JEE Physics Thermodynamics Notes Set 05

Thermodynamic System and State

A thermodynamic system exchanges heat and work with its surroundings. Pressure, volume, temperature and internal energy are state functions: they depend only on the present state of the system. Heat and work depend on the path. Work done by a gas is \( W=\int P\,dV \), equal to the area under the curve on a P-V diagram, positive for expansion and negative for compression. A cyclic process returns the gas to its starting state, so \( \Delta U=0 \) and the net work equals the area enclosed by the cycle (positive for clockwise).

First Law of Thermodynamics

Energy is conserved: the heat given to a system is used partly to raise its internal energy and partly to do external work.

\( \delta Q=dU+\delta W,\qquad \Delta U=nC_v\Delta T\ \text{(ideal gas, any process)} \)

Internal energy of an ideal gas depends only on temperature. For a liquid boiling at atmospheric pressure, \( \Delta U=Q-P_0(V_{vapour}-V_{liquid}) \).

Specific Heats and Processes

For an adiabatic change of an ideal gas: \( PV^{\gamma}=TV^{\gamma-1}=P^{1-\gamma}T^{\gamma}=\text{const} \). The adiabatic curve is steeper than the isothermal by the factor \( \gamma \) (slope \( -\gamma P/V \)). Sudden compression or expansion is adiabatic. For a general polytropic process \( PV^{n}= \) const: \( W=\dfrac{nR\,\Delta T}{1-n} \) and \( C=C_v+\dfrac{R}{1-n} \). When the bulk kinetic energy of a moving gas is stopped suddenly, it is converted into internal energy: \( \tfrac12Mv^{2}=C_v\Delta T \) per mole.

For a process in which \( P\propto V \) the work from temperature \( T_1 \) to \( T_2 \) is \( \tfrac12R(T_2-T_1) \) per mole; in general for \( P\propto V^{m} \) it is \( \dfrac{R\Delta T}{m+1} \). To find the maximum temperature on a straight-line P-V path, find where \( PV \) is largest, which is usually inside the path and not at an end point.

Second Law, Heat Engines and Refrigerators

The second law forbids a cyclic engine that converts all the heat it takes into work, and forbids heat flowing unaided from a cold body to a hot body. For an engine taking \( Q_1 \) from the source, rejecting \( Q_2 \) to the sink and doing work \( W=Q_1-Q_2 \):

\( \eta=\dfrac WQ_1=1-\dfrac{Q_2}{Q_1} \)

The Carnot cycle consists of two isothermal and two adiabatic steps and is the most efficient engine between two temperatures: \( \eta=1-\dfrac{T_2}{T_1} \) (kelvin). Its efficiency is never 1 because that would need a sink at absolute zero. A refrigerator is an engine run backwards. Its coefficient of performance is \( \beta=\dfrac{Q_2}{W}=\dfrac{Q_2}{Q_1-Q_2}=\dfrac{T_2}{T_1-T_2}=\dfrac{1-\eta}{\eta} \).

Entropy

Entropy measures disorder. For a reversible change \( dS=\dfrac{dQ}{T} \); for heating a body of heat capacity \( C \) from \( T_1 \) to \( T_2 \) the change is \( \Delta S=C\ln\dfrac{T_2}{T_1} \), and for a phase change it is \( \pm\dfrac{mL}{T} \). For an ideal gas \( \Delta S=nC_v\ln\dfrac{T_2}{T_1}+nR\ln\dfrac{V_2}{V_1} \). Entropy is a state function, and the entropy of an isolated system never decreases.

Useful Results

1. Mixing two gases in an insulated container without work: total internal energy is conserved, so \( T=\dfrac{n_1T_1+n_2T_2}{n_1+n_2} \) for the same type of gas.
2. Adiabatic oscillations of a piston of mass \( M \) and area \( A \) on a gas of volume \( V_0 \) and pressure \( P_0 \) have \( \omega^{2}=\dfrac{\gamma P_0A^{2}}{MV_0} \).
3. Free expansion into vacuum: \( Q=W=\Delta U=0 \), so the temperature of an ideal gas is unchanged.
4. The first law holds for every process, cyclic or not.