Problem 68 - Pearson Physics

Question

A system expands by \(0.75 \mathrm{~m}^{3}\) at a constant pressure of \(125 \mathrm{kPa}\). Find the heat that flows into or out of the system if its thermal energy (a) increases by \(65 \mathrm{~J}\) or (b) decreases by \(1850 \mathrm{~J}\). In each case, give the direction of the heat flow.

Step-by-Step Solution

Answer

(a) Heat flows into the system: 93,815 J. (b) Heat flows into the system: 91,900 J.

Step 1: Understanding Work Done

Work done by the system during expansion can be calculated using the formula \( W = P \Delta V \), where \( P \) is the constant pressure and \( \Delta V \) is the change in volume. Here, \( P = 125 \text{ kPa} = 125,000 \text{ Pa} \) and \( \Delta V = 0.75 \text{ m}^3 \).

Step 2: Calculating Work Done

Calculate the work done by substituting the values into the formula: \[ W = 125,000 \times 0.75 = 93,750 \text{ Joules} \]

Step 3: Applying the First Law of Thermodynamics

The First Law of Thermodynamics can be expressed as \( \Delta U = Q - W \), where \( \Delta U \) is the change in thermal energy, \( Q \) is the heat added to the system, and \( W \) is the work done by the system. Rearranging, we find \( Q = \Delta U + W \).

Step 4a: Calculate Heat Flow for Increased Thermal Energy

If the thermal energy increases by \( 65 \text{ J} \), substitute this value and the work done into the equation: \[ Q = 65 + 93,750 = 93,815 \text{ Joules} \]Since \( Q \) is positive, heat flows into the system.

Step 5a: Interpret the Result for Increased Thermal Energy

For case (a), the system absorbs heat to account for the increase in internal energy and the work done in expansion.

Step 4b: Calculate Heat Flow for Decreased Thermal Energy

If the thermal energy decreases by \( 1850 \text{ J} \), substitute this value and the work done into the equation (considering decrease as negative): \[ Q = -1850 + 93,750 = 91,900 \text{ Joules} \]Since \( Q \) is positive, heat also flows into the system.

Step 5b: Interpret the Result for Decreased Thermal Energy

For case (b), the system still absorbs heat, though less is needed, as it compensates for a decrease in internal energy while also performing work.

Key Concepts

First Law of Thermodynamics

The First Law of Thermodynamics is a key principle that relates to the conservation of energy within a thermodynamic system. It essentially states that the energy in a closed system is constant, and energy can neither be created nor destroyed, only transformed or transferred. Mathematically, it can be expressed as:\[ \Delta U = Q - W \]Where:

• \( \Delta U \) is the change in the internal energy of the system.
• \( Q \) represents the heat added to the system.
• \( W \) is the work done by the system.

This equation highlights the balance between the change in internal energy and the heat exchanged, minus the work done. When a system does work, it expends energy; if it absorbs heat, its internal energy increases. Understanding this fundamental principle gives insight into how energy flows and transforms within systems.

Work Done

Work done in a thermodynamic context refers to energy transferred when a force acts over a distance, often involving the expansion or compression of gases. When a system expands at constant pressure, like in our original exercise, the work done is calculated using the formula:\[ W = P \Delta V \]Where:

• \( P \) is the pressure, measured in pascals (Pa).
• \( \Delta V \) is the change in volume, measured in cubic meters (m³).

In this case, the pressure is given as 125 kPa or 125,000 Pa, and the volume change is 0.75 m³. By substituting these values in, the work done is calculated as \( 93,750 \text{ Joules} \). This tells us how much energy the system uses to expand, which is crucial for determining changes in internal energy and heat flow.

Heat Flow

Heat flow is a term used to describe the transfer of thermal energy between systems or from one part of a system to another. In thermodynamics, when calculating heat flow, understanding whether the thermal energy is increasing or decreasing is essential to determine heat direction.In our exercise, we calculated the heat flow in two scenarios:1. When the system's thermal energy increased by 65 J.2. When the system's thermal energy decreased by 1850 J.Using the First Law of Thermodynamics:

• If thermal energy increases, heat flow \( Q \) is calculated as \( 93,815 \text{ Joules} \), indicating heat transfer into the system.
• If thermal energy decreases, \( Q \) becomes \( 91,900 \text{ Joules} \), still positive, meaning heat still enters the system, albeit less so, to compensate for the energy lost.

Recognizing heat flow direction and value helps to understand energy transformations occurring in thermodynamic processes.

Thermal Energy Change

Thermal energy change is a measure of how the internal energy of a thermodynamic system alters due to work done or heat exchanged. It can tell us whether a system is absorbing energy or releasing it. This is integral in processes like expansion where doing work results in potential changes to internal energy. In our scenario, calculations showed how:

• An increase in thermal energy by 65 J meant total internal energy grew, translating into a necessity for additional heat input to sustain expansion.
• A decrease by 1850 J meant the system lost energy internally, but still required heat to maintain the work output.

This insight into thermal energy helps in predicting system behavior under various conditions, reinforcing fundamental thermodynamics understanding. Knowing these energy shifts allows us to appreciate the energy journey, from input to transformation and eventual output in work.