Step 1: Understanding the Concept:
An isobaric process is a thermodynamic process in which the pressure remains constant.
When heat is supplied to a gas at constant pressure, it performs work by expanding and also increases its internal energy.
We need to find the ratio of the total heat supplied ($Q$) to the work done ($W$).
Step 2: Key Formula or Approach:
For an ideal gas of $n$ moles undergoing a temperature change $\Delta T$:
Heat supplied at constant pressure:
\[ Q = n C_P \Delta T \]
Work done by the gas at constant pressure:
\[ W = P \Delta V = n R \Delta T \]
where $C_P$ is the molar specific heat at constant pressure and $R$ is the universal gas constant.
Mayer's relation gives:
\[ C_P - C_V = R \]
and the adiabatic index is:
\[ \gamma = \frac{C_P}{C_V} \]
Step 3: Detailed Explanation:
The required ratio is $\frac{Q}{W}$.
Substitute the expressions for $Q$ and $W$:
\[ \frac{Q}{W} = \frac{n C_P \Delta T}{n R \Delta T} \]
Cancel the common terms $n$ and $\Delta T$:
\[ \frac{Q}{W} = \frac{C_P}{R} \]
Now, use Mayer's relation ($R = C_P - C_V$) to substitute for $R$:
\[ \frac{Q}{W} = \frac{C_P}{C_P - C_V} \]
To introduce $\gamma$ into the expression, divide the numerator and the denominator by $C_V$:
\[ \frac{Q}{W} = \frac{\frac{C_P}{C_V}}{\frac{C_P}{C_V} - \frac{C_V}{C_V}} \]
Substitute $\gamma = \frac{C_P}{C_V}$:
\[ \frac{Q}{W} = \frac{\gamma}{\gamma - 1} \]
Step 4: Final Answer:
The ratio of heat supplied to work done is $\frac{\gamma}{\gamma-1}$.