An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Calculate P 1 using the ideal gas law: P 1 = V 1 n R T 1 = 0.003 m 3 ( 1 mol ) ( 8.0 J/mol*K ) ( 300 K ) = 800000 Pa ≈ 7.895 atm .
Assume an isochoric process from state 1 to state 2.
Calculate P 2 using the relation T 1 P 1 = T 2 P 2 , so P 2 = P 1 T 1 T 2 = 800000 Pa × 300 K 50 K ≈ 133333.33 Pa ≈ 1.316 atm .
The pressures are P 1 ≈ 7.895 atm and P 2 ≈ 1.316 atm .
Explanation
Problem Setup and Given Information We are given a problem involving a diatomic gas undergoing a process. We are given the initial and final temperatures and volumes, and we need to find the initial and intermediate pressures, P 1 and P 2 . We will use the ideal gas law and the information provided to determine these pressures. We are given:
Number of moles, n = 1 mol
Initial temperature, T 1 = 300 K
Intermediate temperature, T 2 = 50 K
Final temperature, T 3 = 300 K
Initial volume, V 1 = 3 liters
Final volume, V 3 = 9 liters
Gas constant, R = 8.0 J/mol*K
Conversion factor, 1 atm = 101325 Pa
We need to find P 1 and P 2 .
Calculating P1 using the Ideal Gas Law First, we will calculate P 1 using the ideal gas law, P V = n RT . We have P 1 V 1 = n R T 1 , so P 1 = V 1 n R T 1 .
We need to convert V 1 from liters to m 3 . Since 1 liter = 0.001 m 3 , we have V 1 = 3 liters = 0.003 m 3 .
Now we can calculate P 1 in Pascals:
P 1 = V 1 n R T 1 = 0.003 m 3 ( 1 mol ) ( 8.0 J/mol*K ) ( 300 K ) = 0.003 2400 Pa = 800000 Pa
Now, we convert P 1 from Pascals to atm:
P 1 = 101325 Pa/atm 800000 Pa ≈ 7.895 atm
So, P 1 ≈ 7.895 atm.
Calculating P2 assuming an Isochoric Process To find P 2 , we need information about the process from state 1 to state 2. Since the diagram is not provided, we will assume the process is isochoric (constant volume), meaning V 1 = V 2 . In this case, we can use the relation T 1 P 1 = T 2 P 2 , so P 2 = P 1 T 1 T 2 .
P 2 = P 1 T 1 T 2 = 800000 Pa × 300 K 50 K = 800000 Pa × 6 1 ≈ 133333.33 Pa
Converting P 2 to atm:
P 2 = 101325 Pa/atm 133333.33 Pa ≈ 1.316 atm
So, P 2 ≈ 1.316 atm.
Final Answer Therefore, based on the given information and assuming an isochoric process from state 1 to state 2, we have:
P 1 ≈ 7.895 atm P 2 ≈ 1.316 atm
Examples
Understanding the behavior of gases is crucial in many real-world applications, such as designing internal combustion engines, refrigeration systems, and even weather forecasting models. In an internal combustion engine, the compression and expansion of gases within the cylinders directly impact the engine's efficiency and power output. By applying the principles of thermodynamics and the ideal gas law, engineers can optimize engine designs to maximize performance and minimize fuel consumption. Similarly, in refrigeration systems, understanding the properties of refrigerants and their phase transitions is essential for creating efficient cooling cycles. These principles also extend to larger-scale applications, such as predicting atmospheric conditions and designing efficient power plants.
About 2.81 x 10^{21} electrons flow through the device when a current of 15.0 A is delivered for 30 seconds. This was calculated by determining the total charge flow and then dividing by the charge of a single electron. The total charge is 450 C, which corresponds to that many electrons.
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