Measuring the heat capacity of a calorimeter

Question

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?

Answer 1

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):

$$m_wc_wT_w+CT_c=m_wc_wT_e+CT_e$$

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

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

Answer 2

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 $n$.