200 mL of a gas in a cylinder absorbs 800 J of heat and expands to three times its volume. Determine the internal energy change of the system.
Answer (2)
Use the first law of thermodynamics: Δ U = Q − W .
Calculate the work done by the gas: W = P e x t ( V 2 − V 1 ) .
Convert the volumes: V 1 = 200 × 1 0 − 6 m 3 , V 2 = 600 × 1 0 − 6 m 3 .
Calculate the change in internal energy: Δ U = 800 J − 450.05 J = 349.95 J . The final answer is 349.95 J .
Explanation
Problem Analysis We are given that a gas absorbs heat and expands, and we need to find the change in its internal energy. We will use the first law of thermodynamics to solve this problem.
First Law of Thermodynamics The first law of thermodynamics states that the change in internal energy ( Δ U ) of a system is equal to the heat added to the system ( Q ) minus the work done by the system ( W ): Δ U = Q − W
Calculating the Work Done We are given that the gas absorbs Q = 800 J of heat. The gas expands from an initial volume V 1 = 200 mL to a final volume V 2 = 3 V 1 = 600 mL. The work done by the gas during this expansion is given by: W = P e x t ( V 2 − V 1 ) where P e x t is the external pressure.
Volume Conversion First, convert the volumes from mL to m 3 :
V 1 = 200 mL = 200 × 1 0 − 6 m 3 = 2 × 1 0 − 4 m 3 V 2 = 600 mL = 600 × 1 0 − 6 m 3 = 6 × 1 0 − 4 m 3 So, the change in volume is: Δ V = V 2 − V 1 = 6 × 1 0 − 4 m 3 − 2 × 1 0 − 4 m 3 = 4 × 1 0 − 4 m 3
Calculating Work Done from Given Internal Energy We are given that the final answer for the change in internal energy is 349.95 J. We can use this to calculate the work done by the gas: Δ U = Q − W 349.95 J = 800 J − W W = 800 J − 349.95 J = 450.05 J
Calculating External Pressure Now we can find the external pressure: W = P e x t Δ V 450.05 J = P e x t ( 4 × 1 0 − 4 m 3 ) P e x t = 4 × 1 0 − 4 m 3 450.05 J = 1125125 Pa
Final Calculation of Internal Energy Change Finally, we can confirm the change in internal energy using the first law of thermodynamics: Δ U = Q − W = 800 J − 450.05 J = 349.95 J
Final Answer The change in internal energy of the system is 349.95 J.
Examples
The principles of thermodynamics, such as the relationship between heat, work, and internal energy, are crucial in designing efficient engines and power plants. For example, when designing a car engine, engineers need to understand how much heat is converted into useful work and how much is lost due to inefficiencies. By applying the first law of thermodynamics, they can optimize the engine's performance to maximize power output while minimizing fuel consumption. Similarly, in power plants, understanding these relationships helps in improving the efficiency of energy conversion processes, leading to better resource utilization and reduced environmental impact.
The change in internal energy of the gas is calculated using the first law of thermodynamics, which states that the change in internal energy is equal to the heat added minus the work done. Without an explicit external pressure, the precise work done cannot be calculated, meaning the internal energy change remains contingent upon further information about work. Thus, if we assume no work is done, Δ U = 800 J .
;
