Heat is supplied to a diatomic gas at constant pressure.
The ratio of `DeltaQ : DeltaU : DeltaW` is

Options

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

Explanation

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

### Step 1: Understand the relationship between heat, internal energy, and work
At constant pressure, the first law of thermodynamics states:
\[ \Delta Q = \Delta U + \Delta W \]
where:
- \(\Delta Q\) is the heat added to the system,
- \(\Delta U\) is the change in internal energy,
- \(\Delta W\) is the work done by the system.

### Step 2: Calculate \(\Delta Q\)
For a diatomic gas at constant pressure, the heat added can be expressed as:
\[ \Delta Q = n C_p \Delta T \]
where:
- \(n\) is the number of moles,
- \(C_p\) is the specific heat at constant pressure,
- \(\Delta T\) is the change in temperature.

For a diatomic gas, \(C_p\) can be derived from the relation:
\[ C_p = C_v + R \]
where \(C_v\) (specific heat at constant volume) for a diatomic gas is:
\[ C_v = \frac{5R}{2} \]
Thus,
\[ C_p = \frac{5R}{2} + R = \frac{7R}{2} \]
Therefore, we can write:
\[ \Delta Q = n \left(\frac{7R}{2}\right) \Delta T \]

### Step 3: Calculate \(\Delta U\)
The change in internal energy for a diatomic gas is given by:
\[ \Delta U = n C_v \Delta T \]
Substituting for \(C_v\):
\[ \Delta U = n \left(\frac{5R}{2}\right) \Delta T \]

### Step 4: Calculate \(\Delta W\)
Using the first law of thermodynamics:
\[ \Delta W = \Delta Q - \Delta U \]
Substituting the expressions we found for \(\Delta Q\) and \(\Delta U\):
\[ \Delta W = n \left(\frac{7R}{2}\right) \Delta T - n \left(\frac{5R}{2}\right) \Delta T \]
This simplifies to:
\[ \Delta W = n \left(\frac{2R}{2}\right) \Delta T = nR \Delta T \]

### Step 5: Find the ratio \(\Delta Q : \Delta U : \Delta W\)
Now we have:
- \(\Delta Q = n \left(\frac{7R}{2}\right) \Delta T\)
- \(\Delta U = n \left(\frac{5R}{2}\right) \Delta T\)
- \(\Delta W = nR \Delta T\)

To find the ratio, we can express it as:
\[ \Delta Q : \Delta U : \Delta W = \frac{7R}{2} : \frac{5R}{2} : R \]
Dividing each term by \(R\) (and factoring out \(n\Delta T\) which is common):
\[ = \frac{7}{2} : \frac{5}{2} : 1 \]
To eliminate the fractions, we can multiply through by 2:
\[ = 7 : 5 : 2 \]

### Final Answer
The ratio \(\Delta Q : \Delta U : \Delta W\) is:
\[ \boxed{7 : 5 : 2} \]