Calorimetry: Heat Capacity, Latent Heat and Entropy Change

The method of mixtures, with a proper cooling correction, and the entropy generated by irreversible heat flow.

Objectives

On completing this experiment you should be able to:

Determine the specific heat capacity of metal specimens by the method of mixtures.
Apply a Newton's-law cooling correction to a temperature-time record and justify it.
Verify the Dulong-Petit rule and identify the specimens from their molar heat capacities.
Determine the specific latent heat of fusion of ice and of vaporisation of water.
Compute the entropy change of the system, the surroundings and the universe for each irreversible process, and confirm that the total increases.

Theory

In the method of mixtures a hot body is dropped into a calorimeter and the energy it gives up is equated to the energy gained by the calorimeter and its contents. The calculation assumes that the calorimeter is isolated, which it never is: it exchanges heat with the laboratory at a rate proportional to the excess temperature. The observed maximum is therefore lower than the true equilibrium temperature, and a cooling correction must be applied.

Because entropy is a function of state, the entropy change of each body can be computed along any convenient reversible path even though the actual process is violently irreversible:

dS = \int \frac{C\,dT}{T} = C \ln(T_f / T_i)

for a body of constant heat capacity, and $dS = \frac{mL}{T}$ for an isothermal phase change. The sum over all bodies is the entropy generated, and the second law requires it to be positive.

Key relations

$m_s c_s (T_h - T_f) = (m_{cal}c_{cal} + m_w c_w)(T_f - T_i)$
Energy balance for the method of mixtures.
$\Delta S = \sum_i C_i \ln\frac{T_f}{T_i} + \sum_j \frac{m_j L_j}{T_j} > 0$
Entropy generated by irreversible mixing.
$C\frac{\mathrm{d}T}{\mathrm{d}t} = -U\,(T - T_{\mathrm{env}})$
Newton's law of cooling, the basis of the correction.

Procedure by part

Part A - Specific heat capacity by the method of mixtures

Heat an unlabelled specimen in the steam jacket, drop it into the calorimeter, and record the temperature-time curve.

Weigh the specimen and the water. Set the initial water temperature a few degrees below room temperature so that the losses partly cancel.
Take the reading. The apparatus returns the complete temperature-time record, including a sixty-second pre-period.
Apply a cooling correction to obtain the temperature the calorimeter would have reached with no losses.
Compute the specific heat capacity, then the molar heat capacity, and identify the specimen.
Repeat for all three unlabelled specimens, at least twice each.

Part B - Specific latent heat of fusion of ice

Add dried melting ice to warm water in the calorimeter and follow the temperature until all the ice has melted.

Start with the water several degrees above room temperature so that the mean excess temperature over the run is small.
Add the ice, which is at exactly 0 °C, and record the trace.
Apply a cooling correction and compute the latent heat of fusion.
Compute the entropy change of the ice, of the water and of the universe.

Part C - Specific latent heat of vaporisation of water

Admit dry steam at 100 °C into cool water in the calorimeter for a measured time and weigh the condensate.

Start with the water well below room temperature.
Admit steam for the chosen time; the apparatus reports the mass of condensate from the change in weight.
Apply a cooling correction and compute the latent heat of vaporisation.
Estimate the entropy change of the universe and compare with the value for Part B.

Part D - Entropy generated by irreversible heat flow

Two lagged metal blocks at different temperatures are clamped together; follow both temperatures until they equalise.

Choose the two blocks, their masses and their initial temperatures.
Record the trace of both temperatures and note the common final temperature.
Verify that the energy lost by the hot block equals that gained by the cold one.
Compute the entropy change of each block and their sum.
Repeat with a much smaller initial temperature difference and comment on how the entropy generated scales.

Questions to answer in your report

Explain the cooling correction you applied. Derive it from Newton's law of cooling and state the assumption that makes it valid. [8]
Compute the molar heat capacity of each specimen and compare with 3R. Which specimen departs most from the Dulong-Petit value, and why? [6]
For your ice experiment, compute the entropy change of the ice, of the water and of the universe. Explain why the last is positive, and estimate how it would change if the ice were added in many small pieces over a long time. [8]
Your two determinations of the entropy generated in Part D used different initial temperature differences. Show that for small differences the entropy generated grows as the square of the temperature difference. [6]