A closed vessel 10L in volume contains a diatomic gas under a pressure of `10^5N//m^2`. What amount of heat should be imparted to the gas to increase the pressure in the vessel five times?

Explanation

To solve the problem, we need to determine the amount of heat that must be imparted to a diatomic gas in a closed vessel to increase its pressure from \(10^5 \, \text{N/m}^2\) to \(5 \times 10^5 \, \text{N/m}^2\).

### Step-by-Step Solution:

1. **Convert Volume to Cubic Meters**:
- The volume of the gas is given as 10 liters. We need to convert this to cubic meters.
\[ V = 10 \, \text{L} = 10 \times 10^{-3} \, \text{m}^3 = 0.01 \, \text{m}^3 \]

**Hint**: Remember that 1 liter = \(10^{-3}\) cubic meters.

2. **Determine Initial and Final Pressure**:
- The initial pressure \(P_1\) is \(10^5 \, \text{N/m}^2\).
- The final pressure \(P_2\) is \(5 \times 10^5 \, \text{N/m}^2\).

3. **Calculate Change in Pressure**:
- The change in pressure \(\Delta P\) is given by:
\[ \Delta P = P_2 - P_1 = 5 \times 10^5 - 10^5 = 4 \times 10^5 \, \text{N/m}^2 \]

**Hint**: Subtract the initial pressure from the final pressure to find the change.

4. **Use the Ideal Gas Law**:
- The ideal gas equation is given by \(PV = nRT\). Since the volume is constant, we can relate the change in pressure to the change in temperature:
\[ V \Delta P = nR \Delta T \]
Rearranging gives:
\[ \Delta T = \frac{V \Delta P}{nR} \]

**Hint**: Keep in mind that \(n\) (number of moles) and \(R\) (gas constant) are required for this equation.

5. **Calculate Heat Added**:
- The heat added \(Q\) to the gas can be expressed as:
\[ Q = nC_v \Delta T \]
For a diatomic gas, \(C_v = \frac{5}{2}R\). Therefore:
\[ Q = n \left(\frac{5}{2}R\right) \Delta T \]
Substituting \(\Delta T\) from the previous step:
\[ Q = n \left(\frac{5}{2}R\right) \left(\frac{V \Delta P}{nR}\right) = \frac{5}{2} V \Delta P \]

**Hint**: Notice how the number of moles \(n\) cancels out.

6. **Substitute Values**:
- Now we can substitute the values:
\[ Q = \frac{5}{2} \times 0.01 \, \text{m}^3 \times 4 \times 10^5 \, \text{N/m}^2 \]
\[ Q = \frac{5}{2} \times 0.01 \times 4 \times 10^5 = 10^4 \, \text{J} \]

**Hint**: Ensure units are consistent when performing calculations.

7. **Final Answer**:
- The amount of heat that should be imparted to the gas is:
\[ Q = 10^4 \, \text{J} \]

### Summary:
The amount of heat that should be imparted to the gas to increase the pressure in the vessel five times is \(10^4 \, \text{J}\).