The ideal-gas law relates the state variables $p$, $V$, and $T$, but it tells us nothing about how these variables change if we do something to the gas. The first law of thermodynamics is an additional law that gases must obey, one focused specifically on the process by which a gas is changed. In general, you need to consider both laws to solve thermodynamics problems about gases.

There are three ideal-gas processes in which one of the three terms in the first law—$\text{Δ}{E}_{\text{th}}$, $W$, or Q—is zero:

• Isothermal process $\left(\text{}\text{}\text{Δ}{E}_{th}=0\right)$: If the temperature of a gas doesn’t change, neither does its thermal energy. In this case, the first law is $W+Q=0$. For example, a gas can be compressed $(W>0)$ without a temperature increase if all the energy is returned to the environment as heat $(Q<0$ for heat energy leaving the system). An isothermal process exchanges work and heat (one entering the system, the other leaving) without changing the gas temperature. But pressure and volume do change. Equation 19.13 for the work done in an isothermal process, together with the ideal-gas law, will allow you to calculate what happens to $p$ and $V$. FIGURE 19.12a shows an energy bar chart for an isothermal process.

• Isochoric process $\left(W=0\right)$: We saw in Section 19.2 that work is done on a gas when its volume changes. An isochoric process has $\text{Δ}V=0;$ thus no work is done and the first law can be written $\text{Δ}{E}_{\text{th}}=Q$. This is a process in which the system is mechanically isolated from its environment. Heating (or cooling) the gas increases (or decreases) its thermal energy, so the temperature goes up (or down). You’ll learn later in this chapter how to calculate the temperature change $\text{Δ}T$. After using the first law to find $\text{Δ}T$, then you can use the ideal-gas law to calculate the pressure change. FIGURE 19.12b is an energy bar chart for a cooling process that lowers the temperature.

• Adiabatic process $\left(Q=0\right)$: A process in which no heat is transferred—perhaps the system is extremely well insulated—is called an adiabatic process. The system is thermally isolated from its environment. With $Q=0$, the first law is $\text{Δ}{E}_{\text{th}}=W$. But just because there’s no heat does not mean that the temperature remains constant. Compressing an insulated gas—a mechanical interaction with $W>0$—raises its temperature! Similarly, an adiabatic expansion (one with no heat exchanged) lowers the gas temperature. $Q=0$ does not mean $\text{Δ}T=0.$ We’ll examine adiabatic processes and their pV curves later in the chapter, but notice that all three state variables $p$, $V$, and $T$ change during an adiabatic process. FIGURE 19.12c is an energy bar chart for an adiabatic process with an increasing temperature.

  FIGURE 19.12 First-law bar charts for three special ideal-gas processes.

  

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