In an isobaric process, `Delta Q = (K gamma)/(gamma - 1)` where `gamma = C_(P)//C_(V)`. What is ` K`?

Options

A. Pressure
B. Volume
C. `Delta U`
D. `Delta W`

Explanation

To find the value of \( K \) in the equation for an isobaric process, we start with the given relationship:

\[ \Delta Q = K \frac{\gamma}{\gamma - 1} \]

where \( \gamma = \frac{C_P}{C_V} \).

### Step 1: Substitute the expression for \( \gamma \)

We know that:

\[ \gamma = \frac{C_P}{C_V} \]

Substituting this into the equation gives:

\[ \Delta Q = K \frac{\frac{C_P}{C_V}}{\frac{C_P}{C_V} - 1} \]

### Step 2: Simplify the denominator

The denominator can be simplified as follows:

\[ \frac{C_P}{C_V} - 1 = \frac{C_P - C_V}{C_V} \]

Thus, we can rewrite the equation as:

\[ \Delta Q = K \frac{C_P}{C_V} \cdot \frac{C_V}{C_P - C_V} \]

This simplifies to:

\[ \Delta Q = K \frac{C_P}{C_P - C_V} \]

### Step 3: Multiply and divide by \( n \Delta T \)

Now, we multiply and divide the numerator by \( n \Delta T \):

\[ \Delta Q = K \frac{n C_P \Delta T}{n C_P \Delta T - n C_V \Delta T} \]

### Step 4: Identify terms

Recognizing that \( n C_P \Delta T = \Delta Q \) (the heat added in an isobaric process) and \( n C_V \Delta T = \Delta U \) (the change in internal energy), we can rewrite the equation:

\[ \Delta Q = K \frac{\Delta Q}{\Delta Q - \Delta U} \]

### Step 5: Apply the First Law of Thermodynamics

According to the First Law of Thermodynamics, we have:

\[ \Delta Q = \Delta U + \Delta W \]

Thus, we can rewrite \( \Delta Q - \Delta U \) as \( \Delta W \):

\[ \Delta Q = K \frac{\Delta Q}{\Delta W} \]

### Step 6: Solve for \( K \)

Now, we can cancel \( \Delta Q \) from both sides (assuming \( \Delta Q \neq 0 \)):

\[ 1 = K \frac{1}{\Delta W} \]

This implies:

\[ K = \Delta W \]

### Conclusion

Thus, the value of \( K \) in the isobaric process is:

\[ K = \Delta W \]