21.3.1 Ideal-Gas Summary

We’ve learned a lot about ideal gases in the last three chapters. All gas processes obey the ideal-gas law $p V = n R T$ and the first law of thermodynamics $Δ E_{th} = Q - W_{s}$ . TABLE 21.1 summarizes the results for specific gas processes. This table shows $W_{s}$ , the work done by the system, so the signs are opposite those in Chapter 19.

TABLE 21.1 Summary of ideal-gas processes

Process	Gas law	Work W_s	Heat Q	Thermal energy
Isochoric	$p_{i} / T_{i} = p_{f} / T_{f}$	0	$n C_{V} Δ T$	$Δ E_{th} = Q$
Isobaric	$V_{i} / T_{i} = V_{f} / T_{f}$	$p Δ V$	$n C_{P} Δ T$	$Δ E_{th} = Q - W_{s}$
Isothermal	$p_{i} V_{i} = p_{f} V_{f}$	$n R T ln (V_{f} / V_{i})$	$Q = W_{s}$	$Δ E_{th} = 0$
		$p V ln (V_{f} / V_{i})$
Adiabatic	$p_{i} V_{i}^{γ} = p_{f} V_{f}^{γ}$	$(p_{f} V_{f} - p_{i} V_{i}) / (1 - γ)$	0	$Δ E_{th} = - W_{s}$
	$T_{i} V_{i}^{γ - 1} = T_{f} V_{f}^{γ - 1}$	$- n C_{V} Δ T$
Any	$p_{i} V_{i} / T_{i} = p_{f} V_{f} / T_{f}$	area under curve		$Δ E_{th} = n C_{V} Δ T$

There is one entry in this table that you haven’t seen before. The expression

W_{s} = \frac{p_{f} V_{f} - p_{i} V_{i}}{1 - γ} (work in an adiabatic process)

(21.10)

for the work done in an adiabatic process follows from writing $W_{s} = - Δ E_{th} =_{} - n C_{V} Δ T$ , which you learned in Chapter 19, then using $Δ T = Δ (p V) / n R$ and the definition of $γ$ . The proof will be left for a homework problem.

You learned in Chapter 20 that the thermal energy of an ideal gas depends only on its temperature. TABLE 21.2 lists the thermal energy, molar specific heats, and specific heat ratio $γ = C_{P} / C_{V}$ for monatomic and diatomic gases.

TABLE 21.2 Properties of monatomic and diatomic gases

	Monatomic	Diatomic
$E_{th}$	$\frac{3}{2} n R T$	$\frac{5}{2} n R T$
$C_{V}$	$\frac{3}{2} R$	$\frac{5}{2} R$
$C_{P}$	$\frac{5}{2} R$	$\frac{7}{2} R$
$γ$	$\frac{5}{3} = 1.67$	$\frac{7}{5} = 1.40$