Question

If the internal energy of a thermodynamic system is increased by \(300 .\) \(\mathrm{J}\) while \(75 \mathrm{J}\) of expansion work is done, how much heat was transferred and in which direction, to or from the system?

Short answer

The heat transferred to the thermodynamic system is \(375 \text{ J}\) and the direction is into the system.

Step-by-step solution

First Law of Thermodynamics

First, let's recall the First Law of Thermodynamics, which states that the change in internal energy of a system (∆U) is equal to the heat (Q) added to the system minus the work (W) done by the system on its surroundings: \[\Delta U = Q - W\]

Plug in the given values

Now, we are given that the change in internal energy, ∆U, is 300 J, and the work done, W, is 75 J. We can plug these values into the equation to solve for the heat transfer (Q): \[\Delta U = Q - W\] \[300 \text{ J} = Q - 75 \text{ J}\]

Solve for the heat transfer, Q

Now, solving for Q, we get: \[Q = 300 \text{ J} + 75 \text{ J}\] \[Q = 375 \text{ J}\]

Determine the direction of heat transfer

Since Q is positive, this means that there was a transfer of heat energy into the system, as the system gained 375 J of energy. So, the heat transferred to the system is 375 J and the direction is into the system.

Key concepts

Internal Energy

Internal energy is a crucial concept in thermodynamics. It is the total energy stored within a system. This energy encompasses various forms such as kinetic energy from the motion of particles and potential energy from molecular interactions. Internal energy changes when heat is transferred into or out of the system, or when work is done by or on the system.
For instance, in the problem provided, the internal energy increase of \(300 \, \text{J}\) signifies that the system's energy content has grown. This change can be due to heat entering the system, work being done on it, or both. However, knowing the specific contributions of heat and work requires examining the First Law of Thermodynamics. Understanding internal energy is vital, as it helps determine how a system reacts to energy exchanges.

Work Done

Work done in the context of thermodynamics refers to the energy transferred when a force is applied over a distance, often illustrated by the expanding or compressing of gases. It's inextricable from the First Law of Thermodynamics.
In this particular exercise, \(75 \, \text{J}\) of expansion work implies energy was used by the system to expand or push against external pressure. When work is done by the system, it means energy is leaving the system, which tends to reduce its internal energy, unless aided by an equivalent or greater amount of heat transfer into the system.
Understanding work done is critical, as it affects the overall internal energy, helping predict how a system will change or behave under different conditions.

Heat Transfer

Heat transfer is the process by which thermal energy is exchanged between a system and its surroundings, primarily due to a temperature difference. According to the First Law of Thermodynamics, this transfer is a key determinant of changes in a system's internal energy.
In the problem, calculating the heat transfer value as \(375 \, \text{J}\) indicates that this amount of energy entered the system, offsetting the energy lost through expansion work. Since the calculated \(Q\) is positive, it confirms that the heat flows "into" the system. This inward transfer of heat increases the internal energy and enables the system to perform work.
Recognizing the direction and magnitude of heat transfer facilitates a comprehensive understanding of energy dynamics within a thermodynamic system.