Question

A system undergoes a process consisting of the following two steps: Step 1: The system absorbs 72 \(\mathrm{J}\) of heat while 35 \(\mathrm{J}\) of work is done on it. Step \(2 :\) The system absorbs 35 \(\mathrm{J}\) of heat while performing 72 \(\mathrm{J}\) of work. Calculate \(\Delta E\) for the overall process.

Short answer

The overall change in internal energy for the process is \(\Delta E_\text{total} = 144 \mathrm{J}\).

Step-by-step solution

Calculate Change in Internal Energy for the First Step

In the first step, the system absorbs 72 J of heat (\(Q_1\)) and has 35 J of work done on it (\(W_1\)). Since work is done on the system, we must use a positive sign for \(W_1\). Thus, we can calculate the change in internal energy for the first step as: $$\Delta E_1 = Q_1 - W_1 = 72 - 35 = 37 \mathrm{J}$$

Calculate Change in Internal Energy for the Second Step

In the second step, the system absorbs 35 J of heat (\(Q_2\)) and performs 72 J of work (\(W_2\)). Since work is done by the system, we must use a negative sign for \(W_2\). Thus, we can calculate the change in internal energy for the second step as: $$\Delta E_2 = Q_2 - W_2 = 35 - (-72) = 35 + 72 = 107 \mathrm{J}$$

Calculate the Overall Change in Internal Energy

Now that we have both ΔE₁ and ΔE₂, we can calculate the overall change in internal energy by combining these values: $$\Delta E_\text{total} = \Delta E_1 + \Delta E_2 = 37 + 107 = 144 \mathrm{J}$$ So, the overall change in internal energy for the process is 144 J.

Key concepts

Internal Energy Change

The concept of internal energy change is central to understanding the First Law of Thermodynamics. Internal energy is the total energy contained within a system, and it can change through heat exchange and work interactions.
The change in internal energy (\( \Delta E \)) can be calculated by considering both the heat added to the system and the work done on or by the system.
In mathematical terms, \( \Delta E \) is expressed as:

• \( \Delta E = Q - W \)

Where \( Q \) represents the heat exchanged and \( W \) represents the work done. Heat added to the system increases its internal energy, while work done by the system decreases it.
For our exercise, when calculating for each step independently, we sum the changes from both steps to get the overall internal energy change of the process.

• In Step 1, \( \Delta E_1 = 72 \mathrm{J} - 35 \mathrm{J} = 37 \mathrm{J} \)
• In Step 2, \( \Delta E_2 = 35 \mathrm{J} - (-72 \mathrm{J}) = 107 \mathrm{J} \)
• Total change, \( \Delta E_{\text{total}} = 37 \mathrm{J} + 107 \mathrm{J} = 144 \mathrm{J} \)

Each term corresponds to contributions to internal energy from heat and work, highlighting the energy conversion within the system.

Heat Exchange in Processes

Heat exchange is a pivotal aspect of thermodynamic processes. It refers to the transfer of energy between a system and its surroundings due to temperature differences.
Heat (\( Q \)) flows from a region of higher temperature to one of lower temperature, and it can be absorbed or released by the system.
In analyzing processes involving heat exchange, consider:

• Heat absorbed increases the system's internal energy.
• Heat released decreases it.

In the given exercise:

• Step 1 absorbs 72 J of heat.
• Step 2 absorbs 35 J of heat.

These absorptions increase the system's internal energy in each step, contributing positively to the overall energy change.
It's critical to track heat exchanges accurately, as they directly affect the internal state of the system.

Work Done On/By the System

Work is another form of energy transfer associated with force applied over a distance. When analyzing work in thermodynamic processes, it's essential to know the direction of work concerning the system:

• Work done on the system adds energy to it, increasing internal energy.
• Work done by the system expends energy, decreasing internal energy.

The sign convention is crucial:

• Positive work is done on the system (energy input).
• Negative sign implies work done by the system (energy output).

In the example:

• Step 1 involves 35 J of work done on the system.
• Step 2 involves 72 J of work done by the system.

By understanding these interactions, students can predict how work alters the energy status of a system.
For successful analysis, use the work component to complement the heat exchange, offering a complete picture of energy transformations.