a system consisting of 1.5 mol CO2(g), initially at 15deg and 10 atm and confined to a cylinder of cross-section 10.0 cm2. it is allowed to expand adibatically against an external pressure of 1.0 atm until the piston has moved through 20 cm. assume that carbon dioxide may be considered a perfect gas with Cv,m = 28.8 J/Kmol
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Abel
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To solve this problem, we need to use the ideal gas law and the first law of thermodynamics. Let's break it down step by step:
Step 1: Find the final temperature using the ideal gas law.
The ideal gas law is given by: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
We are given:
Initial pressure, P1 = 10 atm
Final pressure, P2 = 1.0 atm
Number of moles, n = 1.5 mol
We are looking to find the final temperature, T2. We can rearrange the ideal gas law as follows to solve for T2:
T2 = (P2/P1) * (V1/V2) * T1
The volume at the initial state, V1, can be calculated using the cross-sectional area of the cylinder (A) and the displacement of the piston (d). The initial volume is given by: V1 = A * d1
We are not given the value for d1, so unless it's explicitly stated in the problem, we cannot determine the initial volume. Therefore, we won't be able to calculate the final temperature using the ideal gas law alone.
Step 2: Use the adiabatic expansion to find the final temperature.
Since the expansion is adiabatic, there is no heat transfer to or from the system. This means that the change in internal energy (ΔU) is equal to the work done (W) by the system.
The first law of thermodynamics is given by: ΔU = W + Q, where ΔU is the change in internal energy, W is the work done on or by the system, and Q is the heat transfer.
For an adiabatic process, Q = 0. Therefore, we can simplify the first law of thermodynamics as: ΔU = W
The change in internal energy can be calculated using ΔU = n * Cv,m * ΔT, where Cv,m is the molar heat capacity at constant volume.
Given that ΔU = 0 (since the adiabatic process has no heat transfer), we can solve for ΔT. ΔT = -(ΔU) / (n * Cv,m)
Step 3: Calculate the final temperature, T2.
We're given that the total piston displacement is 20 cm. So, the change in volume can be calculated as: ΔV = A * d2, where d2 is the displacement of the piston.
We're also given the specific heat capacity at constant volume, Cv,m = 28.8 J/Kmol.
Using the relationship between work done and change in volume, W = -P2 * ΔV, we can substitute it into the equation ΔU = W to get:
0 = (-P2 * ΔV) + n * Cv,m * ΔT
Solving for ΔT:
ΔT = (P2 * ΔV) / (n * Cv,m)
Finally, using the initial temperature T1 = 15 °C, we can calculate the final temperature:
T2 = T1 + ΔT
With the given information, we have the necessary formulae to solve for the final temperature, T2.