Consider a system consisting of 2.0 mol Carbon dioxide gas, inititially at 25 deg and 10 atm and confined to a cylinder of cross section 10.0square centimeters. lt is allowed to expand adiabatically against an external pressure of 1.0atm until the piston has moved outwards through 20cm. Assume that Carbon dioxidemay be considered a perfect gas with Cv,m 28.8 JK/mol calculate q, w, delta U, delta T and delta S.
To calculate the values of q, w, delta U, delta T, and delta S, we'll need to use the first law of thermodynamics, which states that the change in internal energy (delta U) of a system is equal to the heat (q) added to the system minus the work (w) done by the system.
We can break down the problem into steps:
Step 1: Determine the final volume (Vf) of the system.
Since the initial volume (Vi) is not given, we need to find it. The cross-sectional area (A) of the cylinder is given as 10.0 square centimeters. The piston has moved outwards through 20 cm, which means the change in volume (delta V) is equal to A multiplied by the distance the piston has moved, i.e., delta V = A * 20 cm.
To determine the final volume, we can use the ideal gas law equation: 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 can rewrite this equation as V = (nRT)/P.
Plugging in the values, we have Vf = (2.0 mol * 0.0821 L·atm/(mol·K) * 298 K) / 1.0 atm.
Step 2: Determine the final temperature (Tf) of the system.
The process is adiabatic, which means there is no heat transfer (q = 0). Therefore, we need to use the adiabatic relation: (Tf / Ti) = (Vf / Vi)^(gamma - 1), where gamma is the heat capacity ratio for the gas. For carbon dioxide, gamma ≈ 1.4.
Rearranging the equation, we have Tf = Ti * (Vf / Vi)^(gamma - 1).
Plugging in the values, we get Tf ≈ 25 °C * [(Vf) / (Vi)]^(1.4 - 1).
Step 3: Calculate the work done by the system (w).
The work done by the system is given by the equation: w = -Pext * delta V, where Pext is the external pressure and delta V is the change in volume.
Plugging in the values, we have w = -1.0 atm * A * 20 cm.
Step 4: Calculate the change in internal energy (delta U).
Since the process is adiabatic (q = 0), the change in internal energy is equal to the negative of the work done (delta U = -w).
Step 5: Calculate the change in temperature (delta T).
The change in temperature can be calculated by subtracting the initial temperature (Ti) from the final temperature (Tf), i.e., delta T = Tf - Ti.
Step 6: Calculate the change in entropy (delta S).
The change in entropy can be calculated using the formula: delta S = n * Cv * ln(Tf / Ti), where n is the number of moles and Cv is the molar heat capacity at constant volume.
Plugging in the values, we have delta S = 2.0 mol * 28.8 J/K * ln(Tf / Ti).
Now, you can substitute the values calculated in the respective steps to find the values of q, w, delta U, delta T, and delta S.