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Chapter 18: Problem 26

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

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

Step by step solution

01

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
02

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
03

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
04

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).

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Most popular questions from this chapter

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