Thermodynamics — JEE Mains Physics

Formulas (2)

• The first law and its sign convention · First law (work done by the gas positive)
  $$\Delta Q = \Delta U + W \qquad W = \int_{V_1}^{V_2} P\,dV$$
• Energy balances: phase changes, rates and pistons · Energy balance at a steady pressure
  $$\Delta U = Q - P\Delta V \qquad \frac{dU}{dt} = \frac{dQ}{dt} - \frac{dW}{dt}$$

Watch out for (4)

• Two sign conventions for work
  → The first law and its sign convention
• Heat added does not mean hotter
  → The first law and its sign convention
• Subtract the work against the atmosphere
  → Energy balances: phase changes, rates and pistons
• Convert volumes before multiplying
  → Energy balances: phase changes, rates and pistons

Formulas (3)

• Internal energy from Cv and the degrees of freedom · Internal energy and heat capacities
  $$\Delta U = nC_V\Delta T \qquad C_V = \frac{f}{2}R \qquad C_P - C_V = R \qquad \gamma = 1 + \frac{2}{f}$$
• How heat splits at constant pressure · Isobaric heat split
  $$Q : \Delta U : W = C_P : C_V : R \qquad Q = \frac{\gamma}{\gamma - 1}W \qquad \Delta U = \frac{Q}{\gamma}$$
• Molar heat capacity of any process · Molar heat capacity of a process
  $$C = \frac{Q}{n\Delta T} = C_V + \frac{W}{n\Delta T} \qquad C_{\text{poly}} = C_V + \frac{R}{1 - x}$$

Watch out for (7)

• ΔU uses Cv even at constant pressure
  → Internal energy from Cv and the degrees of freedom
• Each vibrational mode counts twice
  → Internal energy from Cv and the degrees of freedom
• Calories and joules do not mix
  → Internal energy from Cv and the degrees of freedom
• Only at constant volume does all the heat stay inside
  → How heat splits at constant pressure
• "Rotates but does not oscillate" means f = 5
  → How heat splits at constant pressure
• A heat capacity can be negative
  → Molar heat capacity of any process
• A fraction of Q, not of ΔU
  → Molar heat capacity of any process

Formulas (2)

• Work as the area under a P–V path · Work done by the gas
  $$W = \int_{V_1}^{V_2} P\,dV \qquad W_{\text{iso}} = nRT\ln\frac{V_2}{V_1}$$
• Processes given by an equation such as PV^x = constant · Polytropic process PV^x = constant
  $$W = \frac{P_1V_1 - P_2V_2}{x - 1} = \frac{nR(T_1 - T_2)}{x - 1} \qquad T \propto V^{1 - x}$$

Reference tables (1)

Watch out for (6)

• Isothermal does not mean no heat
  → The standard processes and their W, Q and ΔU
• An isochoric line must pass through the origin
  → The standard processes and their W, Q and ΔU
• A leg to the left is negative work
  → Work as the area under a P–V path
• Read each axis on its own scale
  → Work as the area under a P–V path
• x = 1 needs the logarithm
  → Processes given by an equation such as PV^x = constant
• The index belongs to the process, γ to the gas
  → Processes given by an equation such as PV^x = constant

Formulas (2)

• The adiabatic relations PV^γ, TV^(γ−1) and P^(1−γ)T^γ · Adiabatic relations
  $$PV^{\gamma} = \text{const} \qquad TV^{\gamma - 1} = \text{const} \qquad P^{1 - \gamma}T^{\gamma} = \text{const}$$
• Work done in an adiabatic process · Adiabatic work (done by the gas)
  $$W = \frac{nR(T_1 - T_2)}{\gamma - 1} = \frac{P_1V_1 - P_2V_2}{\gamma - 1} = -\Delta U$$

Reference tables (1)

Watch out for (6)

• A sudden change is adiabatic
  → The adiabatic relations PV^γ, TV^(γ−1) and P^(1−γ)T^γ
• Kelvin in TV^(γ−1)
  → The adiabatic relations PV^γ, TV^(γ−1) and P^(1−γ)T^γ
• The sign in a compression
  → Work done in an adiabatic process
• Divide by γ − 1, not by γ
  → Work done in an adiabatic process
• Adiabatic is not isothermal
  → True and false statements about adiabatic processes
• The steeper curve is the adiabat
  → True and false statements about adiabatic processes

Formulas (2)

• Net work of a cycle as the enclosed area · Net work of a cycle
  $$W_{\text{net}} = Q_{\text{net}} = \oint P\,dV \qquad W_{\text{ellipse}} = \pi ab$$
• A cycle worked leg by leg · Net work leg by leg
  $$W_{\text{net}} = \sum_{\text{legs}} W_i = \sum_{\text{legs}} Q_i$$

Watch out for (4)

• Half the width, not the width
  → Net work of a cycle as the enclosed area
• The direction sets the sign
  → Net work of a cycle as the enclosed area
• Find the corner pressure before using PΔV
  → A cycle worked leg by leg
• The volume ratio in the right order
  → A cycle worked leg by leg

Formulas (3)

• Carnot efficiency η = 1 − T₂/T₁ · Carnot efficiency
  $$\eta = \frac{W}{Q_1} = 1 - \frac{T_2}{T_1}$$
• Heat, work and temperature in engines and refrigerators · Engine and refrigerator
  $$Q_1 = W + Q_2 \qquad \frac{Q_2}{Q_1} = \frac{T_2}{T_1} \qquad \text{COP} = \frac{Q_2}{W} = \frac{T_2}{T_1 - T_2}$$
• Engines in series and entropy · Engines in series and entropy
  $$\eta = \eta_1 + \eta_2 - \eta_1\eta_2 = 1 - \frac{T_3}{T_1} \qquad \Delta S = ms\ln\frac{T_2}{T_1}$$

Watch out for (6)

• Kelvin, not Celsius
  → Carnot efficiency η = 1 − T₂/T₁
• Per cent or percentage points?
  → Carnot efficiency η = 1 − T₂/T₁
• Efficiency divides by the heat taken in
  → Heat, work and temperature in engines and refrigerators
• COP is not an efficiency
  → Heat, work and temperature in engines and refrigerators
• Efficiencies in series do not add
  → Engines in series and entropy
• Mass units in ΔS
  → Engines in series and entropy