Calculating work and heat transfer in this Carnot process

In summary, the conversation is about a problem in thermodynamics involving the calculation of entropy. The person is trying to express entropy as a function of temperature so they can use the ##T-S##-plane for their calculations. They start by expressing the fundamental equation as a function of entropy and then computing the partial derivative with respect to internal energy. This allows them to get an expression for internal energy in terms of temperature, volume, and number of particles. However, they are missing the volume in the problem statement and are unable to calculate entropy. They ask for guidance in the right direction.
  • #1
approx12
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6
Homework Statement
I'm given the fundamental equation ##UN^{1/2}V^{3/2}=\alpha (S-R)^3##, where ##A=2*10^{-2} (K^3 m^{9/2} J^3)##. Two moles of this fluid are used as the auxiliary system in a Carnot cycle, operating between two reservoirs of ##T_1=373K## and ##T_2=273K## In the first isothermal expansion ##10^6 J## is extracted from the high-temperature reservoir.
Find the heat transfer and the work transfer for each of the four processes in the Carnot cycle.
Relevant Equations
##UN^{1/2}V^{3/2}=A(S-R)^3##
Hey guys! This is problem from Callens Thermodynamics textbook and I'm stuck with it.

My goal was to get a expression for the entropy ##S## which is dependent on ##T## so I can move into the ##T-S##-plane to do my calculations:
I startet by expressing the fundamental equation as a function of ##S(U,V,N)## and then computing the partial derivative with respect to ##U##: $$\frac{\partial S}{\partial U}=\frac{1}{T}=\frac{1}{3\alpha}U^{-2/3}N^{1/6}V^{1/2}$$ Through that I'm able to get a expression of ##U(T,V,N)## which helps me to express ##S## as a function of ##T,V## and ##N##. Rearranging the partial derivative and plugging back into the original equation I get: $$S(T,V,N)=R+\sqrt{\frac{T}{3}}\frac{1}{\alpha^{3/2}}V^{3/4}N^{1/4}$$

Now it shouldn't be so hard to calculate the the heat and work transfer but what I'm missing is the volume ##V##. It's not given in the problem statement and without it I can't calculate ##S## so I'm stuck with my calculations in the ##T-S##-plane.

I must be missing something here but I can't see what it is. I would appreciate any guidance in the right direction!
 
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  • #2
I would start out by deriving the equation of state.
 

1. What is a Carnot process?

A Carnot process is a theoretical thermodynamic cycle that describes the most efficient way to convert heat energy into work or vice versa. It involves a reversible cycle between two heat reservoirs at different temperatures.

2. How is work calculated in a Carnot process?

The work done in a Carnot process can be calculated using the formula W = Qh - Qc, where Qh is the heat absorbed from the hot reservoir and Qc is the heat released to the cold reservoir.

3. What is the efficiency of a Carnot process?

The efficiency of a Carnot process is given by the formula η = (Th - Tc)/Th, where Th is the temperature of the hot reservoir and Tc is the temperature of the cold reservoir. This means that the efficiency is dependent on the temperature difference between the two reservoirs.

4. How is heat transfer calculated in a Carnot process?

The heat transfer in a Carnot process can be calculated using the formula Q = mCΔT, where m is the mass of the gas, C is the specific heat capacity, and ΔT is the change in temperature. This formula applies to both the hot and cold reservoirs.

5. What are the limitations of a Carnot process?

A Carnot process is a theoretical concept and cannot be achieved in real-world systems. It assumes that the process is reversible and that there are no energy losses due to friction or other factors. In reality, all processes involve some level of irreversibility and energy losses, making the efficiency of a Carnot process unattainable.

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