There are 1.36 kgm of air at 137.9 Kpaa stirred with internal paddles in an insulated rigid container whose volume is 0.142 m3, until the pressure becomes 689.5 Kpaa. Determine (a) the work input, (b) Δ(pV), and (c) Q. Cp=1.0062 kj/(Kgm.°K), Cv = 0.7186 kj/(Kgm.°K), R= 287.18 j/(Kgm.°K)
To determine the work input, Δ(pV), and Q in this system, we can use the first law of thermodynamics:
ΔU = Q - W
Where ΔU is the change in internal energy, Q is the heat transfer, and W is the work done on the system.
First, let's calculate the change in internal energy, ΔU.
ΔU = m * (Cv) * ΔT
Where m is the mass of the air, (Cv) is the specific heat capacity at constant volume, and ΔT is the change in temperature.
Given:
m = 1.36 kg
(Cv) = 0.7186 kj/(Kgm.°K)
Initial temperature, T1 = 137.9 Kpaa
Final temperature, T2 = ? (since ΔT = T2 - T1)
To convert pressure from Kpaa to Kpa, we divide by 10.
Initial pressure, P1 = 137.9 Kpaa / 10 = 13.79 Kpa
Final pressure, P2 = 689.5 Kpaa / 10 = 68.95 Kpa
Let's calculate T2 using the ideal gas law:
P1 * V1 / T1 = P2 * V2 / T2
Substituting the given values:
13.79 Kpa * 0.142 m^3 / 137.9 K = 68.95 Kpa * 0.142 m^3 / T2
T2 = (68.95 Kpa * 0.142 m^3 * 137.9 K) / (13.79 Kpa)
T2 ≈ 695.91 K
ΔT = T2 - T1
ΔT = 695.91 K - 137.9 K = 558.01 K
Now, we can calculate ΔU:
ΔU = m * (Cv) * ΔT
ΔU = 1.36 kg * 0.7186 kj/(Kgm.°K) * 558.01 K
ΔU ≈ 531.63 kJ
Next, let's calculate the work input, W.
W = Δ(pV)
W = P2 * V2 - P1 * V1
Substituting the given values:
W = 68.95 Kpa * 0.142 m^3 - 13.79 Kpa * 0.142 m^3
W ≈ 7.9656 kJ
Finally, we can determine the heat transfer, Q:
ΔU = Q - W
Q = ΔU + W
Q = 531.63 kJ + 7.9656 kJ
Q ≈ 539.6 kJ
Therefore,
(a) The work input is approximately 7.9656 kJ.
(b) Δ(pV) is approximately 7.9656 kJ.
(c) Q is approximately 539.6 kJ.
To determine the work input (a), Δ(pV) (b), and Q (c), we need to use the first law of thermodynamics, which states:
ΔU = Q - W
where:
ΔU represents the change in internal energy
Q represents the heat transfer
W represents the work input
Step 1: Calculate the change in internal energy (ΔU)
To calculate the change in internal energy, we can use the equation:
ΔU = Cv * m * ΔT
where:
Cv represents the specific heat at constant volume
m represents the mass of the air
ΔT represents the change in temperature
Given values:
Cv = 0.7186 kJ/(kgm.°K)
m = 1.36 kgm
Initial temperature = 137.9 Kpaa
Final temperature = 689.5 Kpaa
Since the pressure remains constant and the volume is constant (insulated rigid container), we can assume the process is isochoric (constant volume). Therefore, ΔT = Tf - Ti.
ΔT = 689.5 Kpaa - 137.9 Kpaa
Now calculate ΔU:
ΔU = Cv * m * ΔT
Step 2: Calculate the work input (W)
To calculate the work input, we can use the equation:
W = Δ(pV)
where:
Δ(pV) represents the change in pressure multiplied by the change in volume
Given values:
Initial pressure = 137.9 Kpaa
Final pressure = 689.5 Kpaa
Volume = 0.142 m3
Calculate Δ(pV):
Δ(pV) = (Pf - Pi) * V
Step 3: Calculate the heat transfer (Q)
To calculate the heat transfer, we can use the equation derived from the first law of thermodynamics:
Q = ΔU + W
Given the values of ΔU and W calculated earlier, we can substitute them to find Q.
Now that we have explained the steps, you can use the provided formulas and values to calculate the work input (a), Δ(pV) (b), and Q (c).