Animated Solution for Physics - Thermodynamics: A heat engine is involved with exchange of heat of $1915\text{ J}$, $-40\text{ J}$, $+125\text{ J}$ and $-Q\text{ J}$, during one cycle achieving an efficiency of $50.0\%$. The value of $Q$ is

Imagine a steam locomotive chugging along the tracks. At its heart is a heat engine, a marvel of thermodynamics that converts thermal energy into mechanical work. But how does it actually do this?

A heat engine operates between two thermal reservoirs: a high-temperature source and a low-temperature sink. It absorbs a certain amount of heat, $Q_{\text{in}}$, from the source. It then uses a portion of this energy to perform useful work, $W$. However, the Second Law of Thermodynamics dictates that no engine can be $100\%$ efficient. Therefore, the remaining energy, $Q_{\text{out}}$, must be rejected to the sink.

In our specific problem, the engine undergoes a cycle with four distinct heat exchanges: $+1915\text{ J}$, $-40\text{ J}$, $+125\text{ J}$, and $-Q\text{ J}$.

What do these signs mean? In thermodynamics, heat added to the system is considered positive, while heat removed from the system is negative.

Therefore, the total heat supplied to our engine is the sum of the positive values:

Conversely, the heat rejected by the engine is the sum of the magnitudes of the negative values:

Efficiency

The efficiency, $\eta$, of a heat engine is the ratio of the net work done to the total heat supplied. Mathematically, it is expressed as:

Since the net work done is the difference between the heat supplied and the heat rejected ($W=Q_{\text{in}}-Q_{\text{out}}$), we can rewrite the efficiency formula as:

We are given that the engine operates at an efficiency of $50.0\%$, which means $\eta =0.5$. Let's substitute our known values into the master equation:

Now, we simply solve for $Q$. Multiplying both sides by $2040$:

Rearranging the terms to isolate $Q$:

And there we have it! The unknown heat exchange $Q$ is $980\text{ J}$. This problem beautifully illustrates the fundamental principles of energy conservation and the inevitable realities of heat rejection in any thermodynamic cycle.