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Thermodynamics formulas

7 formulas and 12 common traps for MHT-CET Physics Thermodynamics, grouped by subtopic.

Full notes with worked examples

The First Law of Thermodynamics

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Q = ΔU + W and its Signs

First law

Q=ΔU+W,W=∫P dVQ = \Delta U + W, \qquad W = \int P\,dV

How Heat Splits at Constant Pressure

Isobaric split

ΔUQ=1γ,WQ=1−1γ\frac{\Delta U}{Q} = \frac{1}{\gamma}, \qquad \frac{W}{Q} = 1 - \frac{1}{\gamma}

Work in an Adiabatic Change

Adiabatic work

W=nR (T1−T2)γ−1W = \frac{nR\,(T_1 - T_2)}{\gamma - 1}

Common traps

Getting the sign of work wrong

W is work done BY the gas. When the gas is compressed, W is negative and the work done ON it adds to its internal energy.

Reading the direction of a cycle

Round a cycle the net work is the enclosed area — positive if the path goes clockwise on a P–V graph, negative if anticlockwise. The same triangle gives +3PV or −3PV.

Inverting the ratio

Q/W is γ/(γ − 1), and W/Q is its reciprocal (γ − 1)/γ. Both appear among the options; read which one the question asks for.

Using C_v for heat at constant pressure

Heat supplied at constant pressure is nC_pΔT; nC_vΔT is only the part that raises the internal energy. For 14 g of nitrogen warmed 48 °C that is 84R, not 60R.

Dividing by γ instead of γ − 1

The adiabatic work is nRΔT/(γ − 1), because it equals nC_vΔT and C_v = R/(γ − 1).

The Four Processes and the Adiabatic Relations

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Recognising Each Process

Process equations

PV=const (isothermal),PVγ=const (adiabatic)PV = \text{const (isothermal)}, \qquad PV^\gamma = \text{const (adiabatic)}

The Adiabatic Relations

Adiabatic

PVγ=const,TVγ−1=constPV^{\gamma} = \text{const}, \qquad TV^{\gamma - 1} = \text{const}

Chaining and Comparing Processes

Chaining

isothermal: P1V1=P2V2,adiabatic: P2V2γ=P3V3γ\text{isothermal: } P_1V_1 = P_2V_2, \qquad \text{adiabatic: } P_2V_2^\gamma = P_3V_3^\gamma

Common traps

Calling the steeper curve isothermal

Adiabatic curves are the steeper ones — γ times the isothermal slope at the same point.

Thinking an adiabatic keeps temperature constant

No heat enters, but work changes the internal energy, so the temperature changes. Constant temperature is the isothermal.

Using γ in the temperature relation

Pressure goes as V^(−γ) but temperature as V^(−(γ−1)). Using γ for the temperature turns 4 into 128.

Forgetting that 'sudden' means adiabatic

A gas compressed suddenly has no time to exchange heat. Treating it as isothermal gives 4P instead of 8P for a four-fold compression with γ = 1.5.

Applying one rule to the whole chain

Each leg has its own rule. PV^γ through an isothermal leg, or PV through an adiabatic one, gives a wrong pressure at the join.

The Carnot Engine and the Refrigerator

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Efficiency and Coefficient of Performance

Carnot

η=1−T2T1,β=T2T1−T2\eta = 1 - \frac{T_2}{T_1}, \qquad \beta = \frac{T_2}{T_1 - T_2}

Common traps

Using degrees Celsius in the efficiency

227 °C and 27 °C give 1 − 300/500 = 40%, not 1 − 27/227. Convert to kelvin first.

Changing the source when the sink moves

'Sink lowered by 57 K' leaves T₁ alone. Write the two efficiencies with the same T₁ and solve.

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