motion machine of the second kind. The old formulation catches it: the first law says you cannot win, and the second says you cannot even break even. That also tells you what the question is really testing. Options (1) and (2) are excluded by the first law and by the definition of efficiency; only option (3) and option (4) are candidates, and the entire content of the second law here is that (3) is unattainable and (4) is the correct description. A candidate who can name what each law forbids does not need the formulae at all. Real engines sit well below the Carnot ceiling. A petrol engine converts roughly a quarter to a third of its fuel energy into useful work, a diesel engine somewhat more, a conventional steam thermal power station in the region of a third to two-fifths, and a modern combined cycle gas plant approaches three-fifths. In every case most of the energy leaves as waste heat — which is not a failure of engineering but the second law showing itself in the exhaust pipe and the cooling tower.

• (1)Zero — Option (1) confuses a prohibition with a prediction. The second law places an upper bound on the efficiency of a heat engine; it does not assert that the efficiency is zero. An engine of zero efficiency would deliver no work at all, which describes a broken machine rather than a law of nature — and real heat engines plainly do work, driving vehicles and generators every day. Zero efficiency is what a heat engine approaches when the two reservoirs are at the same temperature, since η = 1 − T_C/T_H vanishes when T_C equals T_H, but that is a special condition rather than the general statement.
• (2)Infinity — Option (2) is impossible on the definition alone. Efficiency is the ratio of the work obtained to the heat supplied, and by the first law the work obtained cannot exceed the heat supplied, so the ratio cannot exceed one, let alone become infinite. An efficiency greater than one would be a perpetual motion machine of the first kind, creating energy from nothing. Note the contrast with a refrigerator or heat pump, whose coefficient of performance can and routinely does exceed one — but that is a different figure of merit, defined as heat moved per unit of work supplied, and it is not an efficiency.
• (3)One — Option (3) is the serious distractor, because an efficiency of one violates nothing in the first law — energy would still be conserved if a machine turned all the heat it absorbed into work. What forbids it is exactly the second law, in the Kelvin-Planck form: no process can have as its sole result the complete conversion of heat from a single reservoir into work. Such a machine is a perpetual motion machine of the second kind. The whole point of the second law is that it rules out this case and nothing in the first law does, so a candidate who marks 'one' has answered by the first law and ignored the law the question names.

A heat engine is any device that operates in a cycle, absorbing heat from a hot reservoir, converting some of it into work, and rejecting the rest to a cold reservoir. The second law of thermodynamics constrains what such a device can do, and it has two classical statements that are logically equivalent. The Kelvin-Planck statement, framed for engines, says no process is possible whose sole result is the absorption of heat from a reservoir and its complete conversion into work. The Clausius statement, framed for refrigerators, says no process is possible whose sole result is the transfer of heat from a colder body to a hotter one — heat does not flow uphill by itself. Each can be derived from the other, and both are expressions of a deeper idea captured by entropy: the entropy of an isolated system never decreases, so processes have a direction, and the natural direction is towards greater disorder and the degradation of energy into less useful forms. Carnot's theorem completes the picture by fixing the ceiling: of all engines working between two given reservoirs, a reversible one is the most efficient, all reversible engines between those reservoirs have the same efficiency whatever their working substance, and that efficiency is 1 − T_C/T_H with temperatures on the absolute scale.

The four laws of thermodynamics are best held as a set of prohibitions. The zeroth law establishes that temperature is a meaningful quantity, since two bodies each in thermal equilibrium with a third are in equilibrium with each other. The first law is conservation of energy applied to heat and work, and it forbids a machine that creates energy — a perpetual motion machine of the first kind. The second law forbids a machine that converts heat completely into work, or moves heat from cold to hot unaided — a perpetual motion machine of the second kind. The third law states that the entropy of a perfect crystal approaches zero as the temperature approaches absolute zero, and with it that absolute zero cannot be reached in a finite number of operations. The practical consequences are everywhere. Thermal power stations need cooling towers or river water because the rejected heat has to go somewhere. Engine efficiency is raised chiefly by raising the inlet temperature, since that is what widens the gap between T_H and T_C. And combined cycle plants achieve their higher figures by using the waste heat of a gas turbine to run a steam cycle beneath it — not by evading the second law but by using two temperature drops instead of one.

• The thermal efficiency of a heat engine is η = W/Q_H = 1 − Q_C/Q_H. It equals one only if the heat rejected Q_C is zero, which the second law forbids, so the efficiency of a heat engine is always less than one.
• Kelvin-Planck statement of the second law: no process is possible whose sole result is the absorption of heat from a single reservoir and its complete conversion into work. Clausius statement: heat cannot pass by itself from a colder to a hotter body. The two are equivalent.
• Carnot's theorem: the maximum efficiency of an engine working between absolute temperatures T_H and T_C is η = 1 − T_C/T_H, attained only by a reversible engine, and independent of the working substance.
• The first law forbids efficiency greater than one — a perpetual motion machine of the first kind. The second law forbids efficiency equal to one — a perpetual motion machine of the second kind. Only the second law is at issue in this question.
• The coefficient of performance of a refrigerator or heat pump, defined as heat moved per unit of work supplied, can exceed one and routinely does. It is not an efficiency and is not bounded by unity.

• Answering by the first law when the question names the second. Efficiency cannot exceed one because energy is conserved; it cannot equal one because of the second law, and only the second point is what this question asks.
• Reading an upper bound as a value. The second law says the efficiency is less than one; it does not say the efficiency is zero or fix it at any particular figure.
• Confusing thermal efficiency with the coefficient of performance of a refrigerator. The COP can exceed one, which looks paradoxical only if the two quantities are treated as the same thing.
• Believing Carnot efficiency can be reached in practice. It is the ceiling for a reversible engine between two reservoirs; every real engine has irreversibilities and falls short of it.

Thermodynamics appears in MPSC's general science section at higher secondary level, and almost always through the statements of the laws rather than through calculation. The three shapes to expect are: which law forbids a given hypothetical machine; what a named statement of the second law says, with Kelvin-Planck and Clausius offered as alternatives; and simple consequences such as why a heat engine must reject heat, why a refrigerator needs work input, or why efficiency rises with the temperature of the source. Since the papers rarely demand numerical work, the highest-yield preparation is to be able to state each of the four laws in one sentence and to name the machine each one prohibits. That single table answers most questions in this area on sight, and it is what separates option (3) from option (4) here.

No directly related past PYQ was found.

• practice — not a real PYQ

  A machine that would convert all the heat absorbed from a single reservoir completely into work would violate :

  • (a)the zeroth law of thermodynamics
  • (b)the first law of thermodynamics
  • (c)the second law of thermodynamics
  • (d)no law, but is technically impossible to build

  Answer(c) the second law of thermodynamics, in its Kelvin-Planck statement. Such a machine would conserve energy perfectly and so would not offend the first law at all — that is precisely why a separate law was needed. It is called a perpetual motion machine of the second kind, and the second law's whole content in relation to engines is that it cannot exist, which is why the efficiency of a heat engine is always strictly less than one.

• practice — not a real PYQ

  The maximum efficiency of a heat engine working between a source at absolute temperature T_H and a sink at absolute temperature T_C is given by :

  • (a)T_C/T_H
  • (b)1 − T_C/T_H
  • (c)1 + T_C/T_H
  • (d)T_H/(T_H − T_C)

  Answer(b) 1 − T_C/T_H, the Carnot efficiency, attainable only by a reversible engine and independent of the working substance. It approaches one only as the sink temperature approaches absolute zero or the source temperature becomes infinite, neither of which is attainable — which is the reason a real engine's efficiency is not merely below one but usually well below it. Note that the temperatures must be on the absolute (kelvin) scale for the formula to hold.