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CAF123
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Homework Statement
a)What value does ##\left(\frac{\partial U}{\partial V}\right)_T## tend to as T tends to 0?
b)A heat pump delivers 2.9kW of heat to a building maintained at 17oC extracting heat from the sea at 7oC. What is the minimum power consumption of the pump?
c)Explain how a measurement of ##C_v## can be used to determine the difference in entropy between equal volume equilibrium states at different temperatures.
Homework Equations
Cyclic rule, Carnot efficiencies, third law of thermodynamics
The Attempt at a Solution
a)I used the cyclic rule here and wrote $$\left(\frac{\partial U}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_U \left(\frac{\partial T}{\partial U}\right)_V = -1$$ to give $$\left(\frac{\partial U}{\partial V}\right)_T = -\frac{C_v}{V \beta_U}$$ where ##C_v## is the constant volume heat capacity and ##\beta_U## is the thermal expansivity at constant U. I think both the thermal expansivity and heat capacity both tend to 0 as T goes to 0, so overall the quantity of interest goes to 0 too. Is this okay?
b) I am a bit confused of the set up (see attached for what I think is going on). Generally for a heat pump the efficiency is defined as Q1/W, where Q1 is the heat supplied to some region and W is the work you had to do to supply the heat. If the heat pump operates between two reservoirs, then max efficiency is T1/(T1-T2), where T2 is the lower temperature reservoir (the sea in this case).
c) ##Q = \int T dS \Rightarrow## $$\left(\frac{\partial Q}{\partial T}\right)_V \equiv C_v = \frac{\partial}{\partial T} \int T dS = \int dS + \int T \left(\frac{\partial S}{\partial T}\right)_V$$ Is this helpful?
Many thanks.