Animated Solution for Physics - Thermodynamics: For an ideal heat engine, the temperature of the source is $127^{\circ }\text{C}$. In order to have $60\%$ efficiency the temperature of the sink should be ...... ${}^{\circ }\text{C}$. (Round off to the nearest integer)

Imagine you have a heat engine. It's like a magical machine that takes heat from a hot source, uses some of it to do useful work, and dumps the leftover heat into a cold sink. To understand how well this engine performs, we look at its efficiency.

For an ideal Carnot engine, the efficiency depends strictly on the absolute temperatures of the source and the sink. The master formula that governs this relationship is:

Crucial Rule: These temperatures must always be in Kelvin! Using Celsius directly in this ratio is the most common trap students fall into.

Let's carefully substitute the values given in the question. The source temperature is $127^{\circ }\text{C}$. We must add $273$ to convert it to the absolute scale, giving us $T_H=400\text{ K}$.

The efficiency is given as $60\%$, which we write as a decimal: $\eta =0.6$. Plugging these into our master equation:

With our values plugged in, let's do some simple algebra. We want to isolate the temperature ratio. Moving terms around, we get:

To find the sink temperature $T_L$, we just multiply $400$ by $0.4$.

Wait, there is a catch here! The question specifically asks for the temperature in degrees Celsius. If you stop at $160$, you will lose marks.

We must convert the temperature back to the Celsius scale by subtracting $273$:

Watch out for that minus sign! The sink must be extremely cold, well below the freezing point of water, to achieve such a high efficiency.

As a thought experiment, what if we wanted an engine with $100\%$ efficiency? The sink temperature would have to be absolute zero ($0\text{ K}$). Since we can't reach absolute zero, a perfectly efficient engine is impossible. That's the profound beauty of the Second Law of Thermodynamics.