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 (W) to the heat supplied (Q) during an isobaric expansion of an ideal gas. Let's break it down step by step.

### Step 1: Understand the Process
In an isobaric process, the pressure (P) remains constant. For an ideal gas, we can use the ideal gas law:
\[ PV = nRT \]
Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Universal gas constant
- \( T \) = Temperature

### Step 2: Calculate the Work Done (W)
The work done by the gas during an isobaric process can be calculated using the formula:
\[ W = P \Delta V \]
Since \( P \) is constant, we can express this in terms of temperature change. The change in volume can be related to the change in temperature using the ideal gas law:
\[ W = nR(T_f - T_i) \]
Where \( T_f \) is the final temperature and \( T_i \) is the initial temperature.

### Step 3: Calculate the Heat Supplied (Q)
The heat supplied to the gas during an isobaric process can be calculated using:
\[ Q = \Delta U + W \]
Where \( \Delta U \) is the change in internal energy. For an ideal gas, the change in internal energy is given by:
\[ \Delta U = nC_V(T_f - T_i) \]
So, substituting for \( Q \):
\[ Q = nC_V(T_f - T_i) + nR(T_f - T_i) \]
Factoring out \( (T_f - T_i) \):
\[ Q = n(T_f - T_i)(C_V + R) \]

### Step 4: Find the Ratio \( \frac{W}{Q} \)
Now we can find the ratio of work done to heat supplied:
\[ \frac{W}{Q} = \frac{nR(T_f - T_i)}{n(T_f - T_i)(C_V + R)} \]
The \( n \) and \( (T_f - T_i) \) terms cancel out:
\[ \frac{W}{Q} = \frac{R}{C_V + R} \]

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

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