2.6 Isothermal and Adiabatic Expansion for Ideal Gasses and Engine Cycles

(September 1, 2023)

We will consider and ideal gas with constant specific heat, so $U=C_{V}T$ . Then $\gamma=C_{p}/C_{V}=1+Nk_{B}/C_{V}$ is also constant.

Isothermal Expansion: For an isothermal expansion of a gas there is no change in temperature due to influx of heat compensating the expansion. For an ideal gas you should be able to show that

\Delta U=0\qquad Q=W_{\rm out}=\int_{i}^{f}pdV=N{k_{\scriptscriptstyle B}T}\ln% (V_{f}/V_{i})

(2.51)

Adiabatic Expansion: For an adiabatic expansion $Q=0$ , and there is a change in temperature as the system expands.

\Delta U=-\int_{i}^{f}pdV

(2.52)

You should be able to show that during the expansion

pV^{\gamma}={\rm const}\qquad\mbox{or}\qquad\frac{p_{i}}{p_{f}}=\left(\frac{V_% {f}}{V_{i}}\right)^{\gamma}

(2.53)

Or, since $p=N{k_{\scriptscriptstyle B}T}/V$ , we have

TV^{\gamma-1}={\rm const}\qquad\mbox{or}\qquad\frac{T_{i}}{T_{f}}=\left(\frac{% V_{f}}{V_{i}}\right)^{\gamma-1}

(2.54)

Using the fact that $U=C_{V}T$ , one can use Eq. (2.54) to find the change in energy $\Delta U=C_{V}\Delta T$ .

Engines: In a car engine we burn gasoline. This involves chemical transitions of atomic levels, each of which provide somewhat less than an electron-volt of energy. Since there are of order an Avogadro’s number of such transitions we typically get

N_{A}\,{\rm(eV)}\simeq 100\,{\rm kJ}

(2.55)

of energy for every mole. The constant $N_{A}\,{\rm eV}$ is known as Faraday’s constant. This is a lot of energy which is why internal combustion engines have taken over.

In a given closed cycle of an engine we have

\cancelto{0}{\Delta U}=Q-W_{\rm out}

(2.56)

The net heat $Q$ involves positive inputs to the engine $Q_{\rm in}$ , and exhaust $Q_{\rm out}$ which is negative, $Q_{\rm out}=-|Q_{\rm out}|$ . In total

Q=Q_{\rm in}+Q_{\rm out}=Q_{\rm in}-|Q_{\rm out}|.

The efficiency is

\eta=\frac{W_{\rm out}}{Q_{\rm in}}=1-\frac{|Q_{\rm out}|}{Q_{\rm in}}

(2.57)