Question

If the internal energy of a thermodynamic system is increased by \(300 .\) 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 \(375\,\text{J}\).

Step-by-step solution

Write down the equation for the first law of thermodynamics.

The first law of thermodynamics equation can be written as: \[ \Delta U = Q - W \] Where: - \(\Delta U\) is the change in internal energy. - \(Q\) is the amount of heat transferred. - \(W\) is the work done.

Substitute the given values into the equation.

We are given the values for the internal energy increase (\(\Delta U = 300\,\text{J}\)) and the expansion work done (\(W = 75\,\text{J}\)). Plugging these values into the equation, we get: \[ 300\,\text{J} = Q - 75\,\text{J} \]

Solve for the heat transferred.

To find the amount of heat transferred (\(Q\)), we need to isolate \(Q\) in our equation. We do this by adding \(75\,\text{J}\) to both sides of the equation: \[ 300\,\text{J} + 75\,\text{J} = Q \] Now, we simply add the values on the left side: \[ 375\,\text{J} = Q \]

Determine the direction of the heat transfer.

Since the value of \(Q\) is positive (\(375\,\text{J}\)), it means that the heat was transferred to the system. If the value of \(Q\) were negative, then it would have meant that the heat was transferred from the system. So, the heat transferred to the thermodynamic system is \(375\,\text{J}\).

Key concepts

Internal Energy

Understanding internal energy is crucial in thermodynamics. It refers to the total energy stored within a system. This energy is a sum of kinetic and potential energies of all particles in a system. Internal energy is a state function, meaning it depends only on the current state of the system, not how that state was reached.

In this exercise, the internal energy increased by 300 J. When we talk about an increase in internal energy, it means that the system gained additional energy. This can happen through processes like heat transfer or work being done on the system. A key point to remember is that internal energy changes with conditions such as temperature, pressure, and volume, as well as through chemical reactions within the system.

Expansion Work

Expansion work is a term in thermodynamics that refers to the work done when a system changes its volume under pressure. Simply put, when a gas expands, it pushes against external pressure, and this effort is called expansion work. This work can also be quantified as the product of pressure and volume change, commonly given by the formula: \[ W = P imes \Delta V \] where \(P\) stands for pressure and \(\Delta V\) represents the change in volume.

In the given problem, 75 J of expansion work is performed by the system. It indicates that the system used 75 J of energy to accomplish this expansion. This energy expenditure has to be considered when calculating the overall energy changes in the system and plays a crucial role in understanding how energy is conserved as it transitions between different forms.

Heat Transfer

Heat transfer is one of the primary ways energy is exchanged between a system and its surroundings. Within thermodynamics, it is crucial to understand whether energy is entering or leaving the system. The sign of the heat transfer value tells us this — positive for heat entering the system and negative for heat exiting.

In the exercise, calculating the heat transfer is about applying the first law of thermodynamics, which is represented as:\[ \Delta U = Q - W \] Using this formula, we found the heat transfer (\(Q\)) to be 375 J. Since it is a positive quantity, it indicates that the environment supplied 375 J of heat to the system. Recognizing the direction of heat transfer helps in comprehending how systems reach equilibrium and react to changes in their surroundings, which is pivotal in scenarios ranging from engine performance to climate models.