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Step-by-Step Solution
Step 1: Identify the thermodynamic process
The process described is an isobaric expansion of an ideal gas, meaning the pressure remains constant throughout.
Step 2: Write the expression for work done in an isobaric process
For an isobaric process, the work done by the gas is given by
$w = p\,\Delta V$.
However, for an ideal gas undergoing a temperature change of $\Delta T$, the pressure $p$ can be written (for an isobaric process) using the ideal gas law:
$p = \frac{nR\,T}{V}$.
Combining these relations and noting that $\Delta V$ corresponds to a change $\Delta T$ at constant $p$, we obtain:
$w = nR\,\Delta T$.
Step 3: Express the heat supplied in an isobaric process
In an isobaric process, the heat supplied $\Delta H$ is determined by the molar heat capacity at constant pressure. From the given information, if $C_V$ is the total constant-volume heat capacity for the $n$ moles of the gas, then the total constant-pressure heat capacity for the $n$ moles is $C_V + nR$. Hence, for a temperature change $\Delta T$:
$\Delta H = (C_V + nR)\,\Delta T.
Step 4: Compute the required ratio of work done to heat supplied
The ratio asked in the question is
$\frac{w}{\Delta H}$.
Substituting $w = nR\,\Delta T$ and $\Delta H = (C_V + nR)\,\Delta T$:
$\frac{w}{\Delta H}
=
\frac{nR\,\Delta T}{(C_V + nR)\,\Delta T}
=
\frac{nR}{C_V + nR}.
Final Answer
$\displaystyle \frac{nR}{C_V + nR}.
