Measuring the heat capacity of a calorimeter

Heat capacity $$C$$ is by definition the limit $$C=\lim_{\Delta t\to 0}\frac{\Delta Q}{\Delta t}$$. Here $$\Delta Q$$ is the amount of energy transferred as heat onto the material, and $$\Delta t$$ is the change in temperature. Suppose hot water is placed inside a calorimeter. We wait for the temperatures/thermal energies to settle. We assume that all the thermal energy from the water is absorbed by the calorimeter, that is $$\Delta Q_w = \Delta Q_c$$.

We have that $$C_{calorimeter}=\lim_{\Delta t \to 0}\frac{\Delta Q_w}{\Delta t_c}=\lim_{\Delta t\to 0}\frac{m_w \cdot C_w\cdot \Delta t_w}{\Delta t_c}$$

But actually, this is $$C_c=\frac{m_w \cdot C_w\cdot \Delta t_w}{\Delta t_c}$$

the equation by which the calculation is done. The inder $$w$$ denotes the appropriate values for water, $$c$$ for calorimeter. How can I derive the last equation, when the definition is given with a limit?

• My guess is that the exercise silently assumes $C_w$ (and $m_w$) do not change with temperature - then the equations would hold exactly. This is a reasonable assumption under most practical circumstances (isobaric near-room temperature operation), but it should be stated explicitely. Dec 28, 2020 at 10:46
• The equation assumes that, over the temperature range of interest, C is constant. Dec 28, 2020 at 13:03

How can I derive the last equation, when the definition is given with a limit?

Make two basic assumptions:

1. the process is adiabatic: no external heat enters or leaves the system
2. the heat capacities are constant over the relevant temperature intervals

The heat balance is thus, with the $$\text{LHS}$$ the initial heat content and $$\text{RHS}$$ the final heat content (due to assumption $$1$$ both are equal to each other):

$$m_wc_wT_w+CT_c=m_wc_wT_e+CT_e$$

$$\Rightarrow m_wc_wT_w-m_wc_wT_e=CT_e-CT_c$$

$$\boxed{C=m_w c_w\frac{T_w-T_e}{T_e-T_c}}$$

I've always seen the heat capacity as a proportional coefficient between $$\Delta T$$, the temperature change, and $$Q$$, the energy input from heating: $$Q = C \Delta T$$ as far as the process involves only heat transfer and temperature variation.

I would even go further and say that $$C$$ is well defined here only for a monophasic body, without any chemical reactions involved and for a constant amount of material $$n$$.