Specific heat capacity depends on temperature.
When heating a substance, energy change = mass of substance $\times$ specific heat capacity $\times$ temperature change.
In school, we may apply this equation over a change in temperature to find the energy put into the system. At the same time, the value for the specific heat capacity is usually a constant.
Is this actually allowed? Surely if the SHC depends on temperature, then using a constant value for the SHC over a temperature change won't give a correct answer?
How does one choose the value for the SHC to use?
For example, if I want to know the energy required to change water from a temperature of $10$ degrees to $50$ degrees, then which value for the SHC of water would I use from this table (in the pictures)?
[Left column shows values for the SHC (right two columns) at the different temperatures in different units.]
Best Answer
Calculate the weighted average, $SHC_{average}$:
$$SHC_{average}=\frac{(20-10)\times 4.1910 +(25-20)\times 4.1570+(30-25)\times 4.1379 +(40-30)\times 4.1175 +(50-40)\times 4.0737}{10+5+5+10+10}$$ $$SHC_{average}=4.137$$
Or more generally:
$$SHC_{average}=\frac{\displaystyle\sum_{i=1}^n\Big(SHC_i \times \Delta T_i\Big)}{\displaystyle\sum_{i=1}^n \Delta T_i}$$
where $SHC_i$ is the SHC of interval $i$, and $\Delta T_i$ the corresponding temperature interval. $n$ is the number of intervals.
Alternatively (but more laboriously) you could curve-fit a function $f(T)$ to the data, by means of regression analysis. Depending on the data-points a linear ($a_0+a_1 T$) or quadratic ($a_0+a_1 T+a_2 T^2$) model should probably suffice.
The heat required to go from $T_1$ to $T_2$ is then calculated by integration:
$$Q_{1 \to 2}=m\int_{T_1}^{T_2}f(T)\text{d}T$$