Calculate the work done when one mole of an ideal gas at 298 K and 5 atm pressure is allowed to expand irreversibly against the constant outside pressure of 1 atm until the internal pressure becomes the same as the outside pressure.
Answer (2)
Calculate the initial volume using the ideal gas law: V 1 = P 1 n R T 1 = 5 1 × 0.0821 × 298 = 4.89316 L .
Calculate the final volume using the ideal gas law: V 2 = P 2 n R T 2 = 1 1 × 0.0821 × 298 = 24.4658 L .
Calculate the work done: W = − P e x t ( V 2 − V 1 ) = − 1 × ( 24.4658 − 4.89316 ) = − 19.57264 L atm .
Convert the work to Joules: W = − 19.57264 × 101.3 = − 1982.708432 J . Therefore, the work done is − 1982.7 J .
Explanation
Problem Setup and Initial Conditions We are given one mole of an ideal gas at an initial temperature T 1 = 298 K and initial pressure P 1 = 5 atm. The gas expands irreversibly against a constant external pressure P e x t = 1 atm until the internal pressure equals the external pressure, so the final pressure P 2 = 1 atm. Our goal is to calculate the work done during this irreversible expansion.
Calculating Initial Volume First, we need to find the initial volume V 1 using the ideal gas law: P 1 V 1 = n R T 1 . We have n = 1 mole, R = 0.0821 m o l × K L × a t m , T 1 = 298 K, and P 1 = 5 atm. Therefore, V 1 = P 1 n R T 1 = 5 1 × 0.0821 × 298 = 4.89316 L
Calculating Final Volume Next, we find the final volume V 2 . Since the final pressure P 2 is equal to the external pressure, P 2 = 1 atm. We assume the temperature remains constant at T 2 = 298 K. Using the ideal gas law again: V 2 = P 2 n R T 2 = 1 1 × 0.0821 × 298 = 24.4658 L
Calculating Work Done in L atm Now we calculate the work done during the irreversible expansion against a constant external pressure using the formula: W = − P e x t ( V 2 − V 1 ) = − 1 atm × ( 24.4658 L − 4.89316 L ) = − 19.57264 L atm
Converting Work to Joules Finally, we convert the work from L atm to Joules using the conversion factor 1 L atm = 101.3 J :
W = − 19.57264 L atm × 101.3 L atm J = − 1982.708432 J Rounding to one decimal place, we get W = − 1982.7 J.
Final Answer Therefore, the work done during the irreversible expansion is approximately -1982.7 J.
Examples
The expansion of gases, as calculated in this problem, is crucial in understanding the operation of internal combustion engines. When fuel combusts, it creates hot gases that expand and push against a piston, doing work. The amount of work done depends on the pressure difference and volume change, similar to our calculation. Engineers use these principles to optimize engine design for maximum efficiency and power output. For example, consider a car engine where the expanding gas does work on the piston. By understanding the principles in this problem, engineers can calculate and optimize the work done by the gas to improve the engine's performance.
The work done when one mole of an ideal gas expands irreversibly against a constant external pressure of 1 atm is approximately -1982.7 J. The calculation involves determining the initial and final volumes using the Ideal Gas Law and applying the work formula. The negative sign indicates that the gas is doing work on the environment.
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