`Q` cal of heat is required to raise the temperature of `1 "mole"` of a monatomic gas from `20^(@)C to 30^(@)C` at constant pressure. The amount of heat required to raise the temperature of `1 "mole"` of diatomic gas from `20^(@)C to 25^(@)C` at constant pressure is

Options

A. `Q`
B. `(3)/(2)Q`
C. `(5)/(6)Q`
D. `(7)/(10)Q`

Explanation

To solve the problem, we need to calculate the amount of heat required to raise the temperature of a diatomic gas from \(20^\circ C\) to \(25^\circ C\) at constant pressure, given that \(Q\) cal of heat is required for a monatomic gas to raise its temperature from \(20^\circ C\) to \(30^\circ C\).

### Step-by-Step Solution:

1. **Identify the heat required for the monatomic gas:**
The heat required to raise the temperature of 1 mole of a monatomic gas from \(20^\circ C\) to \(30^\circ C\) is given as \(Q\) cal.

\[ Q_{mono} = Q \text{ cal} \]

2. **Use the formula for heat transfer at constant pressure:**
The heat transfer \(Q\) at constant pressure can be calculated using the formula:

\[ Q = nC_p\Delta T \]

where:
- \(n\) = number of moles
- \(C_p\) = molar heat capacity at constant pressure
- \(\Delta T\) = change in temperature

3. **Calculate the change in temperature for the monatomic gas:**
For the monatomic gas, the change in temperature \(\Delta T\) is:

\[ \Delta T_{mono} = 30^\circ C - 20^\circ C = 10^\circ C \]

4. **Determine the molar heat capacity \(C_p\) for monatomic gas:**
The molar heat capacity at constant pressure for a monatomic gas is:

\[ C_{p,mono} = \frac{5}{2}R \]

where \(R\) is the universal gas constant.

5. **Set up the equation for the monatomic gas:**
Substituting the values into the heat transfer equation for the monatomic gas:

\[ Q = nC_{p,mono}\Delta T_{mono} \]
\[ Q = 1 \cdot \frac{5}{2}R \cdot 10 \]
\[ Q = 25R \text{ cal} \]

6. **Calculate the heat required for the diatomic gas:**
For the diatomic gas, we need to find the heat required to raise the temperature from \(20^\circ C\) to \(25^\circ C\):

\[ \Delta T_{diatomic} = 25^\circ C - 20^\circ C = 5^\circ C \]

7. **Determine the molar heat capacity \(C_p\) for diatomic gas:**
The molar heat capacity at constant pressure for a diatomic gas is:

\[ C_{p,diatomic} = \frac{7}{2}R \]

8. **Set up the equation for the diatomic gas:**
Using the heat transfer equation for the diatomic gas:

\[ Q_{diatomic} = nC_{p,diatomic}\Delta T_{diatomic} \]
\[ Q_{diatomic} = 1 \cdot \frac{7}{2}R \cdot 5 \]
\[ Q_{diatomic} = \frac{35}{2}R \text{ cal} \]

9. **Relate the heat required for diatomic gas to the monatomic gas:**
Since we have \(Q = 25R\) for the monatomic gas, we can express the heat required for the diatomic gas in terms of \(Q\):

\[ Q_{diatomic} = \frac{35}{2}R = \frac{35}{2} \cdot \frac{Q}{25} \]

10. **Final calculation:**
To find the heat required for the diatomic gas in terms of \(Q\):

\[ Q_{diatomic} = \frac{35}{2} \cdot \frac{Q}{25} = \frac{35Q}{50} = 0.7Q \]

### Final Answer:
The amount of heat required to raise the temperature of 1 mole of diatomic gas from \(20^\circ C\) to \(25^\circ C\) at constant pressure is \(0.7Q\) cal.