Question

A system is taken from state a to state b along the three paths shown in Fig. Q19.23. (a) Along which path is the work done by the system the greatest? The least? (b) If${\mathit{U}}_{\mathbf{b}}\mathbf{>}{\mathit{U}}_{\mathbf{a}}$ , along which path is the absolute value of the heat transfer,$\left|Q\right|$ , the greatest? For this path, is heat absorbed or liberated by the system? Explain

Short answer

a) For the system taken from a to b along the three paths, work done by path 1 is greatest.

b) For the path 3 the absolute value of heat transfer$\left|Q\right|$ is greatest, also the heat $∆Q$is also positive and the system absorbed the heat.

Step-by-step solution

Determining the greatest work done by the paths.

The area under the curve for$\rho V$graph gives the work done in process.

Therefore, for path 1 the work done is greatest as the area is maximum under the curve.

The work done is least for Path 3 , as the area under the curve is minimum.

$W_1=$greatest

$W_2=$least

Determining the path with the greatest amount of heat transfer.

According to the first law of thermodynamics$∆Q=∆U+W$

We, know that$W_1>W_3$and$U_b>U_a$ .

$$\begin{gathered} ∆U=Ub-Ua \\ ∆U>0 \end{gathered}$$

Since,$W_3$ is greatest the path 3 gives the greatest$\left|Q\right|$ ,

$\Rightarrow {\left|∆Q\right|}_{max}=∆U+W_3$

Therefore, for path 3 ,$∆U$ and$W_3$ both are positive. The heat$∆Q$ is also positive and the system absorbed the heat.