A diatomic gas has an initial internal energy of 80 cal. A work of 18 cal is done on the gas, and the gas releases heat energy of 42 J. The final internal energy of the gas is:
Show Hint
Make sure to convert all energy values to the same units before performing calculations. Remember that \( 1 \, \text{cal} = 4.18 \, \text{J} \).
Step 1: Convert all quantities to consistent units (Joules)
We use the conversion factor: \SI{1}{cal} = \SI{4.184}{J}
\begin{itemize}
\item Initial internal energy ($U_i$):
\[
U_i = \SI{80}{cal} \times \SI{4.184}{J/cal} = \SI{334.72}{J}
\]
\item Work done on the gas ($W$):
\[
W = \SI{18}{cal} \times \SI{4.184}{J/cal} = \SI{75.312}{J}
\]
\item Heat released by the gas ($Q$):
\[
Q = -\SI{42}{J} \quad \text{(negative sign indicates heat released)}
\]
\end{itemize}
Step 2: Apply the First Law of Thermodynamics
The First Law states:
\[
\Delta U = Q + W
\]
Substituting the values:
\[
\Delta U = -\SI{42}{J} + \SI{75.312}{J} = \SI{33.312}{J}
\]
Step 3: Calculate the final internal energy
\[
U_f = U_i + \Delta U = \SI{334.72}{J} + \SI{33.312}{J} = \SI{368.032}{J}
\]
Step 4: Compare with the given options
The closest option to \SI{368.032}{J} is:
\begin{center}
Option 2: \SI{369.6}{J}
\end{center}
Final Answer:
The final internal energy of the gas is \boxed{2}.
