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 given in the question.

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

1. **Identify the initial conditions:**
- Initial temperature, \( T_1 = 27^\circ C = 300 \, K \)
- Initial volume, \( V_1 = 4 \, L \)
- Initial pressure, \( P_1 = 100 \, N/m^2 \)
- Final pressure, \( P_2 = 150 \, N/m^2 \)

2. **Apply 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:
\[ P_1 V_1 = P_2 V_2 \]
Rearranging the equation to find the final volume \( V_2 \):
\[ V_2 = \frac{P_1 V_1}{P_2} \]

3. **Substitute the values:**
\[ V_2 = \frac{100 \, N/m^2 \times 4 \, L}{150 \, N/m^2} \]
\[ V_2 = \frac{400}{150} \approx 2.67 \, L \]

4. **Calculate the change in volume:**
\[ \Delta V = V_2 - V_1 = 2.67 \, L - 4 \, L = -1.33 \, L \]
The change in volume is \( 1.33 \, L \) (decrease).

### Part (ii): Calculate the new pressure when the gas is heated at constant volume.

1. **Identify the new conditions:**
- Final temperature, \( T_2 = 127^\circ C = 400 \, K \)
- Volume remains constant, \( V_2 = 2.67 \, L \)
- Initial pressure, \( P_1 = 100 \, N/m^2 \)

2. **Apply Gay-Lussac's Law:**
Gay-Lussac's Law states that for a given mass of gas at constant volume, the ratio of pressure to temperature is constant:
\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]
Rearranging the equation to find the new pressure \( P_2 \):
\[ P_2 = P_1 \times \frac{T_2}{T_1} \]

3. **Substitute the values:**
\[ P_2 = 100 \, N/m^2 \times \frac{400 \, K}{300 \, K} \]
\[ P_2 = 100 \, N/m^2 \times \frac{4}{3} \approx 133.33 \, N/m^2 \]

4. **Calculate the new pressure:**
\[ P_2 \approx 133.33 \, N/m^2 \]

### Summary of Results:
- Change in volume \( \Delta V = -1.33 \, L \) (decrease).
- New pressure \( P_2 \approx 133.33 \, N/m^2 \).