JEE Physics Thermodynamics Notes Set 01

Basic Terms

A system is the part of the universe under study and everything else is the surroundings. State variables such as pressure P, volume V, temperature T and internal energy U describe the state. They are state functions: their change depends only on the initial and final states. Heat Q and work W depend on the path followed.
The zeroth law says that two bodies each in thermal equilibrium with a third body are in equilibrium with each other, which is the basis of temperature measurement. The internal energy of an ideal gas depends only on temperature: \( U = \dfrac{f}{2}nRT \) and \( \Delta U = nC_V\Delta T \) in every process.

Work and the First Law

When a gas changes its volume the work done by it is \( W = \int P\,dV \), the area under the P-V curve. W is positive in expansion and negative in compression. In a closed loop the net work equals the area enclosed: positive for a clockwise loop and negative for an anticlockwise one.
First law: \( Q = \Delta U + W \). Heat supplied to a system goes partly into raising its internal energy and partly into the work it does. It is the law of conservation of energy. Sign convention used here: Q is positive when heat enters the system and W is positive when the system does work.

Thermodynamic Processes

Adiabatic relations: \( PV^\gamma = \text{constant} \), \( TV^{\gamma-1} = \text{constant} \) and \( P^{1-\gamma}T^\gamma = \text{constant} \). On a P-V diagram the adiabatic curve is steeper than the isothermal curve through the same point, with slope \( \gamma \) times larger. The bulk modulus of a gas is P for an isothermal change and \( \gamma P \) for an adiabatic change. In a polytropic process \( PV^n = \text{constant} \) the molar heat capacity is \( C = C_V + \dfrac{R}{1-n} \).
In free expansion into a vacuum no work is done and, if the container is insulated, \( Q = 0 \) and \( \Delta U = 0 \): an ideal gas keeps its temperature. Sudden compression or expansion is nearly adiabatic, so a pumped tyre warms up and a gas that escapes quickly from a bottle cools.

Specific Heats of Gases

Because a gas expands when heated at constant pressure, \( C_P > C_V \). Mayer's relation: \( C_P - C_V = R \). With f degrees of freedom \( C_V = \dfrac{f}{2}R \) and \( \gamma = \dfrac{C_P}{C_V} = 1 + \dfrac{2}{f} \). At constant pressure the fractions of the heat used are \( \dfrac{\Delta U}{Q} = \dfrac{1}{\gamma} \) and \( \dfrac{W}{Q} = 1 - \dfrac{1}{\gamma} \): 60 % and 40 % for a monatomic gas, 71 % and 29 % for a diatomic gas. The specific heat is zero in an adiabatic process and infinite in an isothermal one.

Second Law, Heat Engines and Refrigerators

Kelvin-Planck statement: no cyclic engine can take heat from a single reservoir and convert all of it into work. Clausius statement: heat cannot flow by itself from a colder body to a hotter body. Both statements are equivalent and they set the direction of natural processes. Real processes are irreversible because of friction and other dissipative effects.
A heat engine takes \( Q_1 \) from a hot source, does work \( W = Q_1 - Q_2 \) and rejects \( Q_2 \) to a sink. Its efficiency is \( \eta = \dfrac{W}{Q_1} = 1 - \dfrac{Q_2}{Q_1} \). A refrigerator works in reverse, with coefficient of performance \( \beta = \dfrac{Q_2}{W} = \dfrac{Q_2}{Q_1 - Q_2} \).
The Carnot engine uses two isothermal and two adiabatic steps, is reversible, and has the greatest possible efficiency between two temperatures: \[ \eta = 1 - \dfrac{T_2}{T_1}, \qquad \beta = \dfrac{T_2}{T_1 - T_2}. \] No engine working between the same two reservoirs can be more efficient. The efficiency is raised by raising the source temperature or lowering the sink temperature; it is always below 100 % because the sink cannot be at absolute zero.

Worked Examples

Example 1. A gas absorbs 400 J of heat and does 150 J of work. Find the change in its internal energy. If it is then compressed isothermally, with 200 J of work done on it, how much heat flows out?
Answer:
\( \Delta U = Q - W = 400 - 150 = \) 250 J.
In an isothermal process \( \Delta U = 0 \), so \( Q = W = -200 \) J: 200 J of heat is released.

Example 2. A monatomic ideal gas at 300 K is compressed adiabatically to 1/8 of its volume. Find the final temperature.
Answer:
\( \gamma = \dfrac{5}{3} \) and \( TV^{\gamma-1} = \text{constant} \).
\( T_2 = 300\times8^{2/3} = 300\times4 = \) 1200 K.

Example 3. 2 mol of a diatomic gas is heated at constant pressure from 300 K to 400 K. Find Q, W and \( \Delta U \) (R = 8.31 J/mol K).
Answer:
\( Q = nC_P\Delta T = 2\times\dfrac{7}{2}\times8.31\times100 = \) 5817 J.
\( W = nR\Delta T = 2\times8.31\times100 = \) 1662 J.
\( \Delta U = Q - W = \) 4155 J.

Example 4. A Carnot engine works between 527°C and 127°C and takes 2000 J of heat per cycle from the source. Find its efficiency, the work per cycle and the heat rejected.
Answer:
\( T_1 = 800 \) K, \( T_2 = 400 \) K, so \( \eta = 1 - \dfrac{400}{800} = \) 50 %.
\( W = 0.5\times2000 = \) 1000 J; \( Q_2 = 2000 - 1000 = \) 1000 J.