A heat engine absorbs heat q1 from a source at temperature T1 and heat q2 from a source at temperatu

Q: Why is the answer not the First Law of Thermodynamics?

While the First Law ($\Delta U = Q - W$) is related to energy conservation, the explicit use of the conversion factor $J$ specifically points to Joule's mechanical equivalent of heat, which historically established the equivalence between work and heat.

Q: What exactly does the constant $J$ represent?

$J$ is Joule's mechanical equivalent of heat. It acts as a conversion factor (approximately $4.184 \text{ J/cal}$) to equate mechanical work measured in Joules to thermal energy measured in calories.

Q: Does this engine violate the Second Law of Thermodynamics?

A real engine must reject some heat to a colder sink to operate in a cycle. This problem presents a simplified, theoretical relationship just to test your knowledge of Joule's law, without detailing the full thermodynamic cycle.

Q: Why are there two heat sources mentioned?

An engine can theoretically absorb heat from multiple sources. The law still holds; you simply use the total net heat absorbed, which is the sum $q_1 + q_2$.

Q: Is $J$ a universal constant?

Yes, $J$ is a constant that defines the conversion rate between two different units of energy (Joules and calories). However, in modern SI units where both heat and work are measured in Joules, $J$ is simply $1$.

Question

A heat engine absorbs heat $q_1$ from a source at temperature $T_1$ and heat $q_2$ from a source at temperature $T_2$. Work done is found to be $J(q_1 + q_2)$. This is in accordance with

Options:

(A) First law of thermodynamics

(B) Second law of thermodynamics

(C) Joules equivalent law

(D) None of the above

Sigma Percentile · Accepted Answer

### The Interplay of Heat and Work

Imagine a steam engine chugging along the tracks. It consumes coal, generates intense heat, and magically transforms that thermal energy into the mechanical motion of the wheels. But how exactly do we quantify this transformation? This is the core question that thermodynamics seeks to answer.

In our specific problem, we are presented with a theoretical heat engine. This engine is quite greedy; it absorbs heat from not just one, but two different sources. Let's say it absorbs an amount of heat $q_1$ from a source at temperature $T_1$, and another amount of heat $q_2$ from a source at temperature $T_2$.

The total heat entering our system is simply the sum of these two inputs:
$$Q_{	ext{net}} = q_1 + q_2$$

### Joule's Mechanical Equivalent of Heat

Before the mid-19th century, heat was thought to be a fluid called "caloric." It was James Prescott Joule who definitively proved that heat is actually a form of energy. He demonstrated that mechanical work could be directly converted into heat, and vice versa.

Joule established a direct proportionality between the mechanical work done ($W$) and the heat absorbed ($Q$). To turn this proportionality into an equation, he introduced a constant, $J$, known as the **mechanical equivalent of heat**.

The relationship is elegantly simple:
$$J = \frac{W}{Q_{	ext{net}}}$$

Here, $J$ acts as a conversion factor. Historically, heat was measured in calories and work in Joules. The constant $J$ (approximately $4.184 	ext{ J/cal}$) bridges these two units.

### Decoding the Engine's Work

Now, let's apply Joule's law to our greedy heat engine. We know the total heat absorbed is $Q_{	ext{net}} = q_1 + q_2$. Substituting this into Joule's equation, we get:
$$J = \frac{W}{q_1 + q_2}$$

If we rearrange this equation to solve for the work done, $W$, we simply multiply both sides by the total heat:
$$W = J(q_1 + q_2)$$

Take a close look at this final expression. It is exactly the relationship provided in the problem statement! The problem tells us that the work done is found to be $J(q_1 + q_2)$.

Therefore, this observation is a direct mathematical manifestation of **Joule's equivalent law**. It beautifully encapsulates the fundamental principle that the work produced by an engine is directly proportional to the total heat it absorbs, linked by the mechanical equivalent of heat.

Animated Solution for Chemistry - Chemical Thermodynamics: A heat engine absorbs heat $q_{1}$ from a source at temperature $T_{1}$ and heat $q_{2}$ from a source at temperature $T_{2}$ . Work done is found to be $J (q_{1} + q_{2})$ . This is in accordance with

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The Sigma Insight: First Law of Thermodynamics

The Interplay of Heat and Work

Joule's Mechanical Equivalent of Heat

Decoding the Engine's Work

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