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A system goes from $P$ to $Q$ by two different paths in the $p$ $V$ diagram as shown in figure. Heat given to the system in path 1 is $1000 \mathrm{~J}$. The work done by the system along path 1 is more than path 2 by $100 \mathrm{~J}$. What is the heat exchanged by the system in path 2 ?
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Verified Answer
For path 1 , as given that
Heat $Q_1=+1000 \mathrm{~J}$
Work done $\left(W_1-W_2\right)=100$
For path 2,
Work done $\left(W_2\right)=\left(W_1-100\right) \mathrm{J}$
Heat given $Q_2=$ ?
where $W_1=$ Work done through path 1
$W_2=$ Work done through path 2
As change in internal energy between two states for different path 1 and 2 are same.
$$
\begin{aligned}
&\Delta U=Q_1-W_1=Q_2-W_2 \\
&1000-W_1=Q_2-\left(W_1-100\right) \\
&Q_2=1000-100=900 \mathrm{~J}
\end{aligned}
$$
Heat $Q_1=+1000 \mathrm{~J}$
Work done $\left(W_1-W_2\right)=100$
For path 2,
Work done $\left(W_2\right)=\left(W_1-100\right) \mathrm{J}$
Heat given $Q_2=$ ?
where $W_1=$ Work done through path 1
$W_2=$ Work done through path 2
As change in internal energy between two states for different path 1 and 2 are same.
$$
\begin{aligned}
&\Delta U=Q_1-W_1=Q_2-W_2 \\
&1000-W_1=Q_2-\left(W_1-100\right) \\
&Q_2=1000-100=900 \mathrm{~J}
\end{aligned}
$$
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