Question

Suppose \(6 \mathrm{~L}\) of a monatomic ideal gas, originally at \(400 . \mathrm{K}\) and a pressure of 3.00 atm (called state 1 ), undergoes the following processes: \(1 \rightarrow 2\) isothermal expansion to \(V_{2}=4 V_{1}\) \(2 \rightarrow 3\) isobaric compression \(3 \rightarrow 1\) adiabatic compression to its original state Find the pressure, volume, and temperature of the gas in states 2 and \(3 .\) How many moles of the gas are there?

Short answer

Answer: The gas has the following states: State 2: - Pressure: 0.75 atm - Volume: 24 L - Temperature: 400 K State 3: - Pressure: 0.75 atm - Volume: 6 L - Temperature: 100 K Number of moles: 0.55 moles.

Step-by-step solution

State 1 Information

Given: - Volume, V1 = 6 L - Temperature, T1 = 400 K - Pressure, P1 = 3.00 atm

State 2 Information

According to the problem, state 2 undergoes isothermal expansion, so T2 = T1 = 400 K and V2 = 4*V1 = 24 L. Let's find P2 next. Isothermal process: Since T2 = T1 - P1V1 = P2V2 - P2 = P1V1/V2 Substitute the values: - P2 = (3 * 6) / 24 - P2 = 0.75 atm Thus, at state 2: - P2 = 0.75 atm - V2 = 24 L - T2 = 400 K

State 3 Information

According to the problem, state 3 undergoes isobaric compression, so P3 = P2 = 0.75 atm. The volume at state 3 is equal to the initial volume, V3 = V1 = 6 L. To find the temperature at state 3, we will use the ideal gas law. Isothermal process: P1V1/T1 = P2V2/T2 = P3V3/T3 Simplify and solve for T3: - T3 = P3V3 / P1V1 * T1 Substitute the values: - T3 = (0.75 * 6) / (3 * 6) * 400 - T3 = 100 K Thus, at state 3: - P3 = 0.75 atm - V3 = 6 L - T3 = 100 K

Number of moles

To calculate the number of moles of the gas, we can use the ideal gas law with any of the states' pressure, volume, and temperature. Let's use state 1 information. PV = nRT n = PV / RT Substitute the state 1 values: n = (3.00 * 6) / (0.0821 * 400) n = 0.55 moles.

Summary

So the gas has the following states: State 2: - Pressure: 0.75 atm - Volume: 24 L - Temperature: 400 K State 3: - Pressure: 0.75 atm - Volume: 6 L - Temperature: 100 K Number of moles: 0.55 moles.