This formula sheet compiles essential concepts, formulas, and common pitfalls in Thermodynamics, specifically tailored for JEE Main aspirants. A strong grasp of these principles is crucial for solving problems related to heat, work, energy, and their transformations.
Key Concepts
- Thermal Expansion: The tendency of matter to change in volume in response to a change in temperature. It can be linear, superficial, or volumetric.
- Calorimetry: The science of measuring heat flow. It's based on the principle of conservation of energy, where heat lost by hotter bodies equals heat gained by colder bodies.
- Heat Transfer: The movement of thermal energy from a hotter system to a cooler system. Mechanisms include conduction, convection, and radiation.
- Thermodynamic Systems: Defined as a quantity of matter or a region in space chosen for study. Types include open, closed, and isolated systems.
- Thermodynamic Processes: Changes in the state of a thermodynamic system. Examples include isothermal, adiabatic, isobaric, isochoric, and cyclic processes.
- First Law of Thermodynamics: A statement of conservation of energy, describing the relationship between heat, work, and internal energy.
- Second Law of Thermodynamics: Defines the direction of spontaneous processes and introduces the concept of entropy. It states that heat cannot entirely be converted into work without some loss.
- Carnot Engine: An idealized heat engine that operates on a reversible cycle, providing the maximum possible efficiency between two given temperature reservoirs. Its efficiency is a theoretical upper limit for all heat engines.
- Refrigerators and Heat Pumps: Devices that transfer heat from a colder body to a hotter body, requiring external work input. Their performance is measured by the coefficient of performance (COP).
Formulas & Results
Thermal Expansion:
- Linear Expansion: $L_T = L_0 (1 + \alpha \Delta T)$
- Superficial Expansion: $A_T = A_0 (1 + \beta \Delta T)$, where $\beta = 2\alpha$
- Volumetric Expansion: $V_T = V_0 (1 + \gamma \Delta T)$, where $\gamma = 3\alpha$
- Relationship between coefficients: $\alpha : \beta : \gamma = 1 : 2 : 3$
Calorimetry:
- Heat absorbed/released: $Q = mc\Delta T$ (for change in temperature)
- Latent heat (phase change): $Q = mL$
- Water Equivalent: $W = mc_w$
- Principle of Calorimetry: Heat Lost = Heat Gained ($m_1c_1\Delta T_1 = m_2c_2\Delta T_2$)
Heat Transfer:
- Conduction: $\frac{dQ}{dt} = -KA\frac{dT}{dx}$ (Fourier's Law)
- Thermal Resistance: $R_{th} = \frac{L}{KA}$
- Series combination: $R_{eq} = R_1 + R_2 + ...$
- Parallel combination: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$
- Convection: $\frac{dQ}{dt} = hA\Delta T$ (Newton's Law of Cooling, typically used for combined convection and radiation)
- Radiation (Stefan-Boltzmann Law): $\frac{dQ}{dt} = e\sigma A T^4$
- Net radiated power: $P_{net} = e\sigma A (T^4 - T_0^4)$ (where $T_0$ is surrounding temperature)
- Newton's Law of Cooling (empirical): $\frac{dT}{dt} = -k(T - T_{surr})$
First Law of Thermodynamics:
- $\Delta U = Q - W$ or $Q = \Delta U + W$ (where $W$ is work done by the system)
- Work done by gas: $W = \int P dV$ (for quasi-static process)
- Work done in specific processes:
- Isobaric: $W = P(V_f - V_i)$
- Isochoric: $W = 0$
- Isothermal: $W = nRT \ln\left(\frac{V_f}{V_i}\right) = nRT \ln\left(\frac{P_i}{P_f}\right)$
- Adiabatic: $W = \frac{P_iV_i - P_fV_f}{\gamma - 1} = \frac{nR(T_i - T_f)}{\gamma - 1}$
Internal Energy ($U$) and Specific Heats:
- Internal energy of an ideal gas: $U = n C_V T$
- For a monatomic gas: $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$, $\gamma = \frac{5}{3}$
- For a diatomic gas: $C_V = \frac{5}{2}R$, $C_P = \frac{7}{2}R$, $\gamma = \frac{7}{5}$ (at moderate temperatures)
- For a polyatomic gas: $C_V = 3R$, $C_P = 4R$, $\gamma = \frac{4}{3}$ (approximation)
- Mayer's Formula: $C_P - C_V = R$
- Relationship for adiabatic process: $PV^\gamma = \text{constant}$, $TV^{\gamma-1} = \text{constant}$, $P^{1-\gamma}T^\gamma = \text{constant}$
Second Law of Thermodynamics:
- Efficiency of a heat engine: $\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$
- Efficiency of Carnot engine: $\eta_{Carnot} = 1 - \frac{T_C}{T_H}$ (where temperatures $T_H, T_C$ are in Kelvin)
- Coefficient of Performance (COP) of refrigerator: $COP_R = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C} = \frac{T_C}{T_H - T_C}$
- Coefficient of Performance (COP) of heat pump: $COP_{HP} = \frac{Q_H}{W} = \frac{Q_H}{Q_H - Q_C} = \frac{T_H}{T_H - T_C}$
- Relationship: $COP_{HP} = 1 + COP_R$
- Change in Entropy: $\Delta S = \int \frac{dQ_{rev}}{T}$
- For an ideal gas undergoing a reversible process: $\Delta S = nC_V \ln\left(\frac{T_f}{T_i}\right) + nR \ln\left(\frac{V_f}{V_i}\right)$
- For an isolated system, $\Delta S \ge 0$ (Entropy can only increase or remain constant).
Worked Examples
Example 1: Thermal Expansion A steel rod has a length of 1.0 m at $20^\circ C$. The coefficient of linear expansion for steel is $1.2 \times 10^{-5} / ^\circ C$. What is the change in its length when its temperature is raised to $120^\circ C$?
Solution: Given: $L_0 = 1.0 m$, $T_i = 20^\circ C$, $T_f = 120^\circ C$, $\alpha = 1.2 \times 10^{-5} / ^\circ C$. We use the formula for linear expansion: $\Delta L = L_0 \alpha \Delta T$. First, calculate the change in temperature: $\Delta T = T_f - T_i = 120^\circ C - 20^\circ C = 100^\circ C$. Now, substitute the values into the formula: $\Delta L = (1.0 \text{ m}) \times (1.2 \times 10^{-5} / ^\circ C) \times (100^\circ C)$ $\Delta L = 1.2 \times 10^{-3} \text{ m}$ The change in length is $1.2 \text{ mm}$.
Example 2: First Law of Thermodynamics A gas is enclosed in a cylinder with a movable piston. When 200 J of heat is added to the gas, the piston moves, and the gas does 80 J of work on the surroundings. Calculate the change in the internal energy of the gas.
Solution: Given: Heat added to the system $Q = +200 \text{ J}$ (since heat is added to the system). Work done by the system $W = +80 \text{ J}$. According to the First Law of Thermodynamics: $\Delta U = Q - W$. Substitute the given values: $\Delta U = 200 \text{ J} - 80 \text{ J}$ $\Delta U = 120 \text{ J}$ The change in the internal energy of the gas is 120 J.
Example 3: Carnot Engine Efficiency A Carnot engine operates between a hot reservoir at $227^\circ C$ and a cold reservoir at $27^\circ C$. a) Calculate the efficiency of the engine. b) If the engine absorbs 600 J of heat from the hot reservoir in each cycle, how much work does it perform?
Solution: First, convert temperatures to Kelvin: $T_H = 227^\circ C + 273 = 500 \text{ K}$ $T_C = 27^\circ C + 273 = 300 \text{ K}$
a) Efficiency of a Carnot engine: $\eta = 1 - \frac{T_C}{T_H}$ $\eta = 1 - \frac{300 \text{ K}}{500 \text{ K}}$ $\eta = 1 - 0.6 = 0.4$ The efficiency of the engine is 0.4 or 40%.
b) The efficiency is also given by $\eta = \frac{W}{Q_H}$. We are given $Q_H = 600 \text{ J}$ and we found $\eta = 0.4$. So, $W = \eta \times Q_H$ $W = 0.4 \times 600 \text{ J}$ $W = 240 \text{ J}$ The engine performs 240 J of work in each cycle.
Common Mistakes
- Units: Not converting temperatures to Kelvin for efficiency and COP calculations. For specific heat $mc\Delta T$, $\Delta T$ can be in $^\circ C$ or Kelvin, but for other formulas, absolute temperature in Kelvin is often mandatory.
- Sign Conventions: Confusing the sign convention for work done. $W$ is positive when work is done by the system (expansion) and negative when work is done on the system (compression). Heat added to the system is positive, heat removed is negative.
- Ideal vs. Real Gases: Applying ideal gas laws and specific heat relations where real gas behavior might be implied or relevant (though JEE Main usually focuses on ideal gases).
- Irreversible Processes: Assuming formulas for reversible processes (e.g., $Q = nRT \ln(V_f/V_i)$ for isothermal work) apply to irreversible processes without modification.
- Adiabatic vs. Isothermal: Mixing up the conditions and equations for adiabatic ($PV^\gamma = \text{constant}$) and isothermal ($PV = \text{constant}$) processes.
- Heat vs. Temperature: Confusing heat (energy transfer) with temperature (measure of hotness).
- Latent Heat vs. Specific Heat: Using $mc\Delta T$ for phase changes or $mL$ for temperature changes.
Quick Revision Checklist
- Understand the definitions of thermal expansion coefficients ($\alpha, \beta, \gamma$) and their relations.
- Master calorimetry principles, including specific heat and latent heat calculations.
- Know the mechanisms of heat transfer: conduction, convection, radiation, and their respective formulas.
- Memorize the First Law of Thermodynamics and its application to different thermodynamic processes (isobaric, isochoric, isothermal, adiabatic).
- Recall the sign conventions for heat (Q) and work (W).
- Remember the expressions for internal energy and work done for an ideal gas in various processes.
- Know the relation between specific heats ($C_P$, $C_V$) for monatomic, diatomic, and polyatomic gases, and Mayer's formula.
- Understand the Carnot cycle, its efficiency, and the Carnot theorem.
- Be familiar with the concepts of COP for refrigerators and heat pumps.
- Review the concept of entropy and its implications for spontaneous processes (Second Law of Thermodynamics).
