# Short thermodynamics questions

1. May 14, 2014

### CAF123

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
Cyclic rule, Carnot efficiencies, third law of thermodynamics

3. 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.

#### Attached Files:

• ###### Heat pump.png
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2. May 14, 2014

### Staff: Mentor

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

dqrev=CvdT

dS = dqrev/T

So,...?

Chet