Heat is supplied to a diatomic gas at constant pressure.
The ratio of `DeltaQ : DeltaU : DeltaW` is a) 5: 3: 2 b) 7: 5: 2 c) 2: 3: 5 d) 2: 5: 7

Options

A. `5:3:2`
B. `7:5:2`
C. `2:3:5`
D. `2:5:7`

Explanation

To solve the problem of finding the ratio of \(\Delta Q\), \(\Delta U\), and \(\Delta W\) for a diatomic gas at constant pressure, we can follow these steps:

### Step 1: Identify the specific heats
For a diatomic gas, the specific heat at constant volume (\(C_V\)) and constant pressure (\(C_P\)) are given as:
- \(C_V = \frac{5}{2} R\)
- \(C_P = \frac{7}{2} R\)

### Step 2: Write the expressions for \(\Delta Q\), \(\Delta U\), and \(\Delta W\)
Using the definitions:
- The heat supplied to the gas at constant pressure is given by:
\[ \Delta Q = n C_P \Delta T \]
- The change in internal energy is given by:
\[ \Delta U = n C_V \Delta T \]
- The work done by the gas at constant pressure can be expressed as:
\[ \Delta W = \Delta Q - \Delta U \]

### Step 3: Substitute the expressions for \(\Delta Q\) and \(\Delta U\)
Substituting the values of \(C_P\) and \(C_V\) into the equations:
- \(\Delta Q = n \left(\frac{7}{2} R\right) \Delta T\)
- \(\Delta U = n \left(\frac{5}{2} R\right) \Delta T\)

### Step 4: Calculate \(\Delta W\)
Now, substituting \(\Delta Q\) and \(\Delta U\) into the equation for \(\Delta W\):
\[ \Delta W = \Delta Q - \Delta U = n \left(\frac{7}{2} R\right) \Delta T - n \left(\frac{5}{2} R\right) \Delta T \]
\[ \Delta W = n \left(\left(\frac{7}{2} R - \frac{5}{2} R\right) \Delta T\right) = n \left(\frac{2}{2} R\right) \Delta T = n R \Delta T \]

### Step 5: Form the ratio \(\Delta Q : \Delta U : \Delta W\)
Now we can express the ratio:
\[ \Delta Q : \Delta U : \Delta W = n \left(\frac{7}{2} R \Delta T\right) : n \left(\frac{5}{2} R \Delta T\right) : n \left(R \Delta T\right) \]
Cancelling \(n\) and \(\Delta T\) from all terms, we have:
\[ \Delta Q : \Delta U : \Delta W = \frac{7}{2} R : \frac{5}{2} R : R \]
This simplifies to:
\[ \Delta Q : \Delta U : \Delta W = 7 : 5 : 2 \]

### Final Answer
Thus, the ratio of \(\Delta Q : \Delta U : \Delta W\) is:
\[ \text{Option (b) } 7 : 5 : 2 \]

---