To calculate q, w, and E for the helium in the balloon, we need to use the first law of thermodynamics, which states that the change in internal energy (ΔE) of a system is equal to the heat (q) added to the system minus the work (w) done by the system:
ΔE = q - w
Let's go step by step to calculate q, w, and E:
Step 1: Calculate the change in temperature (ΔT)
ΔT = T₂ - T₁ = 38.0°C - 0.0°C = 38.0°C
Step 2: Calculate the change in volume (ΔV)
ΔV = V₂ - V₁ = 570 L - 500 L = 70 L
Step 3: Calculate the heat added to the system (q)
q = nCΔT
where n is the number of moles of helium and C is the molar heat capacity for helium gas.
Given:
n = 22.3 mol
C = 20.8 J/°C·mol
ΔT = 38.0°C
q = 22.3 mol × 20.8 J/°C·mol × 38.0°C
q = 18,707 J
Therefore, q = 18,707 J
Step 4: Calculate the work done by the system (w)
Since the pressure remains constant, the work done is given by the equation:
w = -PΔV
where P is the pressure in atm and ΔV is the change in volume in liters.
Given:
P = 1.00 atm
ΔV = 70 L
w = -1.00 atm × 70 L
w = -70 atm·L
Since we need the work done in Joules, we will convert atm·L to J using the conversion factor 1 atm·L = 101.3 J:
w = -70 atm·L × 101.3 J/atm·L
w = -7071 J
Therefore, w = -7071 J
Step 5: Calculate the change in internal energy (ΔE)
ΔE = q - w
ΔE = 18,707 J - (-7071 J)
ΔE = 18,707 J + 7071 J
ΔE = 25,778 J
Therefore, ΔE = 25,778 J
In summary:
q = 18,707 J
w = -7071 J
ΔE = 25,778 J