A gas at `27^@ C` in a cylinder has a volume of 4 litre and pressure `100 Nm^-2`.
(i) Gas is first compressed at constant temperature so that the pressure is `150 Nm^-2` . Calaulate the change in volume.
(ii) It is then heated at constant volume so that temperature becomes `127^@ C`. Calculate the new pressure.

Explanation

To solve the problem step-by-step, we will break it down into two parts as per the question.

### Part (i): Calculate the change in volume when the gas is compressed at constant temperature.

1. **Identify the given values:**
- Initial Pressure (P1) = 100 Nm^-2
- Initial Volume (V1) = 4 liters
- Final Pressure (P2) = 150 Nm^-2

2. **Convert the volume from liters to cubic meters for consistency in SI units:**
\[ V1 = 4 \, \text{liters} = 4 \times 10^{-3} \, \text{m}^3 \]

3. **Use Boyle's Law:**
Boyle's Law states that for a given mass of gas at constant temperature, the product of pressure and volume is constant:
\[ P1 \cdot V1 = P2 \cdot V2 \]
Rearranging the equation to find V2:
\[ V2 = \frac{P1 \cdot V1}{P2} \]

4. **Substitute the values into the equation:**
\[ V2 = \frac{100 \, \text{Nm}^{-2} \cdot 4 \times 10^{-3} \, \text{m}^3}{150 \, \text{Nm}^{-2}} \]

5. **Calculate V2:**
\[ V2 = \frac{0.4 \, \text{Nm}^{-2} \cdot \text{m}^3}{150 \, \text{Nm}^{-2}} = \frac{0.4}{150} \approx 0.00267 \, \text{m}^3 \]
Converting back to liters:
\[ V2 \approx 2.67 \, \text{liters} \]

6. **Calculate the change in volume:**
\[ \Delta V = V2 - V1 = 2.67 \, \text{liters} - 4 \, \text{liters} = -1.33 \, \text{liters} \]

### Part (ii): Calculate the new pressure after heating at constant volume.

1. **Identify the new temperature:**
- Initial Temperature (T1) = 27°C = 300 K
- Final Temperature (T2) = 127°C = 400 K
- Final Pressure (P2) after compression = 150 Nm^-2

2. **Use the ideal gas law relationship at constant volume:**
\[ \frac{P1}{T1} = \frac{P2}{T2} \]
Rearranging to find P2:
\[ P2 = P1 \cdot \frac{T2}{T1} \]

3. **Substitute the values into the equation:**
\[ P2 = 150 \, \text{Nm}^{-2} \cdot \frac{400 \, \text{K}}{300 \, \text{K}} \]

4. **Calculate P2:**
\[ P2 = 150 \cdot \frac{4}{3} = 200 \, \text{Nm}^{-2} \]

### Final Answers:
- (i) The change in volume is **-1.33 liters**.
- (ii) The new pressure is **200 Nm^-2**.