Comment on “Experimental determination of heat capacities and their correlation with theoretical predictions” [Am. J. Phys. 79(11), 1099–1103 (2011)]

If $C P$ is constant, it can be taken out of the integral, from which Eq. (1) is obtained. However, for the materials studied in Ref. 1, copper and aluminum, the temperature dependence of the molar heat capacity in the temperature range 100–295 K is not negligible. For example, the heat capacity of copper decreases from approximately 24.4 J mol⁻¹ K⁻¹ at 295 K to approximately 16 J mol⁻¹ K⁻¹ at 100 K,⁵ a percent change of about 35%. Experimental $C P(T)$ data for copper, as reported by Stevens and Boerio-Goates,⁵ are plotted in Fig. 2. Using the assumption that $C P$ is constant with temperature, along with the actual value of Q (determined from the area under the $C P(T)$ data), results in the flat line shown in Fig. 2. This illustrates that Eq. (1) will in general underestimate the value of C_P at T₁. In the above-mentioned example of copper with T₁ = 295 and T₂ = 77 K, the expected measured value would be 21.0 J mol⁻¹ K⁻¹, and the error introduced by using Eq. (1) is approximately 14%. This reflects a systematic error introduced by the assumptions inherent in Eq. (1), in contrast to measurement uncertainty. In ten room temperature experimental trials using the procedure described in Ref. 1 (representative data shown in Fig. 1), we obtained an average value of 20 J mol⁻¹ K⁻¹ for the heat capacity of copper (standard deviation of the mean of 1 J mol⁻¹ K⁻¹, approximately 5%), which is approximately 18% lower than the value of 24.3 J mol⁻¹ K⁻¹ for copper at 295 K reported by Stevens and Boerio-Goates.⁵ This underestimated value for C_P is in good agreement with the expected measured value deduced in the discussion earlier using C_P data for copper reported in the literature (Fig. 1).⁵ 

We consider again the case of copper. Taking the data for $C P(T)$ for copper reported in Ref. 5 as the accepted (“true”) values, and assuming no other systematic errors other than the use of Eq. (1), Fig. 3 shows the values of $C P(T)$ that would in principle be obtained using the original analysis procedure from Ref. 1. Also plotted in Fig. 3 are the values that would in principle be obtained from the same measurements using the improved analysis approach described here, i.e., Eq. (3). As is apparent from Fig. 3, the improved analysis approach is expected to show much greater accuracy, even with increments between the initial temperatures as large as $30 K$⁠. We note that, although we have not evaluated them systematically, other assumptions made in the measurement procedure, such as neglecting the energy lost in transferring the sample and the assumption of uniform temperature of the sample during the experiment, may introduce additional systematic errors. We also note that the experimental method is likely limited to materials with high thermal conductivities, e.g., metals, which ensures accurate measurement of the interior sample temperature using a sensor attached to the surface, as well as rapid equilibration with the liquid nitrogen or cold nitrogen vapor.

We hope this Comment will be useful to future readers considering using this or similar experiments in the classroom laboratory.