What is the process from B to C called? Calculate the total work done by an ideal gas in the process ABC.

$$T_2 = T_1 + \Delta T = 270 \text{ K} + 70 \text{ K} = 340 \text{ K}$$ $$V_2 = V_1 + \Delta V = V_1 + 0.29 \text{ m}^3$$

Charles's Law states that for a fixed amount of gas at constant pressure, the volume is directly proportional to its absolute temperature:

$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$

$$\frac{V_1}{270 \text{ K}} = \frac{V_1 + 0.29 \text{ m}^3}{340 \text{ K}}$$

Cross-multiply:

$$340 V_1 = 270 (V_1 + 0.29)$$ $$340 V_1 = 270 V_1 + 270 \times 0.29$$ $$340 V_1 = 270 V_1 + 78.3$$ $$340 V_1 - 270 V_1 = 78.3$$ $$70 V_1 = 78.3$$ $$V_1 = \frac{78.3}{70}$$ $$V_1 \approx 1.1186 \text{ m}^3$$

The original volume is $\boxed{\text{1.12 m}^3}$.

7.4 The pressure-volume diagram of two gas processes is shown in FIGURE 4 below. Study the diagram and answer the questions that follow.

7.4.1 What is the process from B to C called?

From the diagram, the volume at point B is $50 \text{ m}^3$ and the volume at point C is also $50 \text{ m}^3$. Since the volume remains constant during the process from B to C, it is an isochoric process.

The process from B to C is called an isochoric process (or isovolumetric process).

7.4.2 Calculate the total work done by an ideal gas in the process ABC.

The work done by a gas in a P-V diagram is the area under the process curve.

Process A to B is a linear change in pressure and volume. The work done is the area of the trapezoid under the line AB.

• Point A: $P_A = 200 \text{ kPa} = 200 \times 10^3 \text{ Pa}$, $V_A = 20 \text{ m}^3$
• Point B: $P_B = 100 \text{ kPa} = 100 \times 10^3 \text{ Pa}$, $V_B = 50 \text{ m}^3$

$$W_{AB} = \text{Area of trapezoid} = \frac{1}{2}(P_A + P_B)(V_B - V_A)$$ $$W_{AB} = \frac{1}{2}(200 \times 10^3 \text{ Pa} + 100 \times 10^3 \text{ Pa})(50 \text{ m}^3 - 20 \text{ m}^3)$$ $$W_{AB} = \frac{1}{2}(300 \times 10^3 \text{ Pa})(30 \text{ m}^3)$$ $$W_{AB} = (150 \times 10^3 \text{ Pa})(30 \text{ m}^3)$$ $$W_{AB} = 4.5 \times 10^6 \text{ J} = 4500 \text{ kJ}$$

Process B to C is an isochoric process (constant volume). In an isochoric process, no work is done by the gas.

$$W_{BC} = P \Delta V = P(V_C - V_B) = P(50 \text{ m}^3 - 50 \text{ m}^3) = 0 \text{ J}$$

$$W_{ABC} = W_{AB} + W_{BC}$$ $$W_{ABC} = 4.5 \times 10^6 \text{ J} + 0 \text{ J}$$ $$W_{ABC} = 4.5 \times 10^6 \text{ J} = 4500 \text{ kJ}$$

The total work done by an ideal gas in the process ABC is $\boxed{\text{4500 kJ}}$.