Please help me with these four questions 1) A monatomic ideal gas is initially a
ID: 1303779 • Letter: P
Question
Please help me with these four questions
1) A monatomic ideal gas is initially at a pressure of 155 kPa and volume of 5.5 L. The gas is then compressed adiabatically.
If Tiis the initial temperature of the gas, how much work must be done to raise the temperature of the gas to Tf= 2.4 Ti?
Give your answer in Joules to at least 3 significant digits. (Your answer should be positive.)
2)
The P-V diagram shows a 4-step process where a monatomic ideal gas goes through a constant pressure compression (1 to 2), a constant volume pressure increase (2 to 3), a constant temperature expansion (3 to 4), and an adiabatic expansion (4 to 1).
In this process, V2 = 0.25 V1 and V4 = 0.90 V1.
What is the ratio of the change in thermal energy of the gas for the second process to the change in thermal energy for the fourth process?
In other words, what is (?E23 / ?E41) for this cycle?
Your answer should be unitless. Give your answer to at least three significant figures.
Hint: It helps to write everything in terms of T1, the temperature at the start, and ratios of volumes.
Warning: Signs matter.
3)
An electrical power plant (a heat engine) has an efficiency of 34% and a cooling system that can handle a maximum heat exhaust of 6.8 MW (6.8 x 106 J/s).
What is the maximum output power that can be obtained from this power plant?
Give your answer in MW to at least 3 signficant digits. Do not include units in your answer.
4)
Consider a heat engine, diagrammed above.
The input energy (heat) is Qin = 1,150 J and the ouput exhaust energy (heat) is Qout = 350J.
If TH = 485 K, what is the largest possible value for TC?
Give your answer in Kelvin to at least three signficant figures. Do not include units with your answer.
Explanation / Answer
1)
W = (P2*V2 - P1*V1) / (k-1)
= nR(T2 - T1) / (k-1)
= nR(2.4*T1 - T1) / (k-1)
= 1.4*nRT1 / (k-1)
= 1.4*P1*V1 / (k-1)
= 1.4*(155*10^3) *(5.5*10^-3) / (1.67-1)
= 1781.343 J
2)
E23 = Cv(T3 - T2)
E41 = Cv(T1 - T4)
E23 / E41 = (T3 - T2) / (T1 - T4)
For process 1-2: V1 / T1 = V2 / T2
T2 = (V2 / V1)*T1
= 0.25*T1
For process 2-3:
P2 / T2 = P3 / T3
T3 / T2 = P3 / P2
T3 / T2 = P3 / P1
For process 3-4:
P3*V3 = P4*V4
P3/P4 = V4 / V3
= V4 / V2
= V4 / (0.25*V1)
= 0.9/ 0.25
= 3.6
P3 / P4 = 3.6
For process 4-1:
T4 / T1 = (V1 / V4)^(k-1)
= (1 / 0.9)^(1.67 - 1)
= 1.073
Also, P4 / P1 = (V1 / V4)^k
= (1 / 0.9)^1.67
= 1.192
P3 / P1 = (P3 / P4)*(P4 / P1)
= 3.6*1.192
= 4.2926
T3 / T2 = 4.2926
T3 / (0.25*T1) = 4.2926
T3 = 1.073*T1
E23 / E41 = (T3 - T2) / (T1 - T4)
= (1.073*T1 - 0.25*T1) / (T1 - 1.073*T1)
= -11.254
3)
Carnot efficiency = Power output / Heat input
= Powe routput / (Heat output + Power output)
0.34 = P / (P + 6.8)
P = 3.503 MW
4)
Carnot efficiency = 1 - Tc / Th = (Qin - Qout) / Qin
1 - Tc / 485 = (1150 - 350) / 1150
Tc = 147.609 K