Question

The internal energy of a gas is \(500 .\) J. The gas is compressed adiabatically, and its volume decreases by \(100 . \mathrm{cm}^{3} .\) If the pressure applied on the gas during compression is \(3.00 \mathrm{~atm},\) what is the internal energy of the gas after the adiabatic compression?

Short answer

Answer: The final internal energy of the gas after the adiabatic compression is 530.4 J.

Step-by-step solution

Understand the First Law of Thermodynamics

The First Law of Thermodynamics states that the change in internal energy (ΔU) of a system is equal to the heat (Q) added to the system minus the work (W) done by the system, or: ΔU = Q - W For an adiabatic process, there is no heat exchange between the system and its surroundings, which means Q = 0. Therefore, the First Law of Thermodynamics for an adiabatic process can be simplified to: ΔU = - W

Calculate the work done during compression

The work (W) done during compression can be found using the formula: W = P * ΔV where P is the pressure applied on the gas during compression and ΔV is the change in volume. The given pressure is in atmospheres, but we need to convert it to Pascals (SI unit). To do this, we use the conversion factor 1 atm = 101325 Pa: P = 3.00 atm * 101325 Pa / 1 atm = 303975 Pa ΔV is given as -100 cm³. We need to convert it to m³ (SI unit) using the conversion factor 1 cm³ = 1e-6 m³: ΔV = -100 cm³ * 1e-6 m³ / 1 cm³ = -1e-4 m³ Now, we can calculate the work done during compression: W = 303975 Pa * -1e-4 m³ = -30.3975 J

Calculate the change in internal energy

Now that we know the work done during compression, we can use the First Law of Thermodynamics for adiabatic processes to calculate the change in internal energy: ΔU = - W = 30.3975 J

Calculate the final internal energy

Finally, we can calculate the internal energy of the gas after the adiabatic compression. The initial internal energy is given as 500 J. Final internal energy (U_final) = Initial internal energy (U_initial) + Change in internal energy (ΔU) U_final = 500 J + 30.3975 J = 530.3975 J The internal energy of the gas after the adiabatic compression is 530.4 J (rounded to one decimal place).

Key concepts

Adiabatic Process

In an adiabatic process, there is no heat exchange between a system and its surroundings, meaning the change in heat ( Q ) is zero. For such processes, the First Law of Thermodynamics simplifies the equation for the change in internal energy ( ΔU ) since Q = 0 :

• ΔU = -W

Here, W represents the work done on or by the system. Adiabatic processes are common in thermodynamic systems where insulation prevents heat transfer. During these processes, changes in temperature occur strictly due to work done rather than heat addition or removal.
Adiabatic processes can be either lubrication aided compressions or cool expansions. When a gas compresses without losing heat to the surrounding, its temperature rises despite no external heat addition. Similarly, when a gas expands, its temperature decreases.
Understanding adiabatic processes is essential for fields that deal with thermodynamic cycles, such as refrigeration, air conditioning, and intricate engine designs.

Internal Energy

Internal energy of a gas is the energy stored within it, which results from the kinetic and potential energy of its molecules. This energy contributes to a gas's temperature and pressure and can be affected by heat transfer and work done. The First Law of Thermodynamics governs the change in internal energy.

• ΔU = Q - W

In an adiabatic process, where Q = 0 , the change in internal energy ( ΔU ) is solely influenced by the work ( W ) done. This gives rise to:

• ΔU = -W

An increase in internal energy typically leads to a rise in temperature and pressure, and vice versa. For the exercise, an initial internal energy of 500 J is adjusted by the work calculated during compression, leading to the final internal energy of 530.4 J.
Mastering internal energy changes is crucial for analyzing energy transformations and efficiency in thermodynamic systems.

Work Done

Work done in thermodynamics involves energy transfer while compressing or expanding a substance. In this context, work done ( W ) is calculated using the equation:

• W = P * ΔV

where P is pressure and ΔV is the change in volume. Work is performed when a force moves an object and in thermodynamics, this often relates to volume changes. In our exercise, compression leads to negative work done on the gas due to volume reduction.
First, you need to ensure that units for pressure and volume are compatible. Converting pressure from atm to Pa and volume from cm³ to m³ standardizes the units for precise calculations. After converting, the work done during gas compression in the exercise was found to be -30.3975 J. This work factor is critical in calculating the change in internal energy during an adiabatic process.
Understanding work done allows you to comprehend how external forces influence energy states in thermodynamic processes.

Unit Conversion

Unit conversion is a vital method in physics and engineering to ensure consistent measurements. It involves converting quantities from one unit system to another, such as Imperial to SI (International System of Units). The exercise shows conversions critical for compatibility between units of pressure and volume.
To convert pressure from atmospheres to Pascals (SI unit), you multiply by 101325 (as 1 atm = 101325 Pa ). Similarly, converting volume from cm³ to m³ requires multiplying by 1e-6 , following the relation 1 cm³ = 1e-6 m³ .
Failure to convert units correctly can lead to calculation errors and misunderstandings. Hence, mastering unit conversion ensures fidelity in scientific calculations and is indispensable in practical work environments where precise measurements are imperative.