Effect of specific heat capacity on heat loss

Assume we want to calculate the heat capacities of two substances with different heat capacities. We heat them both up with an immersion heater delivering a constant rate of energy (constant J/s), and we stop the immersion heater after 100 J of heat has been added. Would the lower heat capacity substance experience more or less heat loss, and would that translate into more or less error in the calculated specific heat?

I assume that less heat loss would mean less error but I am not sure as to which substance would experience less heat loss.

If anyone stumbles across this later, I am now strongly leaning towards more heat loss for lower heat capacity because heat loss is proportional to temperature difference and lower heat capacity will change temperature more easily.

• If you are talking about heat loss to the environment during the test, and the heal loss is dominated by the lower heat transfer coefficient on the air side of the sample boundary, then your assessment is correct. Feb 27, 2021 at 13:10

Assume you add a fixed amount of heat to a substance held in an isothermal bath, so that as soon as you start heating it, the substance gives off heat to the surroundings. As heat is added the substance increases in temperature to a maximum. When the heater is turned off, the substance continues to dissipate heat until it is again in equilibrium with the surroundings.

If the rate at which heat is added is much faster than that of heat dissipation and the heat capacity is approximately constant, then the maximum temperature will be approximately

$$T_{\mathrm{max}}=T_{\mathrm{ini}} + \frac{Q}{C_p}$$

where Q is the total amount of heat added. As expected, a substance with a smaller heat capacity will become hotter.

Now consider the rate of heat dissipation. The rate of heat dissipation can be expressed with the heat equation (in one dimension) as

$$\frac{dQ}{dt} = -k \frac{\partial T}{\partial x}$$ where k is the thermal conductivity.

Therefore if k is the same for two substances, then the rate at which heat is released will differ according to the heat capacity, and the lower heat capacity substance will release heat more quickly (since temperature gradients will be greater in that substance).

This can also be argued based on the value of the thermal diffusivity $$\alpha$$, which is inversely proportional to the heat capacity $$c_p$$:

$$\alpha = \frac{k}{\rho c_p}$$

As the Wikipedia explains,

In a sense, thermal diffusivity is the measure of thermal inertia.[6] In a substance with high thermal diffusivity, heat moves rapidly through it because the substance conducts heat quickly relative to its volumetric heat capacity or 'thermal bulk'.

Since the substance with smaller $$c_p$$ will have a larger thermal diffusivity (assuming all else constant) it will also lose heat more quickly.

This can of course be expected to result in greater uncertainty in a measurement of the heat absorbed by the substance.

• I'm not sure. Thermal effusivity, $\sqrt{\kappa \rho c_p}$, rather than thermal diffusivity, would seem to be closer to what he's looking for; but the latter expression allows one to compare two substances at the same $T$, while here the one with the lower $c_p$ would be at a higher $T$. It's an interesting problem—consider just the effect of $\rho$: OTOH, given two substances with the same $c_p$, the one with lower $\rho$ would have a greater surface area. But, on the other hand, with a lower $\rho$, the thermal energy would have to travel further, and the $T$-gradient would be less. Mar 30, 2021 at 1:45
• ...I think you'd need to solve the 3D heat equation, under the assumption the two bodies have the same shape. The math might be simpler if the both bodies were instead heated to the same temperature. Mar 30, 2021 at 2:05
• @theorist you could have a long rod insulated except at the ends that has the same exposed area and yet different $\rho$. The lengths would differ of course. That is more like what I describe with a 1d heat equation. Since the question was about $c_p$ I did not get into a discussion of the density. Mar 30, 2021 at 6:35
• Yes, the question was about calculating $c_p$ from heat loss. But since density affects this, it must be accounted for in any solution. Apr 4, 2021 at 23:32