# Thermodynamics, Heat capacity.

1. Apr 26, 2010

### Je m'appelle

1. The problem statement, all variables and given/known data
The molar heat capacity at a constant volume of a solid at low temperatures T << Td, where Td is the Debye temperature is given by:

$$C_v = 464(\frac{T}{T_d})^3$$

Consider Td = 281 K for the NaCl.

(a) Calculate the average molar heat capacity $$\bar{C_v}$$ of the NaCl between the temperatures of Ti = 10 K and Tf = 20 K.

(b) Calculate the amount of heat necessary to raise the temperature of 1,000g of NaCl from 10 K to 20 K.

2. Relevant equations

$$C_v = 464(\frac{T}{T_d})^3$$

$$C_v = (\frac{\delta Q}{dT})_v$$

3. The attempt at a solution

(a) So I found the heat capacities Ci and Cf at Ti and Tf, respectively, and then in order to find the average heat capacity I summed them and divided by 2.

$$C_i = 464(\frac{10}{281})^3$$

$$C_i = 0,021$$

$$C_f = 464(\frac{20}{281})^3$$

$$C_f = 0,167$$

$$\bar{C_v} = \frac{C_i + C_f}{2}$$

$$\bar{C_v} = 0,94$$

Is this correct?

(b)

$$(\frac{\delta Q}{dT})_v = C_v$$

$$\delta Q = \bar{C_v} dT$$

$$\delta Q = (0,94)(10) = 9,4$$

Is this also correct?

Thanks.

2. Apr 26, 2010

### zachzach

I don't think the first part is correct since:

$$f_{av} = \frac{1}{b-a} \int_{a}^{b} f(x) dx$$

3. Apr 26, 2010

### Je m'appelle

I see my mistake now, thank you very much kind sir.