n moles of an ideal gas with constant volume heat capacity `C_(V)` undergo an isobaric expansion by certain volumes. The ratio of the work done in the process, to the heat supplied is:

Options

A. `(4 nR)/(C_(V)-nR)`
B. `(nR)/(C_(V)-nR)`
C. `(nR)/(C_(V) + nR)`
D. `(4nR)/(C_(V) + nR)`

Explanation

To solve the problem, we need to find the ratio of the work done during an isobaric expansion of an ideal gas to the heat supplied during the process. Let's break this down step by step.

### Step 1: Understanding the Process
In an isobaric process, the pressure (P) remains constant. For an ideal gas, the work done (W) during an isobaric expansion can be expressed as:
\[ W = P \Delta V \]
where \( \Delta V \) is the change in volume.

### Step 2: Relating Work Done to Ideal Gas Law
Using the ideal gas law, we know:
\[ PV = nRT \]
For an isobaric process, we can express the work done as:
\[ W = P(V_f - V_i) = nR(T_f - T_i) \]
where \( T_f \) and \( T_i \) are the final and initial temperatures, respectively.

### Step 3: Calculating Heat Supplied (Q)
The heat supplied (Q) during an isobaric process can be calculated using the formula:
\[ Q = nC_p \Delta T \]
where \( C_p \) is the molar heat capacity at constant pressure and \( \Delta T = T_f - T_i \).

### Step 4: Relating \( C_p \) to \( C_v \)
For an ideal gas, the relationship between the heat capacities is given by:
\[ C_p = C_v + R \]
Thus, we can rewrite the heat supplied as:
\[ Q = n(C_v + R)(T_f - T_i) \]

### Step 5: Finding the Ratio of Work Done to Heat Supplied
Now, we can find the ratio of the work done to the heat supplied:
\[ \frac{W}{Q} = \frac{nR(T_f - T_i)}{n(C_v + R)(T_f - T_i)} \]

### Step 6: Simplifying the Ratio
The \( n \) and \( (T_f - T_i) \) terms cancel out, leading to:
\[ \frac{W}{Q} = \frac{R}{C_v + R} \]

### Final Answer
Thus, the ratio of the work done in the process to the heat supplied is:
\[ \frac{W}{Q} = \frac{R}{C_v + R} \]