Problem 15 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Calculate the energy transferred as work, \(w\), when a gas is comprossed from \(250 \mathrm{cm}^{3}\) to \(125 \mathrm{cm}^{3}\) by an external prossure of \(10 \mathrm{kPa}\). (Section 13.5 )

Step-by-Step Solution

Answer

The energy transferred as work is \(1.25\ \mathrm{J}\).

Step 1: Understanding the Problem

We need to calculate the work done when a gas is compressed. The formula for work done by an external pressure is given by \[ w = -P \cdot \Delta V \]where \( P \) is the external pressure and \( \Delta V \) is the change in volume. The negative sign indicates that work is done on the system.

Step 2: Identifying Known Values

We have:- Initial volume \( V_i = 250 \ \mathrm{cm}^3 \).- Final volume \( V_f = 125 \ \mathrm{cm}^3 \).- External pressure \( P = 10 \ \mathrm{kPa} \).Convert the volumes from \( \mathrm{cm}^3 \) to \( \mathrm{m}^3 \):- \( V_i = 250 \times 10^{-6} \ \mathrm{m}^3 \).- \( V_f = 125 \times 10^{-6} \ \mathrm{m}^3 \).

Step 3: Calculating Change in Volume

Compute the change in volume (\( \Delta V \)) using the formula:\[ \Delta V = V_f - V_i = 125 \times 10^{-6} - 250 \times 10^{-6} = -125 \times 10^{-6} \ \mathrm{m}^3 \]

Step 4: Calculating Work Done

Now calculate the work done using the formula:\[ w = -P \cdot \Delta V \]Substitute the given values:\[ w = -(10,000 \ \mathrm{Pa}) \times (-125 \times 10^{-6} \ \mathrm{m}^3) \]\[ w = 1.25 \ \mathrm{J} \]

Step 5: Final Answer and Interpretation

The positive value of the work done indicates that energy, equal to \(1.25 \ \mathrm{J}\), is transferred into the gas due to the compression by the external pressure.

Key Concepts

Gas Compression

Gas compression is a fundamental concept in thermodynamics, where the volume of a gas is decreased by an external force. When gas is compressed, its molecules are forced closer together. This process is essential in various natural phenomena and technological applications, like refrigeration and engines. Compression is usually achieved by applying an external pressure which reduces the volume the gas occupies.

• The compression of gas often leads to an increase in temperature due to the internal energy of the system being raised.
• This process is commonly described using the ideal gas law, which correlates pressure, volume, and temperature.

Understanding the effects of gas compression helps us appreciate and design systems that rely on controlled environments, such as air conditioners and car engines.

Work Done

In the context of gas compression, 'work done' refers to the energy transferred when force is applied to compress the gas. It is calculated using the formula \[ w = -P \cdot \Delta V \]where \( P \) is the external pressure and \( \Delta V \) is the change in volume. The negative sign indicates that work is inputted into the system, typically leading to an increase in the system's energy.

• The concept of work done is crucial to understanding how energy is transferred in systems.
• It helps in predicting how systems will behave under certain conditions, especially when forces are applied.

Grasping this formula allows us to easily calculate the work during gas processes, aiding in optimizing energy budgets for various applications.

External Pressure

External pressure is the force exerted on the gas from outside the system, often a critical player in compression processes. It impacts how much work is done on the gas and is critical for calculating the energy changes during compression.

• This pressure can be controlled to achieve desired outcomes in industrial and laboratory settings.
• Measurements of such pressure are typically given in kilopascals (kPa) or pascals (Pa).

Understanding external pressure's role helps identify how different factors can manipulate a gas’s state, essential for maintaining operational standards in scientific and engineering tasks.

Volume Change

Volume change, symbolized as \( \Delta V \), represents the difference between the initial and final volumes of gas. It is essential in calculating the work done during compression, using \[ \Delta V = V_f - V_i \]where \( V_f \) is the final volume and \( V_i \) is the initial volume. This shift in volume is a direct indicator of the extent of compression.

• Volume change is measured in cubic meters (\( \mathrm{m}^3 \)) or similar units.
• Negative volume change indicates that the system's volume has decreased, typical in compression.

Understanding how volume changes contribute to energy transfer in a system is crucial for calculating the system’s efficiency and planning resource allocation in technical operations.