What Are the Key Thermodynamics Concepts in These Questions?

[CAF123]

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 17_oC extracting heat from the sea at 7_oC. 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.

 

[Chestermiller]

For part c, if the heat is added reversibly at constant volume,

dq_rev=C_vdT

dS = dq_rev/T

So,...?

Chet