a) A Carnot's engine has 25% efficiency with a sink at 9°C. By how many degrees should the temperature of the source be increased in order to raise the efficiency to 50%? [3] b) State first law of…

a)

Given:

• Initial efficiency, ${\eta }_1=25\%=0.25$
• Sink temperature, $T_2=9^{\circ }\text{C}=282.15\text{K}$
• Final efficiency, ${\eta }_2=50\%=0.50$

Step 1: Use Carnot efficiency formula The efficiency of a Carnot engine is given by: $$\eta =1-\frac{T_2}{T_1}$$ where $T_1$ is the source temperature (in Kelvin).

Step 2: Find initial source temperature $T_{1i}$ $$0.25=1-\frac{282.15}{T_{1i}}$$ $$\frac{282.15}{T_{1i}}=0.75$$ $$T_{1i}=\frac{282.15}{0.75}=376.2\text{K}$$

Step 3: Find final source temperature $T_{1f}$ for 50% efficiency $$0.50=1-\frac{282.15}{T_{1f}}$$ $$\frac{282.15}{T_{1f}}=0.50$$ $$T_{1f}=\frac{282.15}{0.50}=564.3\text{K}$$

Step 4: Calculate the required increase in source temperature $$\Delta T=T_{1f}-T_{1i}=564.3-376.2=188.1\text{K}$$ Convert back to Celsius: $$\Delta T=188.1\text{K}=188.1^{\circ }\text{C}$$

Answer: The source temperature must be increased by 188.1°C.

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b)

First Law of Thermodynamics: The first law of thermodynamics states that energy cannot be created or destroyed, only transferred or converted from one form to another. Mathematically, for a thermodynamic system: $$\Delta U=Q-W$$ where:

• $\Delta U$ = change in internal energy of the system,
• $Q$ = heat added to the system,
• $W$ = work done by the system.

Does it follow the principle of conservation of energy? Yes, the first law is a restatement of the principle of conservation of energy in the context of thermodynamic processes. It ensures that the total energy of an isolated system remains constant, even as energy is exchanged between different forms (e.g., heat, work, internal energy).

Explanation:

• In any thermodynamic process, the total energy input (heat + work) must equal the total energy output (change in internal energy + work done by the system).
• For example, in a Carnot cycle (as in part a), the heat extracted from the source ($Q_1$) is partially converted to work ($W$) and the rest is rejected to the sink ($Q_2$). The law ensures: $$Q_1=W+Q_2$$ This aligns perfectly with energy conservation, as no energy is lost or gained in the process.