In an isobaric process, the ratio of heat supplied to the system `(dQ)` and work done by the system `(dW)` for diatomic gas is

Options

A. `gamma`
B. `gamma - 1`
C. `(gamma)/( gamma + 1)`
D. `(gamma)/(gamma - 1)`

Answer

To find the ratio of heat supplied to the system (dQ) and work done by the system (dW) for a diatomic gas in an isobaric process, we can follow these steps:

### Step 1: Understand the Isobaric Process
In an isobaric process, the pressure (P) remains constant.

### Step 2: Write the Expression for Heat Supplied (dQ)
For an isobaric process, the heat supplied to the system can be expressed as:
\[ dQ = n C_p \Delta T \]
where:
- \( n \) = number of moles of the gas
- \( C_p \) = molar specific heat at constant pressure
- \( \Delta T \) = change in temperature

### Step 3: Write the Expression for Work Done (dW)
The work done by the gas in an isobaric process is given by:
\[ dW = P \Delta V \]
Using the ideal gas law, we can also express this as:
\[ dW = n R \Delta T \]
where:
- \( R \) = universal gas constant

### Step 4: Find the Ratio \( \frac{dQ}{dW} \)
Now we can find the ratio of heat supplied to the work done:
\[ \frac{dQ}{dW} = \frac{n C_p \Delta T}{n R \Delta T} \]
Here, \( n \) and \( \Delta T \) cancel out:
\[ \frac{dQ}{dW} = \frac{C_p}{R} \]

### Step 5: Relate \( C_p \) to \( C_v \) and \( \gamma \)
We know that:
\[ \gamma = \frac{C_p}{C_v} \]
From this, we can express \( C_p \) as:
\[ C_p = \gamma C_v \]
Also, we have the relation for \( C_v \):
\[ C_v = \frac{R}{\gamma - 1} \]
Substituting this into the expression for \( C_p \):
\[ C_p = \gamma \left(\frac{R}{\gamma - 1}\right) = \frac{\gamma R}{\gamma - 1} \]

### Step 6: Substitute \( C_p \) into the Ratio
Now substituting \( C_p \) back into the ratio:
\[ \frac{dQ}{dW} = \frac{\frac{\gamma R}{\gamma - 1}}{R} \]
The \( R \) cancels out:
\[ \frac{dQ}{dW} = \frac{\gamma}{\gamma - 1} \]

### Conclusion
Thus, the ratio of heat supplied to the work done in an isobaric process for a diatomic gas is:
\[ \frac{dQ}{dW} = \frac{\gamma}{\gamma - 1} \]