Class 11 Physics: Thermodynamics (उष्मागतिकी)

Formula Sheet & Quick Revision Notes (MP Board)

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1. Basic Concepts & Definitions

• Thermodynamic System (उष्मागतिक निकाय): A collection of a large number of particles (atoms/molecules) specified by macroscopic variables like Pressure ($P$), Volume ($V$), and Temperature ($T$).
• Surroundings (परिवेश): Everything outside the system that can interact with it.
• Thermal Equilibrium (तापीय संतुलन): Two systems are in thermal equilibrium if they are at the same temperature and there is no net flow of heat between them.
• State Variables (अवस्था चर):
  • Intensive Variables: Independent of the size/mass of the system (e.g., Temperature $T$, Pressure $P$, Density $\rho$).
  • Extensive Variables: Depend on the size/mass of the system (e.g., Volume $V$, Mass $m$, Internal Energy $U$).

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2. Zeroth Law of Thermodynamics (उष्मागतिकी का शून्यवा नियम)

Statement: If two systems $A$ and $B$ are separately in thermal equilibrium with a third system $C$, then $A$ and $B$ are also in thermal equilibrium with each other.

• Significance: This law introduces and defines the concept of Temperature ($T$).

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3. Work, Heat, and Internal Energy

A. Heat ($Q$)

• Energy transferred between system and surroundings due to a temperature difference.
• Unit: Joule ($J$) or Calorie ($cal$). ($1\text{ cal} = 4.184\text{ J}$)

B. Work Done ($W$)

• Work done by a gas during volume change $dV$ at pressure $P$: W = ∫ P dV
• Work done is equal to the Area under the P-V diagram.

C. Internal Energy ($U$)

• Sum of total kinetic energy and potential energy of all molecules in the system.
• For an Ideal Gas, internal energy depends ONLY on temperature: U = f/2 * n * R * T (where $f$ = degrees of freedom, $n$ = number of moles)

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4. Sign Conventions (चिह्न परिपाटी)

Quantity	Positive ($+$)	Negative ($-$)
Heat ($Q$)	Heat supplied to the system	Heat extracted from the system
Work ($W$)	Work done by the system (Expansion)	Work done on the system (Compression)
Change in $U$ ($\Delta U$)	Increase in Temperature	Decrease in Temperature

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5. First Law of Thermodynamics (FLOT) (प्रथम नियम)

Based on the Law of Conservation of Energy.

ΔQ = ΔU + ΔW

• In differential form: dQ = dU + dW = dU + P dV

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6. Specific Heat Capacities & Mayer's Relation

• Molar Specific Heat at Constant Volume ($C_v$): Cv = (1/n) * (dQ/dT)_v = (f/2) * R
• Molar Specific Heat at Constant Pressure ($C_p$): Cp = (1/n) * (dQ/dT)_p = (f/2 + 1) * R
• Mayer's Formula (मेयर का सूत्र): Cp - Cv = R
• Adiabatic Index / Specific Heat Ratio ($\gamma$): γ = Cp / Cv = 1 + (2 / f)

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7. Thermodynamic Processes (उष्मागतिक प्रक्रम) Summary Table

Process	Condition	Equation of State	Work Done Formula ($W$)	First Law Application
Isothermal (समतापी)	$T = \text{Constant}$ ($\Delta T = 0$)	$P V = \text{Constant}$	W = 2.303 nRT log₁₀(V₂/V₁)	$\Delta U = 0 \implies Q = W$
Adiabatic (रुद्धोष्म)	$Q = \text{Constant}$ ($\Delta Q = 0$)	$P V^{\gamma} = \text{Constant}$	W = nR(T₁ - T₂) / (γ - 1)	$Q = 0 \implies \Delta U = -W$
Isobaric (समदाबी)	$P = \text{Constant}$ ($\Delta P = 0$)	$V / T = \text{Constant}$	W = P(V₂ - V₁) = nR(T₂ - T₁)	$Q = \Delta U + P(V_2 - V_1)$
Isochoric (समआयतनिक)	$V = \text{Constant}$ ($\Delta V = 0$)	$P / T = \text{Constant}$	W = 0	$W = 0 \implies Q = \Delta U = n C_v \Delta T$
Cyclic Process (चक्रीय)	Initial State = Final State	System returns to start	W = Area enclosed by PV curve	$\Delta U = 0 \implies Q_{\text{net}} = W_{\text{net}}$

Important Adiabatic Relations:

1. $P V^\gamma = \text{Constant}$
2. $T V^{\gamma-1} = \text{Constant}$
3. $P^{1-\gamma} T^\gamma = \text{Constant}$

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8. Second Law of Thermodynamics (द्वितीय नियम)

A. Kelvin-Planck Statement

It is impossible to construct an engine operating in a cycle that absorbs heat from a reservoir and converts it completely into work without producing any other effect. (100% efficient heat engine is impossible).

B. Clausius Statement

It is impossible for heat to flow by itself from a cooler body to a hotter body without external work being performed.

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9. Heat Engine (उष्मा इंजन)

A device that converts thermal energy into mechanical work continuously in a cyclic process.

• Components: Source at high temperature ($T_1$), Sink at low temperature ($T_2$), Working substance.
• Thermal Efficiency ($\eta$): η = Work Output / Heat Input = W / Q₁ η = (Q₁ - Q₂) / Q₁ = 1 - (Q₂ / Q₁)

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10. Carnot Engine & Carnot Cycle (कार्नो इंजन)

An ideal reversible heat engine operating between two temperatures $T_1$ (Source) and $T_2$ (Sink).

Four Steps of Carnot Cycle:

1. Isothermal Expansion (at $T_1$)
2. Adiabatic Expansion ($T_1 \rightarrow T_2$)
3. Isothermal Compression (at $T_2$)
4. Adiabatic Compression ($T_2 \rightarrow T_1$)

Efficiency of Carnot Engine:

η = 1 - (T₂ / T₁)

(Note: $T_1$ and $T_2$ must always be in Kelvin).

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11. Refrigerator & Heat Pump (प्रशीतक)

A heat engine operating in the reverse direction. It extracts heat $Q_2$ from a cold body ($T_2$) by doing external work $W$ and releases heat $Q_1$ to a hotter body ($T_1$).

• Coefficient of Performance ($\beta$ or $\alpha$): β = Heat extracted / Work done = Q₂ / W = Q₂ / (Q₁ - Q₂)

• For a Carnot Refrigerator: β = T₂ / (T₁ - T₂)

• Relation between Efficiency ($\eta$) and Coefficient of Performance ($\beta$): β = (1 - η) / η

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💡 Quick Exam Tips for MP Board

1. State Variables Question: Be ready to classify $P, V, T, U$ into Intensive and Extensive variables.
2. Derivations: Practice the derivation of Work Done in an Isothermal Process and Adiabatic Process.
3. Slope Comparison: Slope of an Adiabatic curve on a $P-V$ diagram is $\gamma$ times steeper than the slope of an Isothermal curve. Slope of Adiabatic = γ × Slope of Isothermal
4. Units Check: Always convert temperature to Kelvin ($K = ^\circ\text{C} + 273.15$) before using engine formulas!