Calculating ideal gas compressor work and final temperature

H=Cp(T2-T1)In summary, the problem involves using a compressor to adiabatically pressurize three moles of ideal gas from 5 bar and 150 C to 50 bar. The minimum work that must be supplied to the compressor and the outlet temperature of the ideal gas are being determined, assuming a constant pressure heat capacity of 29 J/(mol*K). To solve for T2, the equation T2/T1=(P2/P1)^((γ-1)/γ) is used, where γ=Cp/Cv. Since the process is adiabatic, Q=0 and therefore work=ΔU=ΔH=Cp(T2-T1).
  • #1
supahtofu
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A compressor is being used to adiabatically pressurize three moles of ideal gas from 5 bar and 150 C to 50 bar. Determine the minimum work that must be supplied to the compressor and the outlet temperature of the ideal gas. Assume the ideal gas has a constant pressure heat capacity of 29 J/(mol*K).

Relevant equations: steady state energy balance: 0=Mk(deltaH)+Q+W
Cp=deltaH/deltaT

Attempt at solution: The only problem I'm having is determining how to find T2. With T2 I can solve for delta H and plug it into the energy balance to find W, but I have no clue how to get T2. I don't think I can do anything with the ideal gas law either so I'm not sure what to do. I was thinking maybe P1/T1=P2/T2 but I don't believe the conditions are quite right for that, pllus we've never used that equation in class. Anyone have any ideas or advice?

Thank you for your time.
 
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  • #2
Cp=Cv+R

γ=Cp/Cv

For adiabatic Processes:
T2/T1=(P2/P1)^((γ-1)/γ)

Solve for T2

Since adiabatic Q=0 therefore work=ΔU
 

Related to Calculating ideal gas compressor work and final temperature

1. What is the ideal gas law and how does it relate to calculating compressor work and final temperature?

The ideal gas law is a fundamental equation in thermodynamics that explains the behavior of gases under different conditions. It states that the product of pressure and volume of an ideal gas is directly proportional to the number of moles and the absolute temperature. This law is used to calculate the work done by a gas compressor and the final temperature of the gas after compression.

2. How do you calculate the work done by a gas compressor?

The work done by a gas compressor can be calculated by using the equation W = PΔV, where W is the work done, P is the pressure, and ΔV is the change in volume. This equation is derived from the ideal gas law, where work is equal to the negative of the change in internal energy (ΔU) of the gas.

3. What factors affect the ideal gas compressor work and final temperature?

The ideal gas compressor work and final temperature are affected by several factors, including the initial pressure and volume of the gas, the ratio of specific heats of the gas, and the efficiency of the compressor. Additionally, the temperature and pressure of the surroundings also play a role in determining the final temperature of the gas after compression.

4. Can the ideal gas law be used for any gas?

The ideal gas law is applicable to any gas that behaves ideally under the given conditions. This means that the gas should be at a low pressure and high temperature, and the molecules should not interact with each other. However, at high pressures and low temperatures, the ideal gas law may not accurately predict the behavior of real gases.

5. How is the final temperature of a gas calculated after compression?

The final temperature of a gas after compression can be calculated using the ideal gas law, where T2 = T1 (P2/P1)^((γ-1)/γ), where T1 and T2 are the initial and final temperatures, P1 and P2 are the initial and final pressures, and γ is the ratio of specific heats of the gas. This equation is derived from the first law of thermodynamics, which states that the change in internal energy of a system is equal to the sum of heat and work exchanged with the surroundings.

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